Imagine that we have a regular plane defined in three-dimensional Euclidean space (“world space”).
On this plane, we trace an arc from point A to point B, like a bullet trajectory, in the plane’s space (“local space”). That is to say, if the plane was transformed in “world space” using for example rotation, the arc would still remain relative to the plane as it is defined in the plane’s space.
We now apply an arbitrary deformation to the plane. It follows that the arc would likewise be deformed in world space.
Which branch of geometry (or mathematics in general) would be used to approach calculating the path of this arc in world space; and for problems like these in general; and how would this problem be approached in software?
Update:
This falls within the domain of Differential geometry.
I believe you are looking for affine transformations, though if your arbitrary deformations to the plane are “strong enough” you might be working with Projective geometry instead.