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Evolute

From Wikipedia, the free encyclopedia
Centers of curvature of a curve
Theevolute of acurve (blue parabola) is the locus of all its centers of curvature (red).
The evolute of a curve (in this case, an ellipse) is the envelope of its normals.

In thedifferential geometry of curves, theevolute of acurve is thelocus of all itscenters of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center.[1] Equivalently, an evolute is theenvelope of thenormals to a curve.

The evolute of a curve, a surface, or more generally asubmanifold, is thecaustic of the normal map. LetM be a smooth, regular submanifold inRn. For each pointp inM and each vectorv, based atp and normal toM, we associate the pointp +v. This defines aLagrangian map, called the normal map. The caustic of the normal map is the evolute ofM.[2]

Evolutes are closely connected toinvolutes: A curve is the evolute of any of its involutes.

History

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Apollonius (c. 200 BC) discussed evolutes in Book V of hisConics. However,Huygens is sometimes credited with being the first to study them (1673). Huygens formulated his theory of evolutes sometime around 1659 to help solve the problem of finding thetautochrone curve, which in turn helped him construct an isochronous pendulum. This was because the tautochrone curve is acycloid, and the cycloid has the unique property that its evolute is also a cycloid. The theory of evolutes, in fact, allowed Huygens to achieve many results that would later be found using calculus.[3]

Evolute of a parametric curve

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Ifx→=c→(t),t∈[t1,t2]{\displaystyle {\vec {x}}={\vec {c}}(t),\;t\in [t_{1},t_{2}]} is the parametric representation of aregular curve in the plane with its curvature nowhere 0 andρ(t){\displaystyle \rho (t)} its curvature radius andn→(t){\displaystyle {\vec {n}}(t)} the unit normal pointing to the curvature center, thenE→(t)=c→(t)+ρ(t)n→(t){\displaystyle {\vec {E}}(t)={\vec {c}}(t)+\rho (t){\vec {n}}(t)}describes theevolute of the given curve.

Forc→(t)=(x(t),y(t))T{\displaystyle {\vec {c}}(t)=(x(t),y(t))^{\mathsf {T}}} andE→=(X,Y)T{\displaystyle {\vec {E}}=(X,Y)^{\mathsf {T}}} one getsX(t)=x(t)−y′(t)(x′(t)2+y′(t)2)x′(t)y″(t)−x″(t)y′(t){\displaystyle X(t)=x(t)-{\frac {y'(t){\Big (}x'(t)^{2}+y'(t)^{2}{\Big )}}{x'(t)y''(t)-x''(t)y'(t)}}} andY(t)=y(t)+x′(t)(x′(t)2+y′(t)2)x′(t)y″(t)−x″(t)y′(t).{\displaystyle Y(t)=y(t)+{\frac {x'(t){\Big (}x'(t)^{2}+y'(t)^{2}{\Big )}}{x'(t)y''(t)-x''(t)y'(t)}}.}

Properties of the evolute

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The normal at point P is the tangent at the curvature center C.

In order to derive properties of a regular curve it is advantageous to use thearc lengths{\displaystyle s} of the given curve as its parameter, because of|c→′|=1{\textstyle \left|{\vec {c}}'\right|=1} andn→′=−c→′/ρ{\textstyle {\vec {n}}'=-{\vec {c}}'/\rho } (seeFrenet–Serret formulas). Hence the tangent vector of the evoluteE→=c→+ρn→{\textstyle {\vec {E}}={\vec {c}}+\rho {\vec {n}}} is:E→′=c→′+ρ′n→+ρn→′=ρ′n→ .{\displaystyle {\vec {E}}'={\vec {c}}'+\rho '{\vec {n}}+\rho {\vec {n}}'=\rho '{\vec {n}}\ .}From this equation one gets the following properties of the evolute:

  • At points withρ′=0{\displaystyle \rho '=0} the evolute isnot regular. That means: at points with maximal or minimal curvature (vertices of the given curve) the evolute hascusps. (See the diagrams of the evolutes of the parabola, the ellipse, the cycloid and the nephroid.)
  • For any arc of the evolute that does not include a cusp, the length of the arc equals the difference between the radii of curvature at its endpoints. This fact leads to an easy proof of theTait–Kneser theorem on nesting ofosculating circles.[4]
  • The normals of the given curve at points of nonzero curvature are tangents to the evolute, and the normals of the curve at points of zero curvature are asymptotes to the evolute. Hence: the evolute is theenvelope of the normals of the given curve.
  • At sections of the curve withρ′>0{\displaystyle \rho '>0} orρ′<0{\displaystyle \rho '<0} the curve is aninvolute of its evolute. (In the diagram: The blue parabola is an involute of the red semicubic parabola, which is actually the evolute of the blue parabola.)

Proof of the last property:
Let beρ′>0{\displaystyle \rho '>0} at the section of consideration. Aninvolute of the evolute can be described as follows:C→0=E→−E→′|E→′|(∫0s|E→′(w)|dw+l0),{\displaystyle {\vec {C}}_{0}={\vec {E}}-{\frac {{\vec {E}}'}{\left|{\vec {E}}'\right|}}\left(\int _{0}^{s}\left|{\vec {E}}'(w)\right|\mathrm {d} w+l_{0}\right),}wherel0{\displaystyle l_{0}} is a fixed string extension (seeInvolute of a parameterized curve ).
WithE→=c→+ρn→,E→′=ρ′n→{\displaystyle {\vec {E}}={\vec {c}}+\rho {\vec {n}}\;,\;{\vec {E}}'=\rho '{\vec {n}}} andρ′>0{\displaystyle \rho '>0} one getsC→0=c→+ρn→−n→(∫0sρ′(w)dw+l0)=c→+(ρ(0)−l0)n→.{\displaystyle {\vec {C}}_{0}={\vec {c}}+\rho {\vec {n}}-{\vec {n}}\left(\int _{0}^{s}\rho '(w)\;\mathrm {d} w\;+l_{0}\right)={\vec {c}}+(\rho (0)-l_{0})\;{\vec {n}}\,.}That means: For the string extensionl0=ρ(0){\displaystyle l_{0}=\rho (0)} the given curve is reproduced.

  • Parallel curves have the same evolute.

Proof: A parallel curve with distanced{\displaystyle d} off the given curve has the parametric representationc→d=c→+dn→{\displaystyle {\vec {c}}_{d}={\vec {c}}+d{\vec {n}}} and the radius of curvatureρd=ρ−d{\displaystyle \rho _{d}=\rho -d} (seeparallel curve). Hence the evolute of the parallel curve isE→d=c→d+ρdn→=c→+dn→+(ρ−d)n→=c→+ρn→=E→.{\displaystyle {\vec {E}}_{d}={\vec {c}}_{d}+\rho _{d}{\vec {n}}={\vec {c}}+d{\vec {n}}+(\rho -d){\vec {n}}={\vec {c}}+\rho {\vec {n}}={\vec {E}}\;.}

Examples

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Evolute of a parabola

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For the parabola with the parametric representation(t,t2){\displaystyle (t,t^{2})} one gets from the formulae above the equations:X=⋯=−4t3{\displaystyle X=\cdots =-4t^{3}}Y=⋯=12+3t2,{\displaystyle Y=\cdots ={\frac {1}{2}}+3t^{2}\,,}which describes asemicubic parabola

Evolute (red) of an ellipse

Evolute of an ellipse

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For the ellipse with the parametric representation(acos⁡t,bsin⁡t){\displaystyle (a\cos t,b\sin t)} one gets:[5]X=⋯=a2−b2acos3⁡t{\displaystyle X=\cdots ={\frac {a^{2}-b^{2}}{a}}\cos ^{3}t}Y=⋯=b2−a2bsin3⁡t.{\displaystyle Y=\cdots ={\frac {b^{2}-a^{2}}{b}}\sin ^{3}t\;.}These are the equations of a non symmetricastroid. Eliminating parametert{\displaystyle t} leads to the implicit representation(aX)23+(bY)23=(a2−b2)23 .{\displaystyle (aX)^{\tfrac {2}{3}}+(bY)^{\tfrac {2}{3}}=(a^{2}-b^{2})^{\tfrac {2}{3}}\ .}

Cycloid (blue), its osculating circle (red) and evolute (green).

Evolute of a cycloid

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For thecycloid with the parametric representation(r(t−sin⁡t),r(1−cos⁡t)){\displaystyle (r(t-\sin t),r(1-\cos t))} the evolute will be:[6]X=⋯=r(t+sin⁡t){\displaystyle X=\cdots =r(t+\sin t)}Y=⋯=r(cos⁡t−1){\displaystyle Y=\cdots =r(\cos t-1)}which describes a transposed replica of itself.

The evolute of the large nephroid (blue) is the small nephroid (red).

Evolute of log-aesthetic curves

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The evolute of a log-aesthetic curve is another log-aesthetic curve.[7] One instance of this relation is that the evolute of anEuler spiral is a spiral withCesàro equationκ(s)=−s−3{\displaystyle \kappa (s)=-s^{-3}}.[8]

Evolutes of some curves

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The evolute

Radial curve

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A curve with a similar definition is theradial of a given curve. For each point on the curve take the vector from the point to the center of curvature and translate it so that it begins at the origin. Then the locus of points at the end of such vectors is called the radial of the curve. The equation for the radial is obtained by removing thex andy terms from the equation of the evolute. This produces(X,Y)=(−y′x′2+y′2x′y″−x″y′,x′x′2+y′2x′y″−x″y′).{\displaystyle (X,Y)=\left(-y'{\frac {{x'}^{2}+{y'}^{2}}{x'y''-x''y'}}\;,\;x'{\frac {{x'}^{2}+{y'}^{2}}{x'y''-x''y'}}\right).}

References

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  1. ^Weisstein, Eric W."Circle Evolute".MathWorld.
  2. ^Arnold, V. I.; Varchenko, A. N.; Gusein-Zade, S. M. (1985).The Classification of Critical Points, Caustics and Wave Fronts: Singularities of Differentiable Maps, Vol 1.Birkhäuser.ISBN 0-8176-3187-9.
  3. ^Yoder, Joella G. (2004).Unrolling Time: Christiaan Huygens and the Mathematization of Nature.Cambridge University Press.
  4. ^Ghys, Étienne;Tabachnikov, Sergei; Timorin, Vladlen (2013). "Osculating curves: around the Tait-Kneser theorem".The Mathematical Intelligencer.35 (1):61–66.arXiv:1207.5662.doi:10.1007/s00283-012-9336-6.MR 3041992.
  5. ^R.Courant:Vorlesungen über Differential- und Integralrechnung. Band 1, Springer-Verlag, 1955, S. 268.
  6. ^Weisstein, Eric W."Cycloid Evolute".MathWorld.
  7. ^Yoshida, N., & Saito, T. (2012). "The Evolutes of Log-Aesthetic Planar Curves and the Drawable Boundaries of the Curve Segments".Computer-Aided Design and Applications.9 (5):721–731.doi:10.3722/cadaps.2012.721-731.{{cite journal}}: CS1 maint: multiple names: authors list (link)
  8. ^"Evolute of the Euler spiral".Linebender wiki. 2024-03-11.
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