Why does skinny triangle is avoided in triangulation algorithm?

Question

I recently learned about Delaunay triangulation algorithm.. This algorithm prevents the generation of skinny triangles.. I haven't really seen any good explanation of why we don't want skinny triangles.. Can anyone explain the reason?

Thanks..

Answer 1

We don't want to avoid skinny triangles as much as we want to avoid really long triangles.

Long triangles make interpolation of properties less accurate because the points in which the data is stored are farther apart. This is equivalent to – when using a 2160p (4K UHD) screen – the preference of watching a 1080p video in fullscreen mode to watching the same video but in 720p, also in fullscreen mode. The 720p video is streched 3 times when the image is rescaled to fullscreen while the 1080p video is only streched 2 times; hence the pixels in the 720p video are streched farther apart compared with the pixels in the 1080p video and the interpolation of the colors in the pixels become less acurate (simply because the 720p video can't contain as fine details as the 1080p video can), which is another way of saying that the 1080p video looks better than the 720p video when viewed in fullscreen on a 2160p screen.

However, the skinnier our triangles are, the longer they have to be in order to cover the same area, unless we use more triangles in total. The former is undesirable because of above mentioned reason. The latter is usually undesirable too since it increases the required memory consumption as well as the computational load on the algorithms that work with them since there are more triangles to loop through. We usually therefore want to avoid skinny triangles.