The evolute of a curve (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 the differential geometry of curves, the evolute of a curve is the locus of all its centers 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 the envelope of the normals to a curve.
The evolute of a curve, a surface, or more generally a submanifold, is the caustic of the normal map. Let M be a smooth, regular submanifold in ℝn. For each point p in M and each vector v, based at p and normal to M, we associate the point p + v. This defines a Lagrangian map, called the normal map. The caustic of the normal map is the evolute of M.[2]
Evolutes are closely connected to involutes: A curve is the evolute of any of its involutes.
Apollonius (c. 200 BC) discussed evolutes in Book V of his Conics. 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 the tautochrone curve, which in turn helped him construct an isochronous pendulum. This was because the tautochrone curve is a cycloid, 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]
If is the parametric representation of a regular curve in the plane with its curvature nowhere 0 and its curvature radius and the unit normal pointing to the curvature center, then
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 the arc length of the given curve as its parameter, because of and (see Frenet–Serret formulas). Hence the tangent vector of the evolute is:
From this equation one gets the following properties of the evolute:
At points with the evolute is not regular. That means: at points with maximal or minimal curvature (vertices of the given curve) the evolute has cusps (s. parabola, ellipse, 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 the Tait–Kneser theorem on nesting of osculating 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 the envelope of the normals of the given curve.
At sections of the curve with or the curve is an involute 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 be at the section of consideration. An involute of the evolute can be described as follows:
That means: For the string extension the given curve is reproduced.
Parallel curves have the same evolute.
Proof: A parallel curve with distance off the given curve has the parametric representation and the radius of curvature (see parallel curve). Hence the evolute of the parallel curve is
For the parabola with the parametric representation one gets from the formulae above the equations:
which describes a semicubic parabola
Evolute (red) of an ellipse
Evolute of an ellipse[edit]
For the ellipse with the parametric representation one gets:[5]
These are the equations of a non symmetric astroid.
Eliminating parameter leads to the implicit representation
Cycloid (blue), its osculating circle (red) and evolute (green).
Evolute of a cycloid[edit]
For the cycloid with the parametric representation the evolute will be:[6]
which describes a transposed replica of itself.
The evolute of the large nephroid (blue) is the small nephroid (red).
Evolutes of some curves[edit]
The evolute
of a parabola is a semicubic parabola (see above),
of an ellipse is a non symmetric astroid (see above),
of a line is an ideal point,
of a nephroid is a nephroid (half as large, see diagram),
of an astroid is an astroid (twice as large),
of a cardioid is a cardioid (one third as large),
of a circle is its center,
of a deltoid is a deltoid (three times as large),
of a cycloid is a congruent cycloid,
of a logarithmic spiral is the same logarithmic spiral,
of a tractrix is a catenary.
Radial curve[edit]
A curve with a similar definition is the radial 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 the x and y terms from the equation of the evolute. This produces
References[edit]
^Weisstein, Eric W. "Circle Evolute". MathWorld.
^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.
^Yoder, Joella G. (2004). Unrolling Time: Christiaan Huygens and the Mathematization of Nature. Cambridge University Press.
^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.
^R.Courant: Vorlesungen über Differential- und Integralrechnung. Band 1, Springer-Verlag, 1955, S. 268.
^Weisstein, Eric W. "Cycloid Evolute". MathWorld.
Weisstein, Eric W. "Evolute". MathWorld.
Sokolov, D.D. (2001) [1994], "Evolute", Encyclopedia of Mathematics, EMS Press
Yates, R. C.: A Handbook on Curves and Their Properties, J. W. Edwards (1952), "Evolutes." pp. 86ff
Evolute on 2d curves.
vteDifferential transforms of plane curves
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vteChristiaan Huygens
Books
Systema Saturnium (1659)
De Vi Centrifiga (1659)
Horologium Oscillatorium (1673)
Traité de la Lumiére (1692)
Cosmotheoros (1698)
In science and natural philosophy
Discoveries by Huygens
Centripetal acceleration
Coupled oscillation
Conception of the standardization of the temperature scale
Early history of classical mechanics
Early history of calculus
Huygens' law
Huygens' lemniscate
Huygens' principle (Huygens–Fresnel principle)
Huygens' construction
Huygens' tritone
Huygens' wavelet
Huygens' wave theory
Huygens–Steiner theorem
Hypothesis of intelligent extraterrestrial life
Isochrone curve (tautochrone curve)
Foundations of differential geometry of curves (mathematical notions of the evolute and involute of the curve)
Scientific foundations of horology
Mathematical and physical investigations of properties of the pendulum
Modern conception of centrifugal and centripetal forces
Music theory of microtones (31 equal temperament)
Polarization of light (Iceland spar)
Rings of Saturn
Titan (moon)
Theoretical foundations of wave optics (wave theory of light)
In technology
Inventions by Huygens
Aerial telescope
Centrifugal governor
Cycloidal pendulum
Huygens' engine 1
Huygens' eyepiece
Magic lantern 2
Spiral balance spring
Precision timekeeping (pendulum clock and spiral-hairspring watch)
Recognitions
List of things named after Christiaan Huygens
2801 Huygens
Cassini–Huygens
Huygens probe
Mons Huygens
MS Christiaan Huygens
Huygens (crater)
Huygens-Fokker Foundation
Huygens Gap
Huygens Ringlet
Horologium (constellation)
Prix Descartes-Huygens
Other topics
Cosmos: A Personal Voyage - Episode 6: "Travellers' Tales" (1980 documentary TV series by Carl Sagan)
Clocks and Culture, 1300–1700 (1967 history book by Carlo Cipolla)
Revolution in Time: Clocks and the Making of the Modern World (1983 history book by David Landes)
Scientific Revolution
Golden Age of Dutch science and technology
Science and technology in the Dutch Republic
Académie des Sciences
Related people
Huygens family
Galileo Galilei
René Descartes
Salomon Coster
Antonie van Leeuwenhoek
Gottfried Wilhelm Leibniz
Isaac Newton
Robert Hooke
Denis Papin
Augustin-Jean Fresnel
Thomas Young
1 A rudimentary prototype of internal combustion piston engine. 2 An early practical type of image projector and a precursor to both the modern slide projector and the movie projector.