The orientation of what is reversed? The shape or the vectors e1, e2, and e3?
@ChrisZzh the orientation may change either due to taking the wrong cross product of vectors e1,e2,e3 or even how you determine the matrix A that encodes the linear map f. However the negative sign is just an indication of the orientation of your final volume but its not necessary that the original orientation of vectors e1, e2 and e3 may have changed. This is my understanding of the slide.
@ChrisZzh The orientation of the volume. Imagine the inside is painted blue and the outside is red. When you apply a map of negative determinant, these colors will be reversed.
Could you give an applicational example of why this is useful in CG?
I'm still a bit confused on the orientation being reversed (the inside becomes the outside?). Is there anywhere online I could go to visualize the effect of a map with a negative determinant?
@ChrisZzh Sure; even something as basic as the sign of the determinant turns out to be super important in graphics. For instance, suppose you're simulating an "elastic body" (something like a rubber block, or human flesh). A typical approach is to split up the shape into many small elements, like cubes or tetrahedra. Physically, none of these things should turn inside out, because compressing a finite volume down to zero volume should take infinite energy (and certainly negative volume is non-physical!). So, people develop numerical schemes that penalize meshes with negative determinant. Here's one fun example in the context of elasticity. A closely related problem is flattening a 3D surface into a 2D texture map in a way that guarantees there are no self-overlaps; this is essential for making sure that texture maps are well-behaved. Here's one (of many) recent papers that takes the sign of the determinant into account to avoid these kinds of flips/overlaps.
@Eaglee Just go into your bathroom, and take a look in the mirror! The map taking each point on your face to the corresponding point on your reflection's face is a linear map, with negative determinant. For this reason, you may notice that orientation is reversed; for instance, text on your t-shirt might be backwards.
In general, orientation is an important concept in graphics; though can be a bit tricky to get your head around at first. You might find this page on vector space orientation helpful.
Thank you so much for answering!