I've come across the term "degenerate triangles" in my studies of mathematics and geometry, and I'm curious about why this concept was invented. From what I understand, a degenerate triangle is a triangle that has collapsed into a line or point due to coinciding vertices or a side length of zero. But why was it necessary to include these degenerate cases in the definition of a triangle?
I've searched online for answers, but I haven't found a clear explanation. Can someone help me understand the motivation behind inventing the concept of degenerate triangles? Are there specific mathematical contexts where these degenerate cases arise frequently? Any insights would be greatly appreciated.
Thanks in advance for your help!
As far as I am concerned the concept of a degenerate triangle wasn‘t invented, but rather is a consequence of being not careful enough when trying to find a mathematically rigorous definition of a triangle. For example the convex hull of three points also encapsulates degenerate cases like some points coinciding or all points being collinear. Either you explicitly exclude these degenerate cases in your definition, or you acknowledge them as degenerate cases…
Regarding applications it can be useful to have degenerate cases incorporated in your definition. For example this way every affine linear map preserves triangles, even if it is the constant map. Preserving non-degenerate triangles then becomes an interesting property of affine linear maps one can study…