1 /*
  2     Copyright 2008-2022
  3         Matthias Ehmann,
  4         Michael Gerhaeuser,
  5         Carsten Miller,
  6         Bianca Valentin,
  7         Alfred Wassermann,
  8         Peter Wilfahrt
  9 
 10     This file is part of JSXGraph.
 11 
 12     JSXGraph is free software dual licensed under the GNU LGPL or MIT License.
 13 
 14     You can redistribute it and/or modify it under the terms of the
 15 
 16       * GNU Lesser General Public License as published by
 17         the Free Software Foundation, either version 3 of the License, or
 18         (at your option) any later version
 19       OR
 20       * MIT License: https://github.com/jsxgraph/jsxgraph/blob/master/LICENSE.MIT
 21 
 22     JSXGraph is distributed in the hope that it will be useful,
 23     but WITHOUT ANY WARRANTY; without even the implied warranty of
 24     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 25     GNU Lesser General Public License for more details.
 26 
 27     You should have received a copy of the GNU Lesser General Public License and
 28     the MIT License along with JSXGraph. If not, see <http://www.gnu.org/licenses/>
 29     and <http://opensource.org/licenses/MIT/>.
 30  */
 31 
 32 
 33 /*global JXG: true, define: true*/
 34 /*jslint nomen: true, plusplus: true*/
 35 
 36 /* depends:
 37  jxg
 38  math/math
 39  math/geometry
 40  math/numerics
 41  math/statistics
 42  base/constants
 43  base/coords
 44  base/element
 45  utils/type
 46   elements:
 47    transform
 48    point
 49    ticks
 50  */
 51 
 52 /**
 53  * @fileoverview The geometry object Line is defined in this file. Line stores all
 54  * style and functional properties that are required to draw and move a line on
 55  * a board.
 56  */
 57 
 58 define([
 59     'jxg', 'math/math', 'math/geometry', 'math/numerics', 'math/statistics', 'base/constants', 'base/coords',
 60     'base/element', 'utils/type', 'base/point'
 61 ], function (JXG, Mat, Geometry, Numerics, Statistics, Const, Coords, GeometryElement, Type, Point) {
 62 
 63     "use strict";
 64 
 65     /**
 66      * The Line class is a basic class for all kind of line objects, e.g. line, arrow, and axis. It is usually defined by two points and can
 67      * be intersected with some other geometry elements.
 68      * @class Creates a new basic line object. Do not use this constructor to create a line.
 69      * Use {@link JXG.Board#create} with
 70      * type {@link Line}, {@link Arrow}, or {@link Axis} instead.
 71      * @constructor
 72      * @augments JXG.GeometryElement
 73      * @param {String,JXG.Board} board The board the new line is drawn on.
 74      * @param {Point} p1 Startpoint of the line.
 75      * @param {Point} p2 Endpoint of the line.
 76      * @param {Object} attributes Javascript object containing attributes like name, id and colors.
 77      */
 78     JXG.Line = function (board, p1, p2, attributes) {
 79         this.constructor(board, attributes, Const.OBJECT_TYPE_LINE, Const.OBJECT_CLASS_LINE);
 80 
 81         /**
 82          * Startpoint of the line. You really should not set this field directly as it may break JSXGraph's
 83          * update system so your construction won't be updated properly.
 84          * @type JXG.Point
 85          */
 86         this.point1 = this.board.select(p1);
 87 
 88         /**
 89          * Endpoint of the line. Just like {@link JXG.Line.point1} you shouldn't write this field directly.
 90          * @type JXG.Point
 91          */
 92         this.point2 = this.board.select(p2);
 93 
 94         /**
 95          * Array of ticks storing all the ticks on this line. Do not set this field directly and use
 96          * {@link JXG.Line#addTicks} and {@link JXG.Line#removeTicks} to add and remove ticks to and from the line.
 97          * @type Array
 98          * @see JXG.Ticks
 99          */
100         this.ticks = [];
101 
102         /**
103          * Reference of the ticks created automatically when constructing an axis.
104          * @type JXG.Ticks
105          * @see JXG.Ticks
106          */
107         this.defaultTicks = null;
108 
109         /**
110          * If the line is the border of a polygon, the polygon object is stored, otherwise null.
111          * @type JXG.Polygon
112          * @default null
113          * @private
114          */
115         this.parentPolygon = null;
116 
117         /* Register line at board */
118         this.id = this.board.setId(this, 'L');
119         this.board.renderer.drawLine(this);
120         this.board.finalizeAdding(this);
121 
122         this.elType = 'line';
123 
124         /* Add line as child to defining points */
125         if (this.point1._is_new) {
126             this.addChild(this.point1);
127             delete this.point1._is_new;
128         } else {
129             this.point1.addChild(this);
130         }
131         if (this.point2._is_new) {
132             this.addChild(this.point2);
133             delete this.point2._is_new;
134         } else {
135             this.point2.addChild(this);
136         }
137 
138         this.inherits.push(this.point1, this.point2);
139 
140         this.updateStdform(); // This is needed in the following situation:
141         // * the line is defined by three coordinates
142         // * and it will have a glider
143         // * and board.suspendUpdate() has been called.
144 
145         // create Label
146         this.createLabel();
147 
148         this.methodMap = JXG.deepCopy(this.methodMap, {
149             point1: 'point1',
150             point2: 'point2',
151             getSlope: 'getSlope',
152             getRise: 'getRise',
153             getYIntersect: 'getRise',
154             getAngle: 'getAngle',
155             L: 'L',
156             length: 'L'
157         });
158     };
159 
160     JXG.Line.prototype = new GeometryElement();
161 
162     JXG.extend(JXG.Line.prototype, /** @lends JXG.Line.prototype */ {
163         /**
164          * Checks whether (x,y) is near the line.
165          * @param {Number} x Coordinate in x direction, screen coordinates.
166          * @param {Number} y Coordinate in y direction, screen coordinates.
167          * @returns {Boolean} True if (x,y) is near the line, False otherwise.
168          */
169         hasPoint: function (x, y) {
170             // Compute the stdform of the line in screen coordinates.
171             var c = [], s,
172                 v = [1, x, y],
173                 vnew,
174                 p1c, p2c, d, pos, i,
175                 prec, type,
176                 sw = Type.evaluate(this.visProp.strokewidth);
177 
178             if (Type.isObject(Type.evaluate(this.visProp.precision))) {
179                 type = this.board._inputDevice;
180                 prec = Type.evaluate(this.visProp.precision[type]);
181             } else {
182                 // 'inherit'
183                 prec = this.board.options.precision.hasPoint;
184             }
185             prec += sw * 0.5;
186 
187             c[0] = this.stdform[0] -
188                 this.stdform[1] * this.board.origin.scrCoords[1] / this.board.unitX +
189                 this.stdform[2] * this.board.origin.scrCoords[2] / this.board.unitY;
190             c[1] = this.stdform[1] / this.board.unitX;
191             c[2] = this.stdform[2] / (-this.board.unitY);
192 
193             s = Geometry.distPointLine(v, c);
194             if (isNaN(s) || s > prec) {
195                 return false;
196             }
197 
198             if (Type.evaluate(this.visProp.straightfirst) &&
199                     Type.evaluate(this.visProp.straightlast)) {
200                 return true;
201             }
202 
203             // If the line is a ray or segment we have to check if the projected point is between P1 and P2.
204             p1c = this.point1.coords;
205             p2c = this.point2.coords;
206 
207             // Project the point orthogonally onto the line
208             vnew = [0, c[1], c[2]];
209             // Orthogonal line to c through v
210             vnew = Mat.crossProduct(vnew, v);
211             // Intersect orthogonal line with line
212             vnew = Mat.crossProduct(vnew, c);
213 
214             // Normalize the projected point
215             vnew[1] /= vnew[0];
216             vnew[2] /= vnew[0];
217             vnew[0] = 1;
218 
219             vnew = (new Coords(Const.COORDS_BY_SCREEN, vnew.slice(1), this.board)).usrCoords;
220             d = p1c.distance(Const.COORDS_BY_USER, p2c);
221             p1c = p1c.usrCoords.slice(0);
222             p2c = p2c.usrCoords.slice(0);
223 
224             // The defining points are identical
225             if (d < Mat.eps) {
226                 pos = 0;
227             } else {
228                 /*
229                  * Handle the cases, where one of the defining points is an ideal point.
230                  * d is set to something close to infinity, namely 1/eps.
231                  * The ideal point is (temporarily) replaced by a finite point which has
232                  * distance d from the other point.
233                  * This is accomplished by extracting the x- and y-coordinates (x,y)=:v of the ideal point.
234                  * v determines the direction of the line. v is normalized, i.e. set to length 1 by dividing through its length.
235                  * Finally, the new point is the sum of the other point and v*d.
236                  *
237                  */
238 
239                 // At least one point is an ideal point
240                 if (d === Number.POSITIVE_INFINITY) {
241                     d = 1 / Mat.eps;
242 
243                     // The second point is an ideal point
244                     if (Math.abs(p2c[0]) < Mat.eps) {
245                         d /= Geometry.distance([0, 0, 0], p2c);
246                         p2c = [1, p1c[1] + p2c[1] * d, p1c[2] + p2c[2] * d];
247                     // The first point is an ideal point
248                     } else {
249                         d /= Geometry.distance([0, 0, 0], p1c);
250                         p1c = [1, p2c[1] + p1c[1] * d, p2c[2] + p1c[2] * d];
251                     }
252                 }
253                 i = 1;
254                 d = p2c[i] - p1c[i];
255 
256                 if (Math.abs(d) < Mat.eps) {
257                     i = 2;
258                     d = p2c[i] - p1c[i];
259                 }
260                 pos = (vnew[i] - p1c[i]) / d;
261             }
262 
263             if (!Type.evaluate(this.visProp.straightfirst) && pos < 0) {
264                 return false;
265             }
266 
267             return !(!Type.evaluate(this.visProp.straightlast) && pos > 1);
268 
269         },
270 
271         // documented in base/element
272         update: function () {
273             var funps;
274 
275             if (!this.needsUpdate) {
276                 return this;
277             }
278 
279             if (this.constrained) {
280                 if (Type.isFunction(this.funps)) {
281                     funps = this.funps();
282                     if (funps && funps.length && funps.length === 2) {
283                         this.point1 = funps[0];
284                         this.point2 = funps[1];
285                     }
286                 } else {
287                     if (Type.isFunction(this.funp1)) {
288                         funps = this.funp1();
289                         if (Type.isPoint(funps)) {
290                             this.point1 = funps;
291                         } else if (funps && funps.length && funps.length === 2) {
292                             this.point1.setPositionDirectly(Const.COORDS_BY_USER, funps);
293                         }
294                     }
295 
296                     if (Type.isFunction(this.funp2)) {
297                         funps = this.funp2();
298                         if (Type.isPoint(funps)) {
299                             this.point2 = funps;
300                         } else if (funps && funps.length && funps.length === 2) {
301                             this.point2.setPositionDirectly(Const.COORDS_BY_USER, funps);
302                         }
303                     }
304                 }
305             }
306 
307             this.updateSegmentFixedLength();
308             this.updateStdform();
309 
310             if (Type.evaluate(this.visProp.trace)) {
311                 this.cloneToBackground(true);
312             }
313 
314             return this;
315         },
316 
317         /**
318          * Update segments with fixed length and at least one movable point.
319          * @private
320          */
321         updateSegmentFixedLength: function () {
322             var d, dnew, d1, d2, drag1, drag2, x, y;
323 
324             if (!this.hasFixedLength) {
325                 return this;
326             }
327 
328             // Compute the actual length of the segment
329             d = this.point1.Dist(this.point2);
330             // Determine the length the segment ought to have
331             dnew = this.fixedLength();
332             // Distances between the two points and their respective
333             // position before the update
334             d1 = this.fixedLengthOldCoords[0].distance(Const.COORDS_BY_USER, this.point1.coords);
335             d2 = this.fixedLengthOldCoords[1].distance(Const.COORDS_BY_USER, this.point2.coords);
336 
337             // If the position of the points or the fixed length function has been changed we have to work.
338             if (d1 > Mat.eps || d2 > Mat.eps || d !== dnew) {
339                 drag1 = this.point1.isDraggable &&
340                             (this.point1.type !== Const.OBJECT_TYPE_GLIDER) &&
341                             !Type.evaluate(this.point1.visProp.fixed);
342                 drag2 = this.point2.isDraggable &&
343                             (this.point2.type !== Const.OBJECT_TYPE_GLIDER) &&
344                             !Type.evaluate(this.point2.visProp.fixed);
345 
346                 // First case: the two points are different
347                 // Then we try to adapt the point that was not dragged
348                 // If this point can not be moved (e.g. because it is a glider)
349                 // we try move the other point
350                 if (d > Mat.eps) {
351                     if ((d1 > d2 && drag2) ||
352                             (d1 <= d2 && drag2 && !drag1)) {
353                         this.point2.setPositionDirectly(Const.COORDS_BY_USER, [
354                             this.point1.X() + (this.point2.X() - this.point1.X()) * dnew / d,
355                             this.point1.Y() + (this.point2.Y() - this.point1.Y()) * dnew / d
356                         ]);
357                         this.point2.fullUpdate();
358                     } else if ((d1 <= d2 && drag1) ||
359                             (d1 > d2 && drag1 && !drag2)) {
360                         this.point1.setPositionDirectly(Const.COORDS_BY_USER, [
361                             this.point2.X() + (this.point1.X() - this.point2.X()) * dnew / d,
362                             this.point2.Y() + (this.point1.Y() - this.point2.Y()) * dnew / d
363                         ]);
364                         this.point1.fullUpdate();
365                     }
366                     // Second case: the two points are identical. In this situation
367                     // we choose a random direction.
368                 } else {
369                     x = Math.random() - 0.5;
370                     y = Math.random() - 0.5;
371                     d = Math.sqrt(x * x + y * y);
372 
373                     if (drag2) {
374                         this.point2.setPositionDirectly(Const.COORDS_BY_USER, [
375                             this.point1.X() + x * dnew / d,
376                             this.point1.Y() + y * dnew / d
377                         ]);
378                         this.point2.fullUpdate();
379                     } else if (drag1) {
380                         this.point1.setPositionDirectly(Const.COORDS_BY_USER, [
381                             this.point2.X() + x * dnew / d,
382                             this.point2.Y() + y * dnew / d
383                         ]);
384                         this.point1.fullUpdate();
385                     }
386                 }
387                 // Finally, we save the position of the two points.
388                 this.fixedLengthOldCoords[0].setCoordinates(Const.COORDS_BY_USER, this.point1.coords.usrCoords);
389                 this.fixedLengthOldCoords[1].setCoordinates(Const.COORDS_BY_USER, this.point2.coords.usrCoords);
390             }
391             return this;
392         },
393 
394         /**
395          * Updates the stdform derived from the parent point positions.
396          * @private
397          */
398         updateStdform: function () {
399             var v = Mat.crossProduct(this.point1.coords.usrCoords, this.point2.coords.usrCoords);
400 
401             this.stdform[0] = v[0];
402             this.stdform[1] = v[1];
403             this.stdform[2] = v[2];
404             this.stdform[3] = 0;
405 
406             this.normalize();
407         },
408 
409         /**
410          * Uses the boards renderer to update the line.
411          * @private
412          */
413         updateRenderer: function () {
414             //var wasReal;
415 
416             if (!this.needsUpdate) {
417                 return this;
418             }
419 
420             if (this.visPropCalc.visible) {
421                 // wasReal = this.isReal;
422                 this.isReal = (!isNaN(this.point1.coords.usrCoords[1] + this.point1.coords.usrCoords[2] +
423                         this.point2.coords.usrCoords[1] + this.point2.coords.usrCoords[2]) &&
424                         (Mat.innerProduct(this.stdform, this.stdform, 3) >= Mat.eps * Mat.eps));
425 
426                 if (//wasReal &&
427                     !this.isReal) {
428                     this.updateVisibility(false);
429                 }
430             }
431 
432             if (this.visPropCalc.visible) {
433                 this.board.renderer.updateLine(this);
434             }
435 
436             /* Update the label if visible. */
437             if (this.hasLabel && this.visPropCalc.visible && this.label &&
438                 this.label.visPropCalc.visible && this.isReal) {
439 
440                 this.label.update();
441                 this.board.renderer.updateText(this.label);
442             }
443 
444             // Update rendNode display
445             this.setDisplayRendNode();
446             // if (this.visPropCalc.visible !== this.visPropOld.visible) {
447             //     this.setDisplayRendNode(this.visPropCalc.visible);
448             //     if (this.hasLabel) {
449             //         this.board.renderer.display(this.label, this.label.visPropCalc.visible);
450             //     }
451             // }
452 
453             this.needsUpdate = false;
454             return this;
455         },
456 
457         /**
458          * Used to generate a polynomial for a point p that lies on this line, i.e. p is collinear to
459          * {@link JXG.Line#point1} and {@link JXG.Line#point2}.
460          *
461          * @param {JXG.Point} p The point for that the polynomial is generated.
462          * @returns {Array} An array containing the generated polynomial.
463          * @private
464          */
465         generatePolynomial: function (p) {
466             var u1 = this.point1.symbolic.x,
467                 u2 = this.point1.symbolic.y,
468                 v1 = this.point2.symbolic.x,
469                 v2 = this.point2.symbolic.y,
470                 w1 = p.symbolic.x,
471                 w2 = p.symbolic.y;
472 
473             /*
474              * The polynomial in this case is determined by three points being collinear:
475              *
476              *      U (u1,u2)      W (w1,w2)                V (v1,v2)
477              *  ----x--------------x------------------------x----------------
478              *
479              *  The collinearity condition is
480              *
481              *      u2-w2       w2-v2
482              *     -------  =  -------           (1)
483              *      u1-w1       w1-v1
484              *
485              * Multiplying (1) with denominators and simplifying is
486              *
487              *    u2w1 - u2v1 + w2v1 - u1w2 + u1v2 - w1v2 = 0
488              */
489 
490             return [['(', u2, ')*(', w1, ')-(', u2, ')*(', v1, ')+(', w2, ')*(', v1, ')-(', u1, ')*(', w2, ')+(', u1, ')*(', v2, ')-(', w1, ')*(', v2, ')'].join('')];
491         },
492 
493         /**
494          * Calculates the y intersect of the line.
495          * @returns {Number} The y intersect.
496          */
497         getRise: function () {
498             if (Math.abs(this.stdform[2]) >= Mat.eps) {
499                 return -this.stdform[0] / this.stdform[2];
500             }
501 
502             return Infinity;
503         },
504 
505         /**
506          * Calculates the slope of the line.
507          * @returns {Number} The slope of the line or Infinity if the line is parallel to the y-axis.
508          */
509         getSlope: function () {
510             if (Math.abs(this.stdform[2]) >= Mat.eps) {
511                 return -this.stdform[1] / this.stdform[2];
512             }
513 
514             return Infinity;
515         },
516 
517         /**
518          * Determines the angle between the positive x axis and the line.
519          * @returns {Number}
520          */
521         getAngle: function () {
522             return Math.atan2(-this.stdform[1], this.stdform[2]);
523         },
524 
525         /**
526          * Determines whether the line is drawn beyond {@link JXG.Line#point1} and
527          * {@link JXG.Line#point2} and updates the line.
528          * @param {Boolean} straightFirst True if the Line shall be drawn beyond
529          * {@link JXG.Line#point1}, false otherwise.
530          * @param {Boolean} straightLast True if the Line shall be drawn beyond
531          * {@link JXG.Line#point2}, false otherwise.
532          * @see #straightFirst
533          * @see #straightLast
534          * @private
535          */
536         setStraight: function (straightFirst, straightLast) {
537             this.visProp.straightfirst = straightFirst;
538             this.visProp.straightlast = straightLast;
539 
540             this.board.renderer.updateLine(this);
541             return this;
542         },
543 
544         // documented in geometry element
545         getTextAnchor: function () {
546             return new Coords(Const.COORDS_BY_USER, [0.5 * (this.point2.X() + this.point1.X()), 0.5 * (this.point2.Y() + this.point1.Y())], this.board);
547         },
548 
549         /**
550          * Adjusts Label coords relative to Anchor. DESCRIPTION
551          * @private
552          */
553         setLabelRelativeCoords: function (relCoords) {
554             if (Type.exists(this.label)) {
555                 this.label.relativeCoords = new Coords(Const.COORDS_BY_SCREEN, [relCoords[0], -relCoords[1]], this.board);
556             }
557         },
558 
559         // documented in geometry element
560         getLabelAnchor: function () {
561             var x, y,
562                 fs = 0,
563                 c1 = new Coords(Const.COORDS_BY_USER, this.point1.coords.usrCoords, this.board),
564                 c2 = new Coords(Const.COORDS_BY_USER, this.point2.coords.usrCoords, this.board),
565                 ev_sf = Type.evaluate(this.visProp.straightfirst),
566                 ev_sl = Type.evaluate(this.visProp.straightlast);
567 
568             if (ev_sf || ev_sl) {
569                 Geometry.calcStraight(this, c1, c2, 0);
570             }
571 
572             c1 = c1.scrCoords;
573             c2 = c2.scrCoords;
574 
575             if (!Type.exists(this.label)) {
576                 return new Coords(Const.COORDS_BY_SCREEN, [NaN, NaN], this.board);
577             }
578 
579             switch (Type.evaluate(this.label.visProp.position)) {
580             case 'lft':
581             case 'llft':
582             case 'ulft':
583                 if (c1[1] <= c2[1]) {
584                     x = c1[1];
585                     y = c1[2];
586                 } else {
587                     x = c2[1];
588                     y = c2[2];
589                 }
590                 break;
591             case 'rt':
592             case 'lrt':
593             case 'urt':
594                 if (c1[1] > c2[1]) {
595                     x = c1[1];
596                     y = c1[2];
597                 } else {
598                     x = c2[1];
599                     y = c2[2];
600                 }
601                 break;
602             default:
603                 x = 0.5 * (c1[1] + c2[1]);
604                 y = 0.5 * (c1[2] + c2[2]);
605             }
606 
607             // Correct coordinates if the label seems to be outside of canvas.
608             if (ev_sf || ev_sl) {
609                 if (Type.exists(this.label)) {  // Does not exist during createLabel
610                     fs = Type.evaluate(this.label.visProp.fontsize);
611                 }
612 
613                 if (Math.abs(x) < Mat.eps) {
614                     x = fs;
615                 } else if (this.board.canvasWidth + Mat.eps > x &&
616                             x > this.board.canvasWidth - fs - Mat.eps) {
617                     x = this.board.canvasWidth - fs;
618                 }
619 
620                 if (Mat.eps + fs > y && y > -Mat.eps) {
621                     y = fs;
622                 } else if (this.board.canvasHeight + Mat.eps > y &&
623                             y > this.board.canvasHeight - fs - Mat.eps) {
624                     y = this.board.canvasHeight - fs;
625                 }
626             }
627 
628             return new Coords(Const.COORDS_BY_SCREEN, [x, y], this.board);
629         },
630 
631         // documented in geometry element
632         cloneToBackground: function () {
633             var copy = {}, r, s, er;
634 
635             copy.id = this.id + 'T' + this.numTraces;
636             copy.elementClass = Const.OBJECT_CLASS_LINE;
637             this.numTraces++;
638             copy.point1 = this.point1;
639             copy.point2 = this.point2;
640 
641             copy.stdform = this.stdform;
642 
643             copy.board = this.board;
644 
645             copy.visProp = Type.deepCopy(this.visProp, this.visProp.traceattributes, true);
646             copy.visProp.layer = this.board.options.layer.trace;
647             Type.clearVisPropOld(copy);
648             copy.visPropCalc = {
649                 visible: Type.evaluate(copy.visProp.visible)
650             };
651 
652             s = this.getSlope();
653             r = this.getRise();
654             copy.getSlope = function () {
655                 return s;
656             };
657             copy.getRise = function () {
658                 return r;
659             };
660 
661             er = this.board.renderer.enhancedRendering;
662             this.board.renderer.enhancedRendering = true;
663             this.board.renderer.drawLine(copy);
664             this.board.renderer.enhancedRendering = er;
665             this.traces[copy.id] = copy.rendNode;
666 
667             return this;
668         },
669 
670         /**
671          * Add transformations to this line.
672          * @param {JXG.Transformation|Array} transform Either one {@link JXG.Transformation} or an array of
673          * {@link JXG.Transformation}s.
674          * @returns {JXG.Line} Reference to this line object.
675          */
676         addTransform: function (transform) {
677             var i,
678                 list = Type.isArray(transform) ? transform : [transform],
679                 len = list.length;
680 
681             for (i = 0; i < len; i++) {
682                 this.point1.transformations.push(list[i]);
683                 this.point2.transformations.push(list[i]);
684             }
685 
686             return this;
687         },
688 
689         // see GeometryElement.js
690         snapToGrid: function (pos) {
691             var c1, c2, dc, t, ticks,
692                 x, y, sX, sY;
693 
694             if (Type.evaluate(this.visProp.snaptogrid)) {
695                 if (this.parents.length < 3) {    // Line through two points
696                     this.point1.handleSnapToGrid(true, true);
697                     this.point2.handleSnapToGrid(true, true);
698                 } else if (Type.exists(pos)) {       // Free line
699                     sX = Type.evaluate(this.visProp.snapsizex);
700                     sY = Type.evaluate(this.visProp.snapsizey);
701 
702                     c1 = new Coords(Const.COORDS_BY_SCREEN, [pos.Xprev, pos.Yprev], this.board);
703 
704                     x = c1.usrCoords[1];
705                     y = c1.usrCoords[2];
706 
707                     if (sX <= 0 && this.board.defaultAxes && this.board.defaultAxes.x.defaultTicks) {
708                         ticks = this.board.defaultAxes.x.defaultTicks;
709                         sX = ticks.ticksDelta * (Type.evaluate(ticks.visProp.minorticks) + 1);
710                     }
711                     if (sY <= 0 && this.board.defaultAxes && this.board.defaultAxes.y.defaultTicks) {
712                         ticks = this.board.defaultAxes.y.defaultTicks;
713                         sY = ticks.ticksDelta * (Type.evaluate(ticks.visProp.minorticks) + 1);
714                     }
715 
716                     // if no valid snap sizes are available, don't change the coords.
717                     if (sX > 0 && sY > 0) {
718                         // projectCoordsToLine
719                         /*
720                         v = [0, this.stdform[1], this.stdform[2]];
721                         v = Mat.crossProduct(v, c1.usrCoords);
722                         c2 = Geometry.meetLineLine(v, this.stdform, 0, this.board);
723                         */
724                         c2 = Geometry.projectPointToLine({coords: c1}, this, this.board);
725 
726                         dc = Statistics.subtract([1, Math.round(x / sX) * sX, Math.round(y / sY) * sY], c2.usrCoords);
727                         t = this.board.create('transform', dc.slice(1), {type: 'translate'});
728                         t.applyOnce([this.point1, this.point2]);
729                     }
730                 }
731             } else {
732                 this.point1.handleSnapToGrid(false, true);
733                 this.point2.handleSnapToGrid(false, true);
734             }
735 
736             return this;
737         },
738 
739         // see element.js
740         snapToPoints: function () {
741             var forceIt = Type.evaluate(this.visProp.snaptopoints);
742 
743             if (this.parents.length < 3) {    // Line through two points
744                 this.point1.handleSnapToPoints(forceIt);
745                 this.point2.handleSnapToPoints(forceIt);
746             }
747 
748             return this;
749         },
750 
751         /**
752          * Treat the line as parametric curve in homogeneous coordinates, where the parameter t runs from 0 to 1.
753          * First we transform the interval [0,1] to [-1,1].
754          * If the line has homogeneous coordinates [c, a, b] = stdform[] then the direction of the line is [b, -a].
755          * Now, we take one finite point that defines the line, i.e. we take either point1 or point2
756          * (in case the line is not the ideal line).
757          * Let the coordinates of that point be [z, x, y].
758          * Then, the curve runs linearly from
759          * [0, b, -a] (t=-1) to [z, x, y] (t=0)
760          * and
761          * [z, x, y] (t=0) to [0, -b, a] (t=1)
762          *
763          * @param {Number} t Parameter running from 0 to 1.
764          * @returns {Number} X(t) x-coordinate of the line treated as parametric curve.
765          * */
766         X: function (t) {
767             var x,
768                 b = this.stdform[2];
769 
770             x = (Math.abs(this.point1.coords.usrCoords[0]) > Mat.eps) ?
771                     this.point1.coords.usrCoords[1] :
772                     this.point2.coords.usrCoords[1];
773 
774             t = (t - 0.5) * 2;
775 
776             return (1 - Math.abs(t)) * x - t * b;
777         },
778 
779         /**
780          * Treat the line as parametric curve in homogeneous coordinates.
781          * See {@link JXG.Line#X} for a detailed description.
782          * @param {Number} t Parameter running from 0 to 1.
783          * @returns {Number} Y(t) y-coordinate of the line treated as parametric curve.
784          */
785         Y: function (t) {
786             var y,
787                 a = this.stdform[1];
788 
789             y = (Math.abs(this.point1.coords.usrCoords[0]) > Mat.eps) ?
790                     this.point1.coords.usrCoords[2] :
791                     this.point2.coords.usrCoords[2];
792 
793             t = (t - 0.5) * 2;
794 
795             return (1 - Math.abs(t)) * y + t * a;
796         },
797 
798         /**
799          * Treat the line as parametric curve in homogeneous coordinates.
800          * See {@link JXG.Line#X} for a detailed description.
801          *
802          * @param {Number} t Parameter running from 0 to 1.
803          * @returns {Number} Z(t) z-coordinate of the line treated as parametric curve.
804          */
805         Z: function (t) {
806             var z = (Math.abs(this.point1.coords.usrCoords[0]) > Mat.eps) ?
807                     this.point1.coords.usrCoords[0] :
808                     this.point2.coords.usrCoords[0];
809 
810             t = (t - 0.5) * 2;
811 
812             return (1 - Math.abs(t)) * z;
813         },
814 
815         /**
816          * The distance between the two points defining the line.
817          * @returns {Number}
818          */
819         L: function () {
820             return this.point1.Dist(this.point2);
821         },
822 
823         /**
824          * Treat the element  as a parametric curve
825          * @private
826          */
827         minX: function () {
828             return 0.0;
829         },
830 
831         /**
832          * Treat the element as parametric curve
833          * @private
834          */
835         maxX: function () {
836             return 1.0;
837         },
838 
839         // documented in geometry element
840         bounds: function () {
841             var p1c = this.point1.coords.usrCoords,
842                 p2c = this.point2.coords.usrCoords;
843 
844             return [Math.min(p1c[1], p2c[1]), Math.max(p1c[2], p2c[2]), Math.max(p1c[1], p2c[1]), Math.min(p1c[2], p2c[2])];
845         },
846 
847         // documented in GeometryElement.js
848         remove: function () {
849             this.removeAllTicks();
850             GeometryElement.prototype.remove.call(this);
851         },
852 
853         // hideElement: function () {
854         //     var i;
855         //
856         //     GeometryElement.prototype.hideElement.call(this);
857         //
858         //     for (i = 0; i < this.ticks.length; i++) {
859         //         this.ticks[i].hideElement();
860         //     }
861         // },
862         //
863         // showElement: function () {
864         //     var i;
865         //     GeometryElement.prototype.showElement.call(this);
866         //
867         //     for (i = 0; i < this.ticks.length; i++) {
868         //         this.ticks[i].showElement();
869         //     }
870         // }
871     });
872 
873     /**
874      * @class This element is used to provide a constructor for a general line. A general line is given by two points. By setting additional properties
875      * a line can be used as an arrow and/or axis.
876      * @pseudo
877      * @description
878      * @name Line
879      * @augments JXG.Line
880      * @constructor
881      * @type JXG.Line
882      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
883      * @param {JXG.Point,array,function_JXG.Point,array,function} point1,point2 Parent elements can be two elements either of type {@link JXG.Point} or array of
884      * numbers describing the coordinates of a point. In the latter case the point will be constructed automatically as a fixed invisible point.
885      * It is possible to provide a function returning an array or a point, instead of providing an array or a point.
886      * @param {Number,function_Number,function_Number,function} a,b,c A line can also be created providing three numbers. The line is then described by
887      * the set of solutions of the equation <tt>a*z+b*x+c*y = 0</tt>. For all finite points, z is normalized to the value 1.
888      * It is possible to provide three functions returning numbers, too.
889      * @param {function} f This function must return an array containing three numbers forming the line's homogeneous coordinates.
890      * <p>
891      * Additionally, a line can be created by providing a line and a transformation (or an array of transformations).
892      * Then, the result is a line which is the transformation of the supplied line.
893      * @example
894      * // Create a line using point and coordinates/
895      * // The second point will be fixed and invisible.
896      * var p1 = board.create('point', [4.5, 2.0]);
897      * var l1 = board.create('line', [p1, [1.0, 1.0]]);
898      * </pre><div class="jxgbox" id="JXGc0ae3461-10c4-4d39-b9be-81d74759d122" style="width: 300px; height: 300px;"></div>
899      * <script type="text/javascript">
900      *   var glex1_board = JXG.JSXGraph.initBoard('JXGc0ae3461-10c4-4d39-b9be-81d74759d122', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
901      *   var glex1_p1 = glex1_board.create('point', [4.5, 2.0]);
902      *   var glex1_l1 = glex1_board.create('line', [glex1_p1, [1.0, 1.0]]);
903      * </script><pre>
904      * @example
905      * // Create a line using three coordinates
906      * var l1 = board.create('line', [1.0, -2.0, 3.0]);
907      * </pre><div class="jxgbox" id="JXGcf45e462-f964-4ba4-be3a-c9db94e2593f" style="width: 300px; height: 300px;"></div>
908      * <script type="text/javascript">
909      *   var glex2_board = JXG.JSXGraph.initBoard('JXGcf45e462-f964-4ba4-be3a-c9db94e2593f', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
910      *   var glex2_l1 = glex2_board.create('line', [1.0, -2.0, 3.0]);
911      * </script><pre>
912      * @example
913      *         // Create a line (l2) as reflection of another line (l1)
914      *         // reflection line
915      *         var li = board.create('line', [1,1,1], {strokeColor: '#aaaaaa'});
916      *         var reflect = board.create('transform', [li], {type: 'reflect'});
917      *
918      *         var l1 = board.create('line', [1,-5,1]);
919      *         var l2 = board.create('line', [l1, reflect]);
920      *
921      * </pre><div id="JXGJXGa00d7dd6-d38c-11e7-93b3-901b0e1b8723" class="jxgbox" style="width: 300px; height: 300px;"></div>
922      * <script type="text/javascript">
923      *     (function() {
924      *         var board = JXG.JSXGraph.initBoard('JXGJXGa00d7dd6-d38c-11e7-93b3-901b0e1b8723',
925      *             {boundingbox: [-8, 8, 8,-8], axis: true, showcopyright: false, shownavigation: false});
926      *             // reflection line
927      *             var li = board.create('line', [1,1,1], {strokeColor: '#aaaaaa'});
928      *             var reflect = board.create('transform', [li], {type: 'reflect'});
929      *
930      *             var l1 = board.create('line', [1,-5,1]);
931      *             var l2 = board.create('line', [l1, reflect]);
932      *     })();
933      *
934      * </script><pre>
935      *
936      * @example
937      * var t = board.create('transform', [2, 1.5], {type: 'scale'});
938      * var l1 = board.create('line', [1, -5, 1]);
939      * var l2 = board.create('line', [l1, t]);
940      *
941      * </pre><div id="d16d5b58-6338-11e8-9fb9-901b0e1b8723" class="jxgbox" style="width: 300px; height: 300px;"></div>
942      * <script type="text/javascript">
943      *     (function() {
944      *         var board = JXG.JSXGraph.initBoard('d16d5b58-6338-11e8-9fb9-901b0e1b8723',
945      *             {boundingbox: [-8, 8, 8,-8], axis: true, showcopyright: false, shownavigation: false});
946      *     var t = board.create('transform', [2, 1.5], {type: 'scale'});
947      *     var l1 = board.create('line', [1, -5, 1]);
948      *     var l2 = board.create('line', [l1, t]);
949      *
950      *     })();
951      *
952      * </script><pre>
953      *
954      * @example
955      * //create line between two points
956      * var p1 = board.create('point', [0,0]);
957      * var p2 = board.create('point', [2,2]);
958      * var l1 = board.create('line', [p1,p2], {straightFirst:false, straightLast:false});
959      * </pre><div id="d21d5b58-6338-11e8-9fb9-901b0e1b8723" class="jxgbox" style="width: 300px; height: 300px;"></div>
960      * <script type="text/javascript">
961      *     (function() {
962      *         var board = JXG.JSXGraph.initBoard('d21d5b58-6338-11e8-9fb9-901b0e1b8723',
963      *             {boundingbox: [-8, 8, 8,-8], axis: true, showcopyright: false, shownavigation: false});
964      *             var ex5p1 = board.create('point', [0,0]);
965      *             var ex5p2 = board.create('point', [2,2]);
966      *             var ex5l1 = board.create('line', [ex5p1,ex5p2], {straightFirst:false, straightLast:false});
967      *     })();
968      *
969      * </script><pre>
970      */
971     JXG.createLine = function (board, parents, attributes) {
972         var ps, el, p1, p2, i, attr,
973             c = [],
974             doTransform = false,
975             constrained = false,
976             isDraggable;
977 
978         /**
979          * The line is defined by two points or coordinates of two points.
980          * In the latter case, the points are created.
981          */
982         if (parents.length === 2) {
983             // point 1 given by coordinates
984             if (Type.isArray(parents[0]) && parents[0].length > 1) {
985                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
986                 p1 = board.create('point', parents[0], attr);
987             } else if (Type.isString(parents[0]) || Type.isPoint(parents[0])) {
988                 p1 =  board.select(parents[0]);
989             } else if (Type.isFunction(parents[0]) && Type.isPoint(parents[0]())) {
990                 p1 = parents[0]();
991                 constrained = true;
992             } else if (Type.isFunction(parents[0]) && parents[0]().length && parents[0]().length >= 2) {
993                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
994                 p1 = Point.createPoint(board, parents[0](), attr);
995                 constrained = true;
996             } else if (Type.isObject(parents[0]) && Type.isTransformationOrArray(parents[1])) {
997                 doTransform = true;
998                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
999                 p1 = board.create('point', [parents[0].point1, parents[1]], attr);
1000             } else {
1001                 throw new Error("JSXGraph: Can't create line with parent types '" +
1002                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1003                     "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
1004             }
1005 
1006             // point 2 given by coordinates
1007             if (doTransform) {
1008                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
1009                 p2 = board.create('point', [parents[0].point2, parents[1]], attr);
1010             } else if (Type.isArray(parents[1]) && parents[1].length > 1) {
1011                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
1012                 p2 = board.create('point', parents[1], attr);
1013             } else if (Type.isString(parents[1]) || Type.isPoint(parents[1])) {
1014                 p2 =  board.select(parents[1]);
1015             } else if (Type.isFunction(parents[1]) &&  Type.isPoint(parents[1]()) ) {
1016                 p2 = parents[1]();
1017                 constrained = true;
1018             } else if (Type.isFunction(parents[1]) && parents[1]().length && parents[1]().length >= 2) {
1019                 attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
1020                 p2 = Point.createPoint(board, parents[1](), attr);
1021                 constrained = true;
1022             } else {
1023                 throw new Error("JSXGraph: Can't create line with parent types '" +
1024                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1025                     "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
1026             }
1027 
1028             attr = Type.copyAttributes(attributes, board.options, 'line');
1029 
1030             el = new JXG.Line(board, p1, p2, attr);
1031 
1032             if (constrained) {
1033                 el.constrained = true;
1034                 el.funp1 = parents[0];
1035                 el.funp2 = parents[1];
1036             } else if (!doTransform) {
1037                 el.isDraggable = true;
1038             }
1039 
1040             //if (!el.constrained) {
1041             el.setParents([p1.id, p2.id]);
1042             //}
1043 
1044          // Line is defined by three homogeneous coordinates.
1045          // Also in this case points are created.
1046         } else if (parents.length === 3) {
1047             // free line
1048             isDraggable = true;
1049             for (i = 0; i < 3; i++) {
1050                 if (Type.isNumber(parents[i])) {
1051                     // createFunction will just wrap a function around our constant number
1052                     // that does nothing else but to return that number.
1053                     c[i] = Type.createFunction(parents[i]);
1054                 } else if (Type.isFunction(parents[i])) {
1055                     c[i] = parents[i];
1056                     isDraggable = false;
1057                 } else {
1058                     throw new Error("JSXGraph: Can't create line with parent types '" +
1059                         (typeof parents[0]) + "' and '" + (typeof parents[1]) + "' and '" + (typeof parents[2]) + "'." +
1060                         "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
1061                 }
1062             }
1063 
1064             // point 1 is the midpoint between (0,c,-b) and point 2. => point1 is finite.
1065             attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
1066             if (isDraggable) {
1067                 p1 = board.create('point', [
1068                     c[2]() * c[2]() + c[1]() * c[1](),
1069                     c[2]() - c[1]() * c[0]() + c[2](),
1070                     -c[1]() - c[2]() * c[0]() - c[1]()
1071                 ], attr);
1072             } else {
1073                 p1 = board.create('point', [
1074                     function () {
1075                         return (c[2]() * c[2]() + c[1]() * c[1]()) * 0.5;
1076                     },
1077                     function () {
1078                         return (c[2]() - c[1]() * c[0]() + c[2]()) * 0.5;
1079                     },
1080                     function () {
1081                         return (-c[1]() - c[2]() * c[0]() - c[1]()) * 0.5;
1082                     }], attr);
1083             }
1084 
1085             // point 2: (b^2+c^2,-ba+c,-ca-b)
1086             attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
1087             if (isDraggable) {
1088                 p2 = board.create('point', [
1089                     c[2]() * c[2]() + c[1]() * c[1](),
1090                     -c[1]() * c[0]() + c[2](),
1091                     -c[2]() * c[0]() - c[1]()
1092                 ], attr);
1093             } else {
1094                 p2 = board.create('point', [
1095                     function () {
1096                         return c[2]() * c[2]() + c[1]() * c[1]();
1097                     },
1098                     function () {
1099                         return -c[1]() * c[0]() + c[2]();
1100                     },
1101                     function () {
1102                         return -c[2]() * c[0]() - c[1]();
1103                     }], attr);
1104             }
1105 
1106             // If the line will have a glider and board.suspendUpdate() has been called, we
1107             // need to compute the initial position of the two points p1 and p2.
1108             p1.prepareUpdate().update();
1109             p2.prepareUpdate().update();
1110             attr = Type.copyAttributes(attributes, board.options, 'line');
1111             el = new JXG.Line(board, p1, p2, attr);
1112             // Not yet working, because the points are not draggable.
1113             el.isDraggable = isDraggable;
1114             el.setParents([p1, p2]);
1115 
1116         // The parent array contains a function which returns two points.
1117         } else if (parents.length === 1 && Type.isFunction(parents[0]) && parents[0]().length === 2 &&
1118                 Type.isPoint(parents[0]()[0]) &&
1119                 Type.isPoint(parents[0]()[1])) {
1120             ps = parents[0]();
1121             attr = Type.copyAttributes(attributes, board.options, 'line');
1122             el = new JXG.Line(board, ps[0], ps[1], attr);
1123             el.constrained = true;
1124             el.funps = parents[0];
1125             el.setParents(ps);
1126 
1127         } else if (parents.length === 1 && Type.isFunction(parents[0]) && parents[0]().length === 3 &&
1128                 Type.isNumber(parents[0]()[0]) &&
1129                 Type.isNumber(parents[0]()[1]) &&
1130                 Type.isNumber(parents[0]()[2])) {
1131             ps = parents[0];
1132 
1133             attr = Type.copyAttributes(attributes, board.options, 'line', 'point1');
1134             p1 = board.create('point', [
1135                 function () {
1136                     var c = ps();
1137 
1138                     return [
1139                         (c[2] * c[2] + c[1] * c[1]) * 0.5,
1140                         (c[2] - c[1] * c[0] + c[2]) * 0.5,
1141                         (-c[1] - c[2] * c[0] - c[1]) * 0.5
1142                     ];
1143                 }], attr);
1144 
1145             attr = Type.copyAttributes(attributes, board.options, 'line', 'point2');
1146             p2 = board.create('point', [
1147                 function () {
1148                     var c = ps();
1149 
1150                     return [
1151                         c[2] * c[2] + c[1] * c[1],
1152                         -c[1] * c[0] + c[2],
1153                         -c[2] * c[0] - c[1]
1154                     ];
1155                 }], attr);
1156 
1157             attr = Type.copyAttributes(attributes, board.options, 'line');
1158             el = new JXG.Line(board, p1, p2, attr);
1159 
1160             el.constrained = true;
1161             el.funps = parents[0];
1162             el.setParents([p1, p2]);
1163 
1164         } else {
1165             throw new Error("JSXGraph: Can't create line with parent types '" +
1166                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1167                 "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]], [a,b,c]");
1168         }
1169 
1170         return el;
1171     };
1172 
1173     JXG.registerElement('line', JXG.createLine);
1174 
1175     /**
1176      * @class This element is used to provide a constructor for a segment.
1177      * It's strictly spoken just a wrapper for element {@link Line} with {@link Line#straightFirst}
1178      * and {@link Line#straightLast} properties set to false. If there is a third variable then the
1179      * segment has a fixed length (which may be a function, too).
1180      * @pseudo
1181      * @description
1182      * @name Segment
1183      * @augments JXG.Line
1184      * @constructor
1185      * @type JXG.Line
1186      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1187      * @param {JXG.Point,array_JXG.Point,array} point1,point2 Parent elements can be two elements either of type {@link JXG.Point}
1188      * or array of numbers describing the
1189      * coordinates of a point. In the latter case the point will be constructed automatically as a fixed invisible point.
1190      * @param {number,function} length (optional) The points are adapted - if possible - such that their distance
1191      * has this value.
1192      * @see Line
1193      * @example
1194      * // Create a segment providing two points.
1195      *   var p1 = board.create('point', [4.5, 2.0]);
1196      *   var p2 = board.create('point', [1.0, 1.0]);
1197      *   var l1 = board.create('segment', [p1, p2]);
1198      * </pre><div class="jxgbox" id="JXGd70e6aac-7c93-4525-a94c-a1820fa38e2f" style="width: 300px; height: 300px;"></div>
1199      * <script type="text/javascript">
1200      *   var slex1_board = JXG.JSXGraph.initBoard('JXGd70e6aac-7c93-4525-a94c-a1820fa38e2f', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1201      *   var slex1_p1 = slex1_board.create('point', [4.5, 2.0]);
1202      *   var slex1_p2 = slex1_board.create('point', [1.0, 1.0]);
1203      *   var slex1_l1 = slex1_board.create('segment', [slex1_p1, slex1_p2]);
1204      * </script><pre>
1205      *
1206      * @example
1207      * // Create a segment providing two points.
1208      *   var p1 = board.create('point', [4.0, 1.0]);
1209      *   var p2 = board.create('point', [1.0, 1.0]);
1210      *   var l1 = board.create('segment', [p1, p2]);
1211      *   var p3 = board.create('point', [4.0, 2.0]);
1212      *   var p4 = board.create('point', [1.0, 2.0]);
1213      *   var l2 = board.create('segment', [p3, p4, 3]);
1214      *   var p5 = board.create('point', [4.0, 3.0]);
1215      *   var p6 = board.create('point', [1.0, 4.0]);
1216      *   var l3 = board.create('segment', [p5, p6, function(){ return l1.L();} ]);
1217      * </pre><div class="jxgbox" id="JXG617336ba-0705-4b2b-a236-c87c28ef25be" style="width: 300px; height: 300px;"></div>
1218      * <script type="text/javascript">
1219      *   var slex2_board = JXG.JSXGraph.initBoard('JXG617336ba-0705-4b2b-a236-c87c28ef25be', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1220      *   var slex2_p1 = slex2_board.create('point', [4.0, 1.0]);
1221      *   var slex2_p2 = slex2_board.create('point', [1.0, 1.0]);
1222      *   var slex2_l1 = slex2_board.create('segment', [slex2_p1, slex2_p2]);
1223      *   var slex2_p3 = slex2_board.create('point', [4.0, 2.0]);
1224      *   var slex2_p4 = slex2_board.create('point', [1.0, 2.0]);
1225      *   var slex2_l2 = slex2_board.create('segment', [slex2_p3, slex2_p4, 3]);
1226      *   var slex2_p5 = slex2_board.create('point', [4.0, 2.0]);
1227      *   var slex2_p6 = slex2_board.create('point', [1.0, 2.0]);
1228      *   var slex2_l3 = slex2_board.create('segment', [slex2_p5, slex2_p6, function(){ return slex2_l1.L();}]);
1229      * </script><pre>
1230      *
1231      */
1232     JXG.createSegment = function (board, parents, attributes) {
1233         var el, attr;
1234 
1235         attributes.straightFirst = false;
1236         attributes.straightLast = false;
1237         attr = Type.copyAttributes(attributes, board.options, 'segment');
1238 
1239         el = board.create('line', parents.slice(0, 2), attr);
1240 
1241         if (parents.length === 3) {
1242             el.hasFixedLength = true;
1243 
1244             if (Type.isNumber(parents[2])) {
1245                 el.fixedLength = function () {
1246                     return parents[2];
1247                 };
1248             } else if (Type.isFunction(parents[2])) {
1249                 el.fixedLength = parents[2];
1250             } else {
1251                 throw new Error("JSXGraph: Can't create segment with third parent type '" +
1252                     (typeof parents[2]) + "'." +
1253                     "\nPossible third parent types: number or function");
1254             }
1255 
1256             el.getParents = function() {
1257                 return this.parents.concat(this.fixedLength());
1258             };
1259 
1260             el.fixedLengthOldCoords = [];
1261             el.fixedLengthOldCoords[0] = new Coords(Const.COORDS_BY_USER, el.point1.coords.usrCoords.slice(1, 3), board);
1262             el.fixedLengthOldCoords[1] = new Coords(Const.COORDS_BY_USER, el.point2.coords.usrCoords.slice(1, 3), board);
1263         }
1264 
1265         el.elType = 'segment';
1266 
1267         return el;
1268     };
1269 
1270     JXG.registerElement('segment', JXG.createSegment);
1271 
1272     /**
1273      * @class This element is used to provide a constructor for arrow, which is just a wrapper for element
1274      * {@link Line} with {@link Line#straightFirst}
1275      * and {@link Line#straightLast} properties set to false and {@link Line#lastArrow} set to true.
1276      * @pseudo
1277      * @description
1278      * @name Arrow
1279      * @augments JXG.Line
1280      * @constructor
1281      * @type JXG.Line
1282      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1283      * @param {JXG.Point,array_JXG.Point,array} point1,point2 Parent elements can be two elements either of type {@link JXG.Point} or array of numbers describing the
1284      * coordinates of a point. In the latter case the point will be constructed automatically as a fixed invisible point.
1285      * @param {Number_Number_Number} a,b,c A line can also be created providing three numbers. The line is then described by the set of solutions
1286      * of the equation <tt>a*x+b*y+c*z = 0</tt>.
1287      * @see Line
1288      * @example
1289      * // Create an arrow providing two points.
1290      *   var p1 = board.create('point', [4.5, 2.0]);
1291      *   var p2 = board.create('point', [1.0, 1.0]);
1292      *   var l1 = board.create('arrow', [p1, p2]);
1293      * </pre><div class="jxgbox" id="JXG1d26bd22-7d6d-4018-b164-4c8bc8d22ccf" style="width: 300px; height: 300px;"></div>
1294      * <script type="text/javascript">
1295      *   var alex1_board = JXG.JSXGraph.initBoard('JXG1d26bd22-7d6d-4018-b164-4c8bc8d22ccf', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1296      *   var alex1_p1 = alex1_board.create('point', [4.5, 2.0]);
1297      *   var alex1_p2 = alex1_board.create('point', [1.0, 1.0]);
1298      *   var alex1_l1 = alex1_board.create('arrow', [alex1_p1, alex1_p2]);
1299      * </script><pre>
1300      */
1301     JXG.createArrow = function (board, parents, attributes) {
1302         var el, attr;
1303 
1304         attributes.straightFirst = false;
1305         attributes.straightLast = false;
1306         attr = Type.copyAttributes(attributes, board.options, 'arrow');
1307         el = board.create('line', parents, attr);
1308         //el.setArrow(false, true);
1309         el.type = Const.OBJECT_TYPE_VECTOR;
1310         el.elType = 'arrow';
1311 
1312         return el;
1313     };
1314 
1315     JXG.registerElement('arrow', JXG.createArrow);
1316 
1317     /**
1318      * @class This element is used to provide a constructor for an axis. It's strictly spoken just a wrapper for element {@link Line} with {@link Line#straightFirst}
1319      * and {@link Line#straightLast} properties set to true. Additionally {@link Line#lastArrow} is set to true and default {@link Ticks} will be created.
1320      * @pseudo
1321      * @description
1322      * @name Axis
1323      * @augments JXG.Line
1324      * @constructor
1325      * @type JXG.Line
1326      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1327      * @param {JXG.Point,array_JXG.Point,array} point1,point2 Parent elements can be two elements either of type {@link JXG.Point} or array of numbers describing the
1328      * coordinates of a point. In the latter case, the point will be constructed automatically as a fixed invisible point.
1329      * @param {Number_Number_Number} a,b,c A line can also be created providing three numbers. The line is then described by the set of solutions
1330      * of the equation <tt>a*x+b*y+c*z = 0</tt>.
1331      * @example
1332      * // Create an axis providing two coord pairs.
1333      *   var l1 = board.create('axis', [[0.0, 1.0], [1.0, 1.3]]);
1334      * </pre><div class="jxgbox" id="JXG4f414733-624c-42e4-855c-11f5530383ae" style="width: 300px; height: 300px;"></div>
1335      * <script type="text/javascript">
1336      *   var axex1_board = JXG.JSXGraph.initBoard('JXG4f414733-624c-42e4-855c-11f5530383ae', {boundingbox: [-1, 7, 7, -1], axis: true, showcopyright: false, shownavigation: false});
1337      *   var axex1_l1 = axex1_board.create('axis', [[0.0, 1.0], [1.0, 1.3]]);
1338      * </script><pre>
1339      */
1340     JXG.createAxis = function (board, parents, attributes) {
1341         var attr, attr_ticks, el, els, dist;
1342 
1343         // Arrays or points, that is all we need.
1344         if ((Type.isArray(parents[0]) || Type.isPoint(parents[0])) && (Type.isArray(parents[1]) || Type.isPoint(parents[1]))) {
1345 
1346             // Create line
1347             attr = Type.copyAttributes(attributes, board.options, 'axis');
1348             el = board.create('line', parents, attr);
1349             el.type = Const.OBJECT_TYPE_AXIS;
1350             el.isDraggable = false;
1351             el.point1.isDraggable = false;
1352             el.point2.isDraggable = false;
1353 
1354             for (els in el.ancestors) {
1355                 if (el.ancestors.hasOwnProperty(els)) {
1356                     el.ancestors[els].type = Const.OBJECT_TYPE_AXISPOINT;
1357                 }
1358             }
1359 
1360             // Create ticks
1361             attr_ticks = Type.copyAttributes(attributes, board.options, 'axis', 'ticks');
1362             if (Type.exists(attr_ticks.ticksdistance)) {
1363                 dist = attr_ticks.ticksdistance;
1364             } else if (Type.isArray(attr_ticks.ticks)) {
1365                 dist = attr_ticks.ticks;
1366             } else {
1367                 dist = 1.0;
1368             }
1369 
1370             /**
1371              * The ticks attached to the axis.
1372              * @memberOf Axis.prototype
1373              * @name defaultTicks
1374              * @type JXG.Ticks
1375              */
1376             el.defaultTicks = board.create('ticks', [el, dist], attr_ticks);
1377             el.defaultTicks.dump = false;
1378             el.elType = 'axis';
1379             el.subs = {
1380                 ticks: el.defaultTicks
1381             };
1382             el.inherits.push(el.defaultTicks);
1383 
1384         } else {
1385             throw new Error("JSXGraph: Can't create axis with parent types '" +
1386                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1387                 "\nPossible parent types: [point,point], [[x1,y1],[x2,y2]]");
1388         }
1389 
1390         return el;
1391     };
1392 
1393     JXG.registerElement('axis', JXG.createAxis);
1394 
1395     /**
1396      * @class With the element tangent the slope of a line, circle, or curve in a certain point can be visualized. A tangent is always constructed
1397      * by a glider on a line, circle, or curve and describes the tangent in the glider point on that line, circle, or curve.
1398      * @pseudo
1399      * @description
1400      * @name Tangent
1401      * @augments JXG.Line
1402      * @constructor
1403      * @type JXG.Line
1404      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1405      * @param {Glider} g A glider on a line, circle, or curve.
1406      * @example
1407      * // Create a tangent providing a glider on a function graph
1408      *   var c1 = board.create('curve', [function(t){return t},function(t){return t*t*t;}]);
1409      *   var g1 = board.create('glider', [0.6, 1.2, c1]);
1410      *   var t1 = board.create('tangent', [g1]);
1411      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-4018d0d17a98" style="width: 400px; height: 400px;"></div>
1412      * <script type="text/javascript">
1413      *   var tlex1_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-4018d0d17a98', {boundingbox: [-6, 6, 6, -6], axis: true, showcopyright: false, shownavigation: false});
1414      *   var tlex1_c1 = tlex1_board.create('curve', [function(t){return t},function(t){return t*t*t;}]);
1415      *   var tlex1_g1 = tlex1_board.create('glider', [0.6, 1.2, tlex1_c1]);
1416      *   var tlex1_t1 = tlex1_board.create('tangent', [tlex1_g1]);
1417      * </script><pre>
1418      */
1419     JXG.createTangent = function (board, parents, attributes) {
1420         var p, c, j, el, tangent;
1421 
1422         // One argument: glider on line, circle or curve
1423         if (parents.length === 1) {
1424             p = parents[0];
1425             c = p.slideObject;
1426         // Two arguments: (point,F"|conic) or (line|curve|circle|conic,point). // Not yet: curve!
1427         } else if (parents.length === 2) {
1428             // In fact, for circles and conics it is the polar
1429             if (Type.isPoint(parents[0])) {
1430                 p = parents[0];
1431                 c = parents[1];
1432             } else if (Type.isPoint(parents[1])) {
1433                 c = parents[0];
1434                 p = parents[1];
1435             } else {
1436                 throw new Error("JSXGraph: Can't create tangent with parent types '" +
1437                     (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1438                     "\nPossible parent types: [glider], [point,line|curve|circle|conic]");
1439             }
1440         } else {
1441             throw new Error("JSXGraph: Can't create tangent with parent types '" +
1442                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1443                 "\nPossible parent types: [glider], [point,line|curve|circle|conic]");
1444         }
1445 
1446         if (c.elementClass === Const.OBJECT_CLASS_LINE) {
1447             tangent = board.create('line', [c.point1, c.point2], attributes);
1448             tangent.glider = p;
1449         } else if (c.elementClass === Const.OBJECT_CLASS_CURVE && c.type !== Const.OBJECT_TYPE_CONIC) {
1450             if (Type.evaluate(c.visProp.curvetype) !== 'plot') {
1451                 tangent = board.create('line', [
1452                     function () {
1453                         var g = c.X,
1454                             f = c.Y;
1455                         return -p.X() * Numerics.D(f)(p.position) + p.Y() * Numerics.D(g)(p.position);
1456                     },
1457                     function () {
1458                         return Numerics.D(c.Y)(p.position);
1459                     },
1460                     function () {
1461                         return -Numerics.D(c.X)(p.position);
1462                     }
1463                 ], attributes);
1464 
1465                 p.addChild(tangent);
1466                 // this is required for the geogebra reader to display a slope
1467                 tangent.glider = p;
1468             } else {  // curveType 'plot'
1469                 // In case of bezierDegree == 1:
1470                 // Find two points p1, p2 enclosing the glider.
1471                 // Then the equation of the line segment is: 0 = y*(x1-x2) + x*(y2-y1) + y1*x2-x1*y2,
1472                 // which is the cross product of p1 and p2.
1473                 //
1474                 // In case of bezieDegree === 3:
1475                 // The slope dy / dx of the tangent is determined. Then the
1476                 // tangent is computed as cross product between
1477                 // the glider p and [1, p.X() + dx, p.Y() + dy]
1478                 //
1479                 tangent = board.create('line', [
1480                     function () {
1481                         var i = Math.floor(p.position),
1482                             p1, p2, t, A, B, C, D, dx, dy, d;
1483 
1484                         if (c.bezierDegree === 1) {
1485                             if (i === c.numberPoints - 1) {
1486                                 i--;
1487                             }
1488                         } else if (c.bezierDegree === 3) {
1489                             // i is start of the Bezier segment
1490                             // t is the position in the Bezier segment
1491                             i = Math.floor(p.position * (c.numberPoints - 1) / 3) * 3;
1492                             t = (p.position * (c.numberPoints - 1) - i) / 3;
1493                             if (i >= c.numberPoints - 1) {
1494                                 i = c.numberPoints - 4;
1495                                 t = 1;
1496                             }
1497                         } else {
1498                             return 0;
1499                         }
1500 
1501                         if (i < 0) {
1502                             return 1;
1503                         }
1504 
1505                         // The curve points are transformed (if there is a transformation)
1506                         // c.X(i) is not transformed.
1507                         if (c.bezierDegree === 1) {
1508                             p1 = c.points[i].usrCoords;
1509                             p2 = c.points[i + 1].usrCoords;
1510                         } else {
1511                             A = c.points[i].usrCoords;
1512                             B = c.points[i + 1].usrCoords;
1513                             C = c.points[i + 2].usrCoords;
1514                             D = c.points[i + 3].usrCoords;
1515                             dx = (1 - t) * (1 - t) * (B[1] - A[1]) + 2 * (1 - t) * t * (C[1] - B[1]) + t * t * (D[1]- C[1]);
1516                             dy = (1 - t) * (1 - t) * (B[2] - A[2]) + 2 * (1 - t) * t * (C[2] - B[2]) + t * t * (D[2]- C[2]);
1517                             d = Math.sqrt(dx * dx + dy * dy);
1518                             dx /= d;
1519                             dy /= d;
1520                             p1 = p.coords.usrCoords;
1521                             p2 = [1, p1[1] + dx, p1[2] + dy];
1522                       }
1523                       return p1[2] * p2[1] - p1[1] * p2[2];
1524                     },
1525                     function () {
1526                         var i = Math.floor(p.position),
1527                             p1, p2, t, A, B, C, D, dx, dy, d;
1528 
1529                         if (c.bezierDegree === 1) {
1530                             if (i === c.numberPoints - 1) {
1531                                 i--;
1532                             }
1533                         } else if (c.bezierDegree === 3) {
1534                             // i is start of the Bezier segment
1535                             // t is the position in the Bezier segment
1536                             i = Math.floor(p.position * (c.numberPoints - 1) / 3) * 3;
1537                             t = (p.position * (c.numberPoints - 1) - i) / 3;
1538                             if (i >= c.numberPoints - 1) {
1539                                 i = c.numberPoints - 4;
1540                                 t = 1;
1541                             }
1542                         } else {
1543                             return 0;
1544                         }
1545 
1546                         if (i < 0) {
1547                             return 0;
1548                         }
1549 
1550                         // The curve points are transformed (if there is a transformation)
1551                         // c.X(i) is not transformed.
1552                         if (c.bezierDegree === 1) {
1553                             p1 = c.points[i].usrCoords;
1554                             p2 = c.points[i + 1].usrCoords;
1555                         } else {
1556                             A = c.points[i].usrCoords;
1557                             B = c.points[i + 1].usrCoords;
1558                             C = c.points[i + 2].usrCoords;
1559                             D = c.points[i + 3].usrCoords;
1560                             dx = (1 - t) * (1 - t) * (B[1] - A[1]) + 2 * (1 - t) * t * (C[1] - B[1]) + t * t * (D[1]- C[1]);
1561                             dy = (1 - t) * (1 - t) * (B[2] - A[2]) + 2 * (1 - t) * t * (C[2] - B[2]) + t * t * (D[2]- C[2]);
1562                             d = Math.sqrt(dx * dx + dy * dy);
1563                             dx /= d;
1564                             dy /= d;
1565                             p1 = p.coords.usrCoords;
1566                             p2 = [1, p1[1] + dx, p1[2] + dy];
1567                         }
1568                         return p2[2] - p1[2];
1569                     },
1570                     function () {
1571                         var i = Math.floor(p.position),
1572                             p1, p2, t, A, B, C, D, dx, dy, d;
1573 
1574                         if (c.bezierDegree === 1) {
1575                             if (i === c.numberPoints - 1) {
1576                                 i--;
1577                             }
1578                         } else if (c.bezierDegree === 3) {
1579                             // i is start of the Bezier segment
1580                             // t is the position in the Bezier segment
1581                             i = Math.floor(p.position * (c.numberPoints - 1) / 3) * 3;
1582                             t = (p.position * (c.numberPoints - 1) - i) / 3;
1583                             if (i >= c.numberPoints - 1) {
1584                                 i = c.numberPoints - 4;
1585                                 t = 1;
1586                             }
1587                         } else {
1588                             return 0;
1589                         }
1590 
1591                         if (i < 0) {
1592                             return 0.0;
1593                         }
1594 
1595                         // The curve points are transformed (if there is a transformation)
1596                         // c.X(i) is not transformed.
1597                         if (c.bezierDegree === 1) {
1598                             p1 = c.points[i].usrCoords;
1599                             p2 = c.points[i + 1].usrCoords;
1600                         } else {
1601                             A = c.points[i].usrCoords;
1602                             B = c.points[i + 1].usrCoords;
1603                             C = c.points[i + 2].usrCoords;
1604                             D = c.points[i + 3].usrCoords;
1605                             dx = (1 - t) * (1 - t) * (B[1] - A[1]) + 2 * (1 - t) * t * (C[1] - B[1]) + t * t * (D[1]- C[1]);
1606                             dy = (1 - t) * (1 - t) * (B[2] - A[2]) + 2 * (1 - t) * t * (C[2] - B[2]) + t * t * (D[2]- C[2]);
1607                             d = Math.sqrt(dx * dx + dy * dy);
1608                             dx /= d;
1609                             dy /= d;
1610                             p1 = p.coords.usrCoords;
1611                             p2 = [1, p1[1] + dx, p1[2] + dy];
1612                         }
1613                         return p1[1] - p2[1];
1614                     }], attributes);
1615 
1616                 p.addChild(tangent);
1617                 // this is required for the geogebra reader to display a slope
1618                 tangent.glider = p;
1619             }
1620         } else if (c.type === Const.OBJECT_TYPE_TURTLE) {
1621             tangent = board.create('line', [
1622                 function () {
1623                     var i = Math.floor(p.position);
1624 
1625                     // run through all curves of this turtle
1626                     for (j = 0; j < c.objects.length; j++) {
1627                         el = c.objects[j];
1628 
1629                         if (el.type === Const.OBJECT_TYPE_CURVE) {
1630                             if (i < el.numberPoints) {
1631                                 break;
1632                             }
1633 
1634                             i -= el.numberPoints;
1635                         }
1636                     }
1637 
1638                     if (i === el.numberPoints - 1) {
1639                         i--;
1640                     }
1641 
1642                     if (i < 0) {
1643                         return 1;
1644                     }
1645 
1646                     return el.Y(i) * el.X(i + 1) - el.X(i) * el.Y(i + 1);
1647                 },
1648                 function () {
1649                     var i = Math.floor(p.position);
1650 
1651                     // run through all curves of this turtle
1652                     for (j = 0; j < c.objects.length; j++) {
1653                         el = c.objects[j];
1654 
1655                         if (el.type === Const.OBJECT_TYPE_CURVE) {
1656                             if (i < el.numberPoints) {
1657                                 break;
1658                             }
1659 
1660                             i -= el.numberPoints;
1661                         }
1662                     }
1663 
1664                     if (i === el.numberPoints - 1) {
1665                         i--;
1666                     }
1667                     if (i < 0) {
1668                         return 0;
1669                     }
1670 
1671                     return el.Y(i + 1) - el.Y(i);
1672                 },
1673                 function () {
1674                     var i = Math.floor(p.position);
1675 
1676                     // run through all curves of this turtle
1677                     for (j = 0; j < c.objects.length; j++) {
1678                         el = c.objects[j];
1679                         if (el.type === Const.OBJECT_TYPE_CURVE) {
1680                             if (i < el.numberPoints) {
1681                                 break;
1682                             }
1683                             i -= el.numberPoints;
1684                         }
1685                     }
1686                     if (i === el.numberPoints - 1) {
1687                         i--;
1688                     }
1689 
1690                     if (i < 0) {
1691                         return 0;
1692                     }
1693 
1694                     return el.X(i) - el.X(i + 1);
1695                 }], attributes);
1696             p.addChild(tangent);
1697 
1698             // this is required for the geogebra reader to display a slope
1699             tangent.glider = p;
1700         } else if (c.elementClass === Const.OBJECT_CLASS_CIRCLE || c.type === Const.OBJECT_TYPE_CONIC) {
1701             // If p is not on c, the tangent is the polar.
1702             // This construction should work on conics, too. p has to lie on c.
1703             tangent = board.create('line', [
1704                 function () {
1705                     return Mat.matVecMult(c.quadraticform, p.coords.usrCoords)[0];
1706                 },
1707                 function () {
1708                     return Mat.matVecMult(c.quadraticform, p.coords.usrCoords)[1];
1709                 },
1710                 function () {
1711                     return Mat.matVecMult(c.quadraticform, p.coords.usrCoords)[2];
1712                 }], attributes);
1713 
1714             p.addChild(tangent);
1715             // this is required for the geogebra reader to display a slope
1716             tangent.glider = p;
1717         }
1718 
1719         if (!Type.exists(tangent)) {
1720             throw new Error('JSXGraph: Couldn\'t create tangent with the given parents.');
1721         }
1722 
1723         tangent.elType = 'tangent';
1724         tangent.type = Const.OBJECT_TYPE_TANGENT;
1725         tangent.setParents(parents);
1726 
1727         return tangent;
1728     };
1729 
1730     /**
1731      * @class This element is used to provide a constructor for the radical axis with respect to two circles with distinct centers.
1732      * The angular bisector of the polar lines of the circle centers with respect to the other circle is always the radical axis.
1733      * The radical axis passes through the intersection points when the circles intersect.
1734      * When a circle about the midpoint of circle centers, passing through the circle centers, intersects the circles, the polar lines pass through those intersection points.
1735      * @pseudo
1736      * @description
1737      * @name RadicalAxis
1738      * @augments JXG.Line
1739      * @constructor
1740      * @type JXG.Line
1741      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1742      * @param {JXG.Circle} circle Circle one of the two respective circles.
1743      * @param {JXG.Circle} circle Circle the other of the two respective circles.
1744      * @example
1745      * // Create the radical axis line with respect to two circles
1746      *   var board = JXG.JSXGraph.initBoard('7b7233a0-f363-47dd-9df5-5018d0d17a98', {boundingbox: [-1, 9, 9, -1], axis: true, showcopyright: false, shownavigation: false});
1747      *   var p1 = board.create('point', [2, 3]);
1748      *   var p2 = board.create('point', [1, 4]);
1749      *   var c1 = board.create('circle', [p1, p2]);
1750      *   var p3 = board.create('point', [6, 5]);
1751      *   var p4 = board.create('point', [8, 6]);
1752      *   var c2 = board.create('circle', [p3, p4]);
1753      *   var r1 = board.create('radicalaxis', [c1, c2]);
1754      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-5018d0d17a98" class="jxgbox" style="width:400px; height:400px;"></div>
1755      * <script type='text/javascript'>
1756      *   var rlex1_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-5018d0d17a98', {boundingbox: [-1, 9, 9, -1], axis: true, showcopyright: false, shownavigation: false});
1757      *   var rlex1_p1 = rlex1_board.create('point', [2, 3]);
1758      *   var rlex1_p2 = rlex1_board.create('point', [1, 4]);
1759      *   var rlex1_c1 = rlex1_board.create('circle', [rlex1_p1, rlex1_p2]);
1760      *   var rlex1_p3 = rlex1_board.create('point', [6, 5]);
1761      *   var rlex1_p4 = rlex1_board.create('point', [8, 6]);
1762      *   var rlex1_c2 = rlex1_board.create('circle', [rlex1_p3, rlex1_p4]);
1763      *   var rlex1_r1 = rlex1_board.create('radicalaxis', [rlex1_c1, rlex1_c2]);
1764      * </script><pre>
1765      */
1766     JXG.createRadicalAxis = function (board, parents, attributes) {
1767         var el, el1, el2;
1768 
1769         if (parents.length !== 2 ||
1770                 parents[0].elementClass !== Const.OBJECT_CLASS_CIRCLE ||
1771                 parents[1].elementClass !== Const.OBJECT_CLASS_CIRCLE) {
1772             // Failure
1773             throw new Error("JSXGraph: Can't create 'radical axis' with parent types '" +
1774                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1775                 "\nPossible parent type: [circle,circle]");
1776         }
1777 
1778         el1 = board.select(parents[0]);
1779         el2 = board.select(parents[1]);
1780 
1781         el = board.create('line', [function () {
1782             var a = el1.stdform,
1783                 b = el2.stdform;
1784 
1785             return Mat.matVecMult(Mat.transpose([a.slice(0, 3), b.slice(0, 3)]), [b[3], -a[3]]);
1786         }], attributes);
1787 
1788         el.elType = 'radicalaxis';
1789         el.setParents([el1.id, el2.id]);
1790 
1791         el1.addChild(el);
1792         el2.addChild(el);
1793 
1794         return el;
1795     };
1796 
1797     /**
1798      * @class This element is used to provide a constructor for the polar line of a point with respect to a conic or a circle.
1799      * @pseudo
1800      * @description The polar line is the unique reciprocal relationship of a point with respect to a conic.
1801      * The lines through the intersections of a conic and the polar line of a point with respect to that conic and through that point are tangent to the conic.
1802      * A point on a conic has the polar line of that point with respect to that conic as the tangent line to that conic at that point.
1803      * See {@link http://en.wikipedia.org/wiki/Pole_and_polar} for more information on pole and polar.
1804      * @name PolarLine
1805      * @augments JXG.Line
1806      * @constructor
1807      * @type JXG.Line
1808      * @throws {Exception} If the element cannot be constructed with the given parent objects an exception is thrown.
1809      * @param {JXG.Conic,JXG.Circle_JXG.Point} el1,el2 or
1810      * @param {JXG.Point_JXG.Conic,JXG.Circle} el1,el2 The result will be the polar line of the point with respect to the conic or the circle.
1811      * @example
1812      * // Create the polar line of a point with respect to a conic
1813      * var p1 = board.create('point', [-1, 2]);
1814      * var p2 = board.create('point', [ 1, 4]);
1815      * var p3 = board.create('point', [-1,-2]);
1816      * var p4 = board.create('point', [ 0, 0]);
1817      * var p5 = board.create('point', [ 4,-2]);
1818      * var c1 = board.create('conic',[p1,p2,p3,p4,p5]);
1819      * var p6 = board.create('point', [-1, 1]);
1820      * var l1 = board.create('polarline', [c1, p6]);
1821      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-6018d0d17a98" class="jxgbox" style="width:400px; height:400px;"></div>
1822      * <script type='text/javascript'>
1823      * var plex1_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-6018d0d17a98', {boundingbox: [-3, 5, 5, -3], axis: true, showcopyright: false, shownavigation: false});
1824      * var plex1_p1 = plex1_board.create('point', [-1, 2]);
1825      * var plex1_p2 = plex1_board.create('point', [ 1, 4]);
1826      * var plex1_p3 = plex1_board.create('point', [-1,-2]);
1827      * var plex1_p4 = plex1_board.create('point', [ 0, 0]);
1828      * var plex1_p5 = plex1_board.create('point', [ 4,-2]);
1829      * var plex1_c1 = plex1_board.create('conic',[plex1_p1,plex1_p2,plex1_p3,plex1_p4,plex1_p5]);
1830      * var plex1_p6 = plex1_board.create('point', [-1, 1]);
1831      * var plex1_l1 = plex1_board.create('polarline', [plex1_c1, plex1_p6]);
1832      * </script><pre>
1833      * @example
1834      * // Create the polar line of a point with respect to a circle.
1835      * var p1 = board.create('point', [ 1, 1]);
1836      * var p2 = board.create('point', [ 2, 3]);
1837      * var c1 = board.create('circle',[p1,p2]);
1838      * var p3 = board.create('point', [ 6, 6]);
1839      * var l1 = board.create('polarline', [c1, p3]);
1840      * </pre><div class="jxgbox" id="JXG7b7233a0-f363-47dd-9df5-7018d0d17a98" class="jxgbox" style="width:400px; height:400px;"></div>
1841      * <script type='text/javascript'>
1842      * var plex2_board = JXG.JSXGraph.initBoard('JXG7b7233a0-f363-47dd-9df5-7018d0d17a98', {boundingbox: [-3, 7, 7, -3], axis: true, showcopyright: false, shownavigation: false});
1843      * var plex2_p1 = plex2_board.create('point', [ 1, 1]);
1844      * var plex2_p2 = plex2_board.create('point', [ 2, 3]);
1845      * var plex2_c1 = plex2_board.create('circle',[plex2_p1,plex2_p2]);
1846      * var plex2_p3 = plex2_board.create('point', [ 6, 6]);
1847      * var plex2_l1 = plex2_board.create('polarline', [plex2_c1, plex2_p3]);
1848      * </script><pre>
1849      */
1850     JXG.createPolarLine = function (board, parents, attributes) {
1851         var el, el1, el2,
1852             firstParentIsConic, secondParentIsConic,
1853             firstParentIsPoint, secondParentIsPoint;
1854 
1855         if (parents.length > 1) {
1856             firstParentIsConic = (parents[0].type === Const.OBJECT_TYPE_CONIC ||
1857                 parents[0].elementClass === Const.OBJECT_CLASS_CIRCLE);
1858             secondParentIsConic = (parents[1].type === Const.OBJECT_TYPE_CONIC ||
1859                 parents[1].elementClass === Const.OBJECT_CLASS_CIRCLE);
1860 
1861             firstParentIsPoint = (Type.isPoint(parents[0]));
1862             secondParentIsPoint = (Type.isPoint(parents[1]));
1863         }
1864 
1865         if (parents.length !== 2 ||
1866                 !((firstParentIsConic && secondParentIsPoint) ||
1867                     (firstParentIsPoint && secondParentIsConic))) {
1868             // Failure
1869             throw new Error("JSXGraph: Can't create 'polar line' with parent types '" +
1870                 (typeof parents[0]) + "' and '" + (typeof parents[1]) + "'." +
1871                 "\nPossible parent type: [conic|circle,point], [point,conic|circle]");
1872         }
1873 
1874         if (secondParentIsPoint) {
1875             el1 = board.select(parents[0]);
1876             el2 = board.select(parents[1]);
1877         } else {
1878             el1 = board.select(parents[1]);
1879             el2 = board.select(parents[0]);
1880         }
1881 
1882         // Polar lines have been already provided in the tangent element.
1883         el = board.create('tangent', [el1, el2], attributes);
1884 
1885         el.elType = 'polarline';
1886         return el;
1887     };
1888 
1889     /**
1890      * Register the element type tangent at JSXGraph
1891      * @private
1892      */
1893     JXG.registerElement('tangent', JXG.createTangent);
1894     JXG.registerElement('polar', JXG.createTangent);
1895     JXG.registerElement('radicalaxis', JXG.createRadicalAxis);
1896     JXG.registerElement('polarline', JXG.createPolarLine);
1897 
1898     return {
1899         Line: JXG.Line,
1900         createLine: JXG.createLine,
1901         createTangent: JXG.createTangent,
1902         createPolar: JXG.createTangent,
1903         createSegment: JXG.createSegment,
1904         createAxis: JXG.createAxis,
1905         createArrow: JXG.createArrow,
1906         createRadicalAxis: JXG.createRadicalAxis,
1907         createPolarLine: JXG.createPolarLine
1908     };
1909 });
1910