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EVOLUTOID OF A PLANE CURVE
Notion studied by Réaumur in 1709 and Lancret
in 1811.
bouasse 450, loria 297 See also this PhD thesis. 
For an initial curve with current point , the evolutoid is the set of points . Complex parametrization: . If the initial curve's intrinsic equation 2: , we get: Parametric intrinsic equation 1: . Intrinsic equation 2: . 
The evolutoids of a curve (G)
are the envelopes of the lines
(D) forming a constant angle a with the
initial curve; when a = p/2,
we get the evolute, and when a
is 0 or 180°, we get the curve itself.
The characteristic point of the envelope can be constructed
very simply by projecting the centre of curvature of the initial curve
on the line (D) (Réaumur's theorem).
Examples:
 the evolutoid of a line is the point
at infinity of the varying line.
 the evolutoids of a circle are concentric
circles:
 the evolutoids of a logarithmic
spiral are logarithmic spirals.
 the evolutoids of a cycloidal
curve (intrinsic equation )
are similar to the initial curve (intrinsic equation
with ).
The evolutoids of the cycloid are isometric cycloids.
Below, the evolutoid at 45°. 
Below, the evolutoid at 60° of a cardioid.

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© Robert FERRÉOL
2017