envelopekurven

Envelopekurven, or envelope curve, is a concept in mathematics describing a curve that is tangent to a whole family of curves at some point, effectively forming a boundary or limit position of the family as its parameter varies.

Definition and construction

Given a family of curves described by F(x, y, t) = 0 with a parameter t, the envelope is the set of points (x, y) that satisfy both F(x, y, t) = 0 and the partial derivative ∂F/∂t(x, y, t) = 0. Solving these equations for x and y in terms of t and then eliminating t yields the envelope equation. Equivalently, if you have a family g_t(x, y) = 0, differentiate with respect to t, set ∂g_t/∂t = 0, and solve together with g_t = 0 to obtain the envelope.

Example

Consider the family of lines y = m x + b(m). Let F(x, y, m) = y − m x − b(m). The envelope is found from F = 0 and ∂F/∂m = −x − b′(m) = 0, giving x = −b′(m). Substituting back gives y = b(m) − m b′(m). Thus the envelope is the parametric curve (x(m), y(m)) = (−b′(m), b(m) − m b′(m)).

Applications and related concepts

In optics, the envelope of light rays forms a caustic. In differential geometry, the evolute of a curve is the envelope of its normals. Envelopes can reveal boundaries of swept regions and guide design in engineering and analysis in geometry and physics.