# Is geodesic distance between two similar photos less than the Euclidean distance between them? If so, why?

This is from a ML book:

"Principal component analysis, which we discussed in section 6.3, works when the data lies in a linear subspace. However, this may not hold in many applications. Take, for example, face recognition where a face is represented as a two-dimensional, say 100 × 100 image. In this case, each face is a point in 10, 000 dimensions. Now let us say that we take a series of pictures as a person slowly rotates his or her head from right to left. The sequence of face images we capture follows a trajectory in the 10, 000-dimensional space, and this is not linear. Now consider the faces of many people. The trajectories of all their faces as they rotate their faces define a manifold in the 10, 000-dimensional space, and this is what we want to model. The similarity between two faces cannot simply be written in terms of the sum of the pixel differences, and hence Euclidean distance is not a good metric. It may even be the case that images of two different people with the same pose have smaller Euclidean distance between them than the images of two different poses of the same person. This is not what we want. What should count is the distance along the manifold, which is called the geodesic distance."

Based on this explanation, I have the following question.

Is geodesic distance between two similar photos less than the Euclidean distance between them? If so, why?

Imagine your images are embedded in a space that is a circle. Now, to simplify, we put an extra condition in which each embedding (i.e., an image represented somehow in 2 dimensions) must lie on the circumference of the circle. Then you want to measure the distance between two embeddings, i.e., two distinct points on the space: say $$u$$ and $$v$$, as in the picture below.
According to our assumptions, we can say that the Euclidean distance is always smaller than the Geodesic, for whatever pair of points $$(u,v)\in C$$ we pick: this may also include dissimilar images, as you mentioned.