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Envelope of a family of curves

Definition (Envelope) The envelope of a family of planar curves is a curve that is tangent, at each of its points, to a member of the family.

Suppose we are given a family of curves described by

C_{t} = \{(x, y) \in \mathbb{ R}^{2}, f_{t}(x, y) = 0\}

for all $t \in \mathbb{ R}$ . For convenience, consider the function $F: (t, x, y) \mapsto f_{t}(x, y)$ and assume $F$ is differentiable. A point $(x, y) \in \mathbb{ R}^{2}$ belongs to the envelope of the family $(C_{t})$ , if and only if there exists $t \in \mathbb{ R}$ such that

F(t, x, y) = \partial_{t} F(t, x, y) = 0

simultaneously.

An important and easy special case is when the family of curves are lines. Suppose $f_{t}(x, y) = a(t) x + b(t) y - c(t)$ for some differentiable functions $a, b, c: \mathbb{ R} \to \mathbb{ R}$ . A point $(x, y) \in \mathbb{ R}^{2}$ belongs to the envelope if there exists $t \in \mathbb{ R}$ such that

a(t) x + b(t) y = c(t), \qquad a'(t) x + b'(t) y = c'(t).

This is a linear system in $(x, y)$ , which we can solve if it is not singular, that is, if $a(t) b'(t) \neq a'(t) b(t)$ . In that case, we find

(x, y) = \left(\frac{b' c - b c'}{a b' - a' b}, \frac{a' c - a c'}{a' b - a b'}\right),

where we omitted the parameter $t$ for cleaner notations.

Example Let $k \geq 1$ and consider connecting the lines connecting the complex points $e^{i t}$ and $e^{k i t}$ , for all $t \in \left[ 0, 2\pi\right)$ . The equation of these lines correspond to the coefficients $a(t) = \sin(k t) - \sin(t)$ , $b(t) = \cos(t) - \cos(k t)$ and $c(t) = \sin((k - 1) t)$ . Noticing that $a$ and $b$ can be factored by $\sin([(k - 1) / 2] t)$ , we are left with new coefficients $A(t) = \cos([(k+1)/2] t)$ , $B(t) = \sin([(k+1)/2] t)$ and $C(t) = \cos([(k-1)/2] t)$ after simplifying. Applying the parameterisation of the envelope established above, we obtain

\begin{align*} x(t) &= \cos\left(\frac{k+1}{2}t\right) \cos\left(\frac{k-1}{2} t\right) + \frac{k-1}{k+1} \sin\left(\frac{k+1}{2} t\right) \sin\left(\frac{k-1}{2}t\right), \\ y(t) &= \sin\left(\frac{k+1}{2}t\right) \cos\left(\frac{k-1}{2} t\right) - \frac{k-1}{k+1} \cos\left(\frac{k+1}{2} t\right) \sin\left(\frac{k-1}{2}t\right), \end{align*}

which simplifies into

\begin{align*} x(t) &= \frac{1}{k+1} [k \cos(t) + \cos(kt)], \\ y(t) &= \frac{1}{k+1} [k \sin(t) + \sin(kt)]. \end{align*}

This is the equation of an epicycloid; it can also be obtained by rolling (without slipping) a small circle of radius $1/(k+1)$ around a circle of radius $1 - 2/(k+1)$ .

CC BY-SA 4.0 Alexandre Magueresse. Last modified: March 04, 2026.
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