Name of method for finding envelope of curve using expanding circles

Question

In page-10 of this pdf, a method to find envelope using circles is described. Is there a name for this method and what are sources to read more on it?

Answer 1

First of all, see this alternative document and its extension here with very sound explanations.

This method could be called "the wavefront method" or the "Huygens-Fresnel" method (as Snake707 has remarked) which is classical in optics. See for example here. See as well fig. 2 in this past answer of mine about sound envelopes.

Here is a way to interpret the issue in this case using the metaphor of fireworks.

Imagine fireworks being ignited from a unique spot and sent in all directions. Let us assume that each individual rocket emits light all the time plus a more intense light at every tenth of second for example. I have seen such fireworks and it is striking to observe the formation of circles (observable due to persistence on light on our retina) as illustrated on the figure here (obtained by a quickly written Matlab simulation program), in perfect agreement with the figure in the document you give.

Edit: Starting from cinematic equations (1) and (2) in the document:

$$\begin{cases}x&=&vt \cos \theta \\ y&=&-\tfrac12gt^2+vt\sin \theta +h \end{cases}$$

that we can write:

$$\begin{cases}x&=&vt \cos \theta \\ y-(h-\tfrac12gt^2)&=&vt\sin \theta \end{cases}$$

if we square and add them, we get equation (8) giving the equation of the generic circle $(C_t)$, for a fixed value of $t$. Written it under the classical following form, where $(x_0,y_0)$ are the center's coordinates and $R$ the radius:

$$(x-x_t)^2 +(y-y_t)^2=R_t^2$$

we have: $$x_t=0, \ \ \ y_t=h-\tfrac12 gt^2, \ \ \ R_t=vt$$

The expression of $y_t$ shows that the center of $(C_t)$ is, right from the beginning, going down and down, crossing the $x$ axis for $t = \dfrac{2h}{g}$.

Answer 2

It reminds me of the Huygens–Fresnel principle.