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.

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MATH-189: Mathematics
Ce cours a pour but de donner les fondements de mathématiques nécessaires à l'architecte contemporain évoluant dans une école polytechnique.
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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.
Archimedean spiral
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.
Logarithmic spiral
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