Envelopes of straight line families in the plane

TL;DR

This paper discusses the limitations of traditional methods for representing envelopes of straight line families in the plane and introduces a correct, precise method for such representations, explaining when and why common methods fail.

Contribution

It provides a clear explanation of the recently discovered correct method for representing envelopes of straight line families and clarifies the failure cases of traditional methods.

Findings

Traditional methods can fail to accurately represent envelopes.

The paper explains the correct method for envelope representation.

Conditions under which widespread methods fail are clarified.

Abstract

There is a widespread method to represent the envelope when a given hyperplane family creates an envelope. However, one sometimes encounters cases when the widespread method fails to represent the desired envelope precisely, and is confused. At the same time, one wants to find a correct method to draw the envelope precisely. In this article, focused on straight line families in the plane, an easy to understand explanation is given on the recently discovered correct method to represent the envelope precisely. Moreover, it is explained when and why the widespread method fails to represent the precise shape of envelope as well.