**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Involute

Summary

In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from or wrapped around the curve.
The evolute of an involute is the original curve.
It is generalized by the roulette family of curves. That is, the involutes of a curve are the roulettes of the curve generated by a straight line.
The notions of the involute and evolute of a curve were introduced by Christiaan Huygens in his work titled Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (1673), where he showed that the involute of a cycloid is still a cycloid, thus providing a method for constructing the cycloidal pendulum, which has the useful property that its period is independent of the amplitude of oscillation.
Arc length
Let be a regular curve in the plane with its curvature nowhere 0 and , then the curve with the parametric representation
is an involute of the given curve.
Adding an arbitrary but fixed number to the integral results in an involute corresponding to a string extended by (like a ball of wool yarn having some length of thread already hanging before it is unwound). Hence, the involute can be varied by constant and/or adding a number to the integral (see Involutes of a semicubic parabola).
If one gets
In order to derive properties of a regular curve it is advantageous to suppose the arc length to be the parameter of the given curve, which lead to the following simplifications: and , with the curvature and the unit normal. One gets for the involute:
and
and the statement:
At point the involute is not regular (because ),
and from follows:
The normal of the involute at point is the tangent of the given curve at point .
The involutes are parallel curves, because of and the fact, that is the unit normal at .
The family of involutes and the family of tangents to the original curve makes up an orthogonal coordinate system.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (1)

Related concepts (5)

Related publications (3)

Related lectures (10)

Ontological neighbourhood

MATH-189: Mathematics

Ce cours a pour but de donner les fondements de mathématiques nécessaires à l'architecte contemporain évoluant dans une école polytechnique.

Arc length is the distance between two points along a section of a curve. Determining the length of an irregular arc segment by approximating the arc segment as connected (straight) line segments is also called curve rectification. A rectifiable curve has a finite number of segments in its rectification (so the curve has a finite length). If a curve can be parameterized as an injective and continuously differentiable function (i.e., the derivative is a continuous function) , then the curve is rectifiable (i.

The Archimedean spiral (also known as the arithmetic spiral) is a spiral named after the 3rd-century BC Greek mathematician Archimedes. It is the locus corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line that rotates with constant angular velocity. Equivalently, in polar coordinates (r, θ) it can be described by the equation with real numbers a and b.

A logarithmic spiral, equiangular spiral, or growth spiral is a self-similar spiral curve that often appears in nature. The first to describe a logarithmic spiral was Albrecht Dürer (1525) who called it an "eternal line" ("ewige Linie"). More than a century later, the curve was discussed by Descartes (1638), and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".

Involutes and Bezier Curves

Explores involutes and Bezier curves, essential in computer-aided design.

Curvature and Osculating Circle

Covers curvature, osculating circles, and the evolute of plane curves, with examples and equations.

Fourier Series Interpretation

Explores the interpretation of Fourier series from basic to complex signals, demonstrating the concept through animations and explaining the relationship between sine waves and circles.

Jürg Alexander Schiffmann, Luis Carlos Mendoza Toledo

Efficient compressed air energy storage requires reversible isothermal compression and expansion devices. The isothermal compression and expansion processes can either be approached by several stages with intercooling or by the more convenient injection of ...

Annalisa Buffa, Pablo Antolin Sanchez, Xiaodong Wei

We present a novel method to perform numerical integration over curved polyhedra enclosed by high-order parametric surfaces. Such a polyhedron is first decomposed into a set of triangular and/or rectangular pyramids, whose certain faces correspond to the g ...

Michaël Unser, Philippe Thévenaz

Traditional snakes, or active contours, are planar parametric curves. Their parameters are determined by optimizing the weighted sum of three energy terms: one depending on the data (typically on the integral of its gradient under the curve, or on its inte ...