Roulettes (curves traced by rolling curves)

The parabola $4ay=x^{2}$ rolls along the $Ox$ axis. Prove that the focus of the parabola traces out a catenary.

The parabola $y=-ax^{2}$ rolls without slipping along the parabola $y=ax^{2}$. Find the locus of its vertex, assuming that at the initial moment the vertices coincided.

Prove that during the same motion of the parabola, its focus describes a straight line, namely the directrix of the fixed parabola.

Along the ellipse $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$, another, equal ellipse rolls without slipping. In the initial position their major axes touched at the ends. Find the locus of the center of the rolling ellipse.

The parabola $4ay=x^{2}$ rolls along the $Ox$ axis. Prove that its focus traces out the catenary $y=a\ch \frac{x}{a}$.

Prove that when the cardioid $r=a(1-\sin \varphi )$ rolls along the cycloid

its cusp traces out a straight line.

Prove that when a parabola $x^{2} = 4 a y$ rolls (without slipping) along the axis $O x$, its focus moves along the catenary $y = a \operatorname {ch}(x / a)$.

Prove that if a catenary rolls without slipping along a line, then the center of curvature of the point of tangency moves along a parabola.

Prove that when a cardioid rolls without slipping

along the cycloid

the cusp of the cardioid moves along a line.

An arc of a logarithmic spiral rolls without slipping along a line. Prove that the center of curvature of the point of tangency moves along a line.

A cycloid rolls without slipping along a line. Prove that the center of curvature of the point of tangency moves along a circle.

(Cycloid.) A circle of radius $a$ rolls without slipping along the axis $O x$. Set up the parametric equation of the curve traced by the point $M$ of the rolling circle that at the initial moment was at the origin, taking as the parameter $t$ the angle from the radius $C M$ of the circle, going to the point $M$, to the radius $C A$, going to the point of tangency $A$.

(Epicycloid.) A circle of radius $r$ rolls without slipping along a circle of radius $R$, remaining outside it. Find the parametric equation of the curve traced by a point of the rolling circle, taking as the origin the center of the fixed circle, and as the parameter the angle $t$ between the positive direction of the abscissa axis and the radius of the fixed circle going to the point of tangency of the two circles. In the initial position, the point of tangency of the circles lies on the abscissa axis.

(Hypocycloid.) A circle of radius $r$ rolls without slipping along a circle of radius $R$, remaining inside it. Write the parametric equation of the curve traced by a point of the rolling circle. The choice of coordinate system and notation is the same as in the preceding problem.