Cardioid and limacon of Pascal

the spiral of Archimedes $r = a \varphi$;

the hyperbolic spiral $r = a / \varphi$;

the logarithmic spiral $r = a e^{b \varphi }$;

the cardioid $r = a(\cos \varphi +1)$;

the arc of the lemniscate of Bernoulli $r^{2} = a^{2} \cos 2 \varphi , 0 \leqslant \varphi < \pi / 4$.

$x = a \cos t, \quad y = b \sin t$ (ellipse);

$x = \frac{c^{2}}{a} \cos^{3} t, \quad y = \frac{c^{2}}{b} \sin^{3} t, \quad c^{2} = a^{2}-b^{2}$ (evolute of the ellipse);

$x = \frac{1-t^{2}}{\left(1+t^{2}\right)^{2}}, \quad y = \frac{2 a t}{\left(1+t^{2}\right)^{2}}$ (limaçon);

$x = r(n \cos t-\cos n t), \quad y = r(n \sin t-\sin n t), \quad n-1 \in \mathbb {N}$ (epicycloid);

$x = r(n \cos t+\cos n t), \quad y = r(n \sin t-\sin n t), \quad n-1 \in \mathbb {N}$ (hypocycloid);

$x = a(1-\cos t) \cos t, \quad y = a(1-\cos t) \sin t$ (cardioid).

$r = \frac{a}{\cos (\varphi -\pi / 3)}, \quad \varphi = 0, \quad \varphi = \frac{\pi }{2}$;

$r = a \cos \varphi$;

$r = a(1+\cos \varphi )$ (cardioid);

$r = b+a \cos \varphi , a \geqslant b > 0$ (limaçon);

$r = a \sin 2 \varphi$ (find the area of one petal);

$r = a \cos 3 \varphi$;

$r = a \sin 5 \varphi$;

$r = a \sin n \varphi , n \in \mathbb {N}$

$r = \frac{p}{1+e \cos \varphi }, \varphi = 0, \varphi = \varphi_{0}, 0 \leqslant \varphi \leqslant \varphi_{0}, 0 < e < 1$ (ellipse);

$r = \frac{p}{1+e \cos \varphi }, \varphi = 0, \varphi = \varphi_{0}, 0 \leqslant \varphi \leqslant \varphi_{0}, e > 1, e \cos \varphi_{0} > -1$ (hyperbola);

$r = \frac{1}{\sqrt{1+\cos^{2} \varphi }}, \varphi = 0, \varphi = \frac{\pi }{4}$

$r = \frac{1}{\sqrt{4-3 \cos^{2}}} = , \varphi = 0, \varphi = \frac{\pi }{2}$;

$r = 2 \sqrt{\varphi \arccos \left(\varphi^{2}-1\right)}, \varphi = 1, \varphi = \sqrt{3 / 2}$;

$r = \frac{\sin \varphi }{\sqrt{\cos \varphi }}, \varphi = 0, \varphi = \frac{\pi }{6}$.

the outer loop of the limaçon $\left|r-1 \cos \varphi \right| = 1$;

its inner loop.

$r = a \sin \varphi$

$r = a e^{k \varphi }, \quad \varphi_{1} \leqslant \varphi \leqslant \varphi_{2}$ (logarithmic spiral);

$r = a(1-\cos \varphi )$ (cardioid);

$r = 2(1+\cos \varphi ), \quad r \leqslant 1$;

$r = a(1-\sin \varphi ), \quad -\pi / 2 \leqslant \varphi \leqslant -\pi / 6$;

$r = a \operatorname {th}(\varphi / 2), \quad 0 \leqslant \varphi \leqslant \varphi_{0}$.

$r = a \cos \varphi +l, l < a$ (outer loop);

$r = a \cos \varphi -l, l < a$ (inner loop);

$r = a \cos \varphi +l, l \geqslant a$.

for $b < a$;

for $b = a$?

the locus of points symmetric to a point $A$, lying on a given circle, with respect to all possible tangents to this circle;

the locus of the feet of the perpendiculars dropped from a point $A$, lying on a given circle, onto all possible tangents to this circle.