Example: Pretzel curve

The pretzel curve is a quartic curve, the shape of which resembles a pretzel, as shown in figures below. The mathematical representations of this curve are as follows:

• Polar equation: $r = a\dfrac{1-2\cos{\varphi}}{1-\cos{\varphi}}; \quad a \in \mathcal{R}\setminus\{0\}.$


• Parametric equations: $x = a\dfrac{-t^4+4t^2-3}{2t^3+2}, \quad y = a\dfrac{t^3-3t}{t^2+1}; \quad a \in \mathcal{R}\setminus\{0\}, \quad t \in \mathcal{R}.$


• Algebraic equation: $y^{4}+x^{2}y^{2}+2ax^{3}+2axy^{2}+3a^{2}x^{2}-a^{2}y^{2}=0; \quad a \in \mathcal{R}\setminus\{0\}.$

Source: Hasek, R. (2017) A Remarkable Quartic Pretzel Curve. Journal for Geomnetry and Graphics. Volume 21, No. 1, 37-44.

Pretzel curve given by its polar equation

$r = a\dfrac{1-2\cos{\varphi}}{1-\cos{\varphi}}; \quad a \in \mathcal{R}\setminus\{0\}$

Pretzel curve given by its parametric equations

$x = a\dfrac{-t^4+4t^2-3}{2t^3+2}, \quad y = a\dfrac{t^3-3t}{t^2+1}; \quad a \in \mathcal{R}\setminus\{0\}, \quad t \in \mathcal{R}$

Pretzel curve as a locus of points

The pretzel curve is part of the solution of the following locus problem (for the meaning of the mentioneg geometric objects, see the attached dynamic figure): Given a circle with a center $A$ and a diameter $MP$. For an arbitrary point $B$ of this circle there is a point $C$ on the line $AB$ so that $$\dfrac{|MO|}{|AO|}=\dfrac{|AB|}{|CB|},$$ where $O$ is the foot of a perpendicular drawn from $B$ to $MP$. Find the locus of the points $C$.

Two curves solve this locus problem, a parabola with the focus $[0, 0]$ and directrix $d:\,y = -a$, where $a=|AM|$, for pont $C1$, and a pretzel curve with a parameter $a=|AM|$ for point $C2$ (See the figure below, point $B$ is movable along the circle.).