Envelope of a One-Parameter Family of Curves

Given a one-parameter family of curves in the plane, written implicitly as

\( \varphi(x, y, c) = 0, \)

a curve \( \gamma \) is called an envelope of the family if through every point of \( \gamma \) there passes a curve of the family that is tangent to \( \gamma \) at that point.

By this definition, the family member and the envelope share the same slope \( y' \) at the common point \( (x,y) \). Therefore, when the family of curves represents the general solution of a first-order ODE, the envelope satisfies the same ODE.

A visual cue
Visually, the graph of an envelope resembles works of String Art: many straight threads, none of which is the curve — yet the curve emerges as the shape they all touch.

Example 1

We find the envelope of the family of lines

\( \frac{x}{c} + \frac{y}{1-c} = 1, \qquad 0 < c < 1. \)

Differentiate the equation with respect to \( c \):

\( \frac{\partial}{\partial c}\!\left( \frac{x}{c} + \frac{y}{1-c} \right) = -\frac{x}{c^2} + \frac{y}{(1-c)^2} = 0. \)

Isolating \( y \):

\( y (1-c)^2 = x c^2 \;\Rightarrow\; y = x\, \frac{(1-c)^2}{c^2}. \)

Substituting this \( y \) back into the original equation:

\( \frac{x}{c} + \frac{x (1-c)}{c^2} = 1 \;\Rightarrow\; \frac{x}{c^2} = 1 \;\Rightarrow\; x = c^2, \)

and then

\( y = x\, \frac{(1-c)^2}{c^2} = (1-c)^2. \)

We obtained the envelope as a parametric curve: \( x = c^2 \), \( y = (1-c)^2 \), i.e. \( \sqrt{x} = c \), \( \sqrt{y} = 1 - c \). That is:

\( \sqrt{x} + \sqrt{y} = 1. \)
String art: the lines \( \frac{x}{c} + \frac{y}{1-c} = 1 \) sweep over \( 0 < c < 1 \), and the envelope \( \sqrt{x}+\sqrt{y}=1 \) emerges

Example 2 — The Astroid

We find the envelope of the family of lines

\( \frac{x}{\cos\alpha} + \frac{y}{\sin\alpha} = 1. \)
Segments of constant length sliding between the axes: their envelope is the astroid \( x^{2/3}+y^{2/3}=1 \)

Denote \( \varphi(x, y, \alpha) = \dfrac{x}{\cos\alpha} + \dfrac{y}{\sin\alpha} - 1 = 0 \) and differentiate with respect to the parameter \( \alpha \):

\( \frac{\partial \varphi}{\partial \alpha} = \frac{x \sin\alpha}{\cos^2\alpha} - \frac{y \cos\alpha}{\sin^2\alpha} = 0. \)

We solve the system of the two equations. From the second equation:

\( x \sin^3\alpha = y \cos^3\alpha. \)

Set \( x = \cos^3\alpha \), \( y = \sin^3\alpha \) and substitute into the first equation:

\( \frac{\cos^3\alpha}{\cos\alpha} + \frac{\sin^3\alpha}{\sin\alpha} = \cos^2\alpha + \sin^2\alpha = 1, \)

which holds for every \( \alpha \). The envelope

\( x = \cos^3\alpha, \qquad y = \sin^3\alpha \)

is a parametrisation of a curve called the astroid. Its Cartesian equation is

\( x^{2/3} + y^{2/3} = 1. \)

The definition, point by point

Given a one-parameter family of curves in the plane, written implicitly as \( \varphi(x,y,c) = 0 \), a curve \( \gamma \) is an envelope of the family if at every point of \( \gamma \) there is a curve of the family passing through that point and tangent to \( \gamma \) there. That is: for every point \( P \) on the envelope there exists a value of the parameter \( c \) such that the corresponding family curve \( \varphi(x,y,c) = 0 \) passes through \( P \), and at that point it has the same tangent as the envelope.

© Shir Sivroni