Derivation of the equation for the envelope

Question

Suppose we have a family of curves on the plane. The equation of the curves is given by

$$ f(x ,y ;t) = 0 . $$

Here $t$ is the parameter. On Wiki, the equations determining the envelope of this family are

$$ f(x, y ; t ) = f_t(x, y; t) =0 . $$

The derivation on wiki sounds reasonable, but I still find it not so intuitive.

Any one has some other method?

Answer 1

There are a number of explanations here: https://en.wikipedia.org/wiki/Envelope_(mathematics)

The mathworld explanation is much too short; I can see how you would be puzzled if you looked there.

Answer 2

Let a family $\Gamma$ of curves $\gamma_c$ in the $(x,y)$-plane be given by an equation of the form $$F(x,y,c)=0\ .\tag{1}$$ A triple $(x_0,y_0,c_0)$ satisfying $(1)$ is regular if $$\eqalign{\bigl(F_x(x_0,y_0,c_0),F_y(x_0,y_0,c_0)\bigr)&\ne(0,0),\cr F_c(x_0,y_0,c_0)&\ne0\ .\cr}\tag{2}$$ The first condition guarantees that the the curve $\gamma_{c_0}$ is a nice level line of the function $(x,y)\mapsto F(x,y,c_0)$ passing through the point $(x_0,y_0)$. The second condition is more crucial. It guarantees that the curves $\gamma_c$ with $c_0-\delta<c<c_0+\delta$ form a "homogeneous curve field in the neighborhood of $(x_0,y_0)$": Up to a differentiable distortion it looks like a field of parallels. For an explanation of this fact see below.

Now to the envelopes: If the point $(x_0,y_0)$ is lying not only on the curve $\gamma_{c_0}$, but also on the envelope $\epsilon$ of the family $\Gamma$ then the curve field in the neighborhood of $(x_0,y_0)$ does definitely not look like a field of parallels, because all $\gamma_c$ with $c$ near $c_0$ intersect $\gamma_{c_0}$ in points near $(x_0,y_0)$. It follows that in all points of $\epsilon$ the second condition of $(2)$ is violated; in other words: The triples $(x,y,t)$ with $(x,y)\in\epsilon$ necessarily satisfy $F_c(x,y,c)=0$.

Coming back to the regular triples: The second condition $(2)$ guarantees that we can solve $(1)$ for $c$ in the neighborhood of $(x_0,y_0,c_0)$: There is a differentiable function $$g:\quad (x,y)\mapsto c=g(x,y)$$ with $g(x_0,y_0)=c_0$ such that in a neighborhood of $(x_0,y_0,c_0)$ the equation $(1)$ is equivalent with $$g(x,y)=c\ .$$ This can be interpreted as follows: The family curves $\gamma_c$ appear as level lines of the function $g$. In order to check that these level lines behave as desired in the neighborhood of $(x_0,y_0)$ we have to make sure that $\nabla g(x_0,y_0)\ne(0,0)$. Now from the formula for the derivative of an implicitly given function we get $$\nabla g(x_0,y_0)=\left(-{F_x\over F_c}, {F_y\over F_c}\right)_{(x_0,y_0,c_0)}\ne(0,0)\ .$$

Answer 3

Make a simple construction and all will be clear. ( That what I did way back)

Plot 3 lines ( 3 graphs) for three values of c for three straight lines in (2). $$ y = 2 x c + c^2 \tag{1} $$

Differentiate with respect to $ c $

$$ 0 = 2 x + 2 c \tag{2} $$

So plug in (1) $ x=-c $

Draw a smooth parabola touching the three straight lines, as an envelope or singular solution of (1).

Called C- discriminant method using Clairaut's differential equation.