Curves, Envelopes, Evolutes

Abstract

Curves appear in a wide variety of mathematical problems and in very many ways. Here is a list of examples (in a real affine space):

• in kinematics, e.g., as integral curves of vector fields, in other words as solutions of differential equations;

• a variant: as envelopes of families of lines, or of families of other curves;

• they also appear, in space, at the intersection of two surfaces (e.g., of a surface and a plane), therefore, they are useful for the study of surfaces (see Chapter VIII);

• as the solutions of an algebraic equation P( x, y) = 0 (as conics in the plane, for instance), or of several algebraic equations (P₁ (x, y, z) = P₂ (x, y, z) = 0 in space⁽¹⁾ of dimension 3, etc.).