Evolutes and involutes of plane curves

Find the equation of the evolute of each of the following curves: the evolute of the parabola $y^{2}=2px$.

Find the equation of the evolute of each of the following curves: the evolute of the ellipse $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}=1$.

Find the equation of the evolute of each of the following curves: the evolute of the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$.

Find the equation of the evolute of each of the following curves: the evolute of the tractrix $x=a\ln \dfrac {a+\sqrt{a^{2}-y^{2}}}{y}-\sqrt{a^{2}-y^{2}}$.

Find the equation of the evolute of each of the following curves: the evolute of the logarithmic spiral $r=ae^{m\varphi }$.

Prove that the evolute of the cycloid

is again a cycloid, differing from the given cycloid only in its position.

The locus of the centers of curvature of a curve is called the evolute of the given curve. The equation of the evolute for the curve $x=\varphi (t)$, $y=\psi (t)$ is obtained by eliminating the parameter $t$ from the equations

Find the evolute of the parabola $y^{2}=2px$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the ellipse $x=a\cos t$, $y=b\sin t$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the hyperbola $x=a\ch t$, $y=b\sh t$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the astroid $x=a\cos^{3}t$, $y=a\sin^{3}t$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the cissoid $y^{2}(2a-x)=x^{3}$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the hyperbola $xy=a^{2}$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the catenary $y=a\ch \dfrac {x}{a}$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the curve

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the curve

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the cardioid $r=a(1+\cos \varphi )$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the hypocycloid

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the tractrix $x = -a\left(\ln \tg \dfrac {t}{2}+\cos t\right)$, $y=a\sin t$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the curve $y=a\ln \cos \dfrac {x}{a}$.

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the involute of a circle

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the logarithmic spiral $r=e^{a\varphi }$.

Prove the equality

where $(x,y)$ is a point of the curve, and $(\xi ,\eta )$ is the corresponding point of the evolute.

Using the properties of the evolute, find the arc length of one arch of the cycloid $x=a(t-\sin t)$, $y=a(1-\cos t)$.

Using the properties of the evolute, find the arc length of the astroid $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ between the points $(a,0)$ and $(0,a)$.

Using the properties of the evolute, find the arc length of the cardioid $r=a(1+\cos \varphi )$.

Using the properties of the evolute, find the arc length of Neile's parabola $y^{2}=x^{3}$ from $(0,0)$ to $(4,8)$.

Using the properties of the evolute, find the arc length of the evolute of the parabola $y^{2}=2px$ from the cusp to the point where the evolute meets the parabola.

Using the properties of the evolute, find the arc length of the logarithmic spiral $r=e^{a\varphi }$ for $\varphi_{0}<\varphi <\varphi_{0}+2\pi$.

Write the parametric equation of the involute of a circle, i.e., the trajectory of the end of a taut thread unwinding from a fixed round flat spool.

Find the evolute of the curve:

What is the evolute of a circle?

Write the equation of the evolute:

Prove that the evolute of a cycloid is also a cycloid, differing from the given one only in position.

Prove that the evolute of a cardioid is also a cardioid.

Prove that the evolute of an astroid is an astroid similar to the given one, with similarity coefficient 2, rotated relative to the given one by an angle $\pi / 4$.

Find the condition on the parameter $a$ of the logarithmic spiral $\rho = c a^{\varphi }$ under which its evolute coincides with the spiral itself.

Prove that the evolutes of the epicycloid

and of the hypocycloid

$x / a = (1-\lambda ) \cos \lambda t+\lambda \cos (\lambda -1) t, y / a = (1-\lambda ) \sin \lambda t+\lambda \sin (\lambda -1) t$,

are again, respectively, an epicycloid and a hypocycloid, obtained from the given ones by a rotation and a similarity transformation.

Prove that under the conditions of Problem 94 the center of curvature moves along a curve tending to coincide with the circle $\rho = 1$.

Prove that the centers of curvature at points of the Archimedean spiral $\rho = a \varphi$ lying on one ray are located on an ellipse whose semi-axes do not depend on the choice of ray.

Using the properties of the evolute, find the lengths of the following curves:

An inextensible thread is unwound from a fixed drum (of radius $R$), being kept taut. The angular velocity of the point where the thread leaves the circle of the drum is known as a function of time $\omega (t), t \geqslant 0$. Find the path traveled by the end of the thread over the time $T$. What path will the end of the thread travel when the point where the thread leaves the drum completes a full revolution?