Section 5.4

Evolutes of Curves

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Problem 2058

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=φ(t)x=\varphi (t), y=ψ(t)y=\psi (t) is obtained by eliminating the parameter tt from the equations

X−x=−y′(x′2+y′2)x′y′′−x′′y′,Y−y=x′(x′2+y′2)x′y′′−x′′y′. X-x = -\frac{y'(x'^{2}+y'^{2})}{x'y''-x''y'}, \qquad Y-y = \frac{x'(x'^{2}+y'^{2})}{x'y''-x''y'}.

Find the evolute of the parabola y2=2pxy^{2}=2px.

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Problem 2059

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=acos⁡tx=a\cos t, y=bsin⁡ty=b\sin t.

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Problem 2060

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=ach⁡tx=a\ch t, y=bsh⁡ty=b\sh t.

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Problem 2061

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=acos⁡3tx=a\cos^{3}t, y=asin⁡3ty=a\sin^{3}t.

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Problem 2062

The locus of the centers of curvature of a curve is called the evolute of the given curve. Find the evolute of the cissoid y2(2a−x)=x3y^{2}(2a-x)=x^{3}.

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Problem 2063

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=a2xy=a^{2}.

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Problem 2064

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=ach⁡xay=a\ch \dfrac {x}{a}.

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Problem 2065

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)x=a(t-\sin t), y=a(1−cos⁡t)y=a(1-\cos t).

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Problem 2066

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

x=a(2tcos⁡t+(t2−2)sin⁡t),y=a(2tsin⁡t−(t2−2)cos⁡t). x = a\bigl(2t\cos t+(t^{2}-2)\sin t\bigr), \qquad y = a\bigl(2t\sin t-(t^{2}-2)\cos t\bigr).
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Problem 2067

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

x=acos⁡t+(at+b)sin⁡t,y=asin⁡t−(at+b)cos⁡t. x = a\cos t+(at+b)\sin t, \qquad y = a\sin t-(at+b)\cos t.
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Problem 2068

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⁡φ)r=a(1+\cos \varphi ).

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Problem 2069

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

x=a(2cos⁡t+cos⁡2t),y=a(2sin⁡t−sin⁡2t). x = a(2\cos t+\cos 2t), \qquad y = a(2\sin t-\sin 2t).
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Problem 2070

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(ln⁡tg⁡t2+cos⁡t)x = -a\left(\ln \tg \dfrac {t}{2}+\cos t\right), y=asin⁡ty=a\sin t.

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Problem 2071

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=aln⁡cos⁡xay=a\ln \cos \dfrac {x}{a}.

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Problem 2072

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

x=a(cos⁡t+tsin⁡t),y=a(sin⁡t−tcos⁡t). x = a(\cos t+t\sin t), \qquad y = a(\sin t-t\cos t).
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Problem 2073

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=eaφr=e^{a\varphi }.

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Problem 2074

Prove the equality

x′2+y′2x′y′′−x′′y′=ξ′2+η′2ξ′η′′−ξ′′η′, \frac{x'^{2}+y'^{2}}{x'y''-x''y'} = \frac{\xi '^{2}+\eta '^{2}}{\xi '\eta ''-\xi ''\eta '},

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

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Problem 2075

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

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Problem 2076

Using the properties of the evolute, find the arc length of the astroid x23+y23=a23x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}} between the points (a,0)(a,0) and (0,a)(0,a).

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Problem 2077

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

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Problem 2078

Using the properties of the evolute, find the arc length of Neile's parabola y2=x3y^{2}=x^{3} from (0,0)(0,0) to (4,8)(4,8).

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Problem 2079

Using the properties of the evolute, find the arc length of the evolute of the parabola y2=2pxy^{2}=2px from the cusp to the point where the evolute meets the parabola.

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Problem 2080

Using the properties of the evolute, find the arc length of the logarithmic spiral r=eaφr=e^{a\varphi } for φ0<φ<φ0+2π\varphi_{0}<\varphi <\varphi_{0}+2\pi.

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Remark.
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Remark. All the curves listed above (problems 2075--2080) are evolutes of other curves, which are easily found using the solutions of the problems given in the first part of the section.