I've defined a sort of 'warp' procedure for 2d shapes, and I'm curious whether it's familiar within the math canon or even has a name. Given its simplicity I'm sure that there's a formal definition somewhere - but I'm unsure of where to even start my exploration of this area.
An initial shape is defined with points inside of a unit square of side length 1. A new 'warped' set of four points is defined as the basis for the transformation.
The coordinates of the shape's new, post-transform points are represented as intersection points between the subdividing lines that connect the opposing faces of the square. Where these bisecting lines appear is defined by the initial x and y values of the initial shape's coordinates.
Here's an example: the point at (0.2,0.5) is represented as the intersection of a horizontal line at y=0.5 and the vertical line 0.2. Alternatively, this is the intersection of line A, which passes through the point 20% of the way from (0,0) to (0,1) and the point 20% of the way from (1,0) to (1,1), with line B, which passes through the point 50% of the way from (0,0) to (1,0) and the point 50% of the way from (0,1) to (1,1). By changing these 'basis' points, the respective bisecting lines change as does their intersection. By performing this 'intersection' operation on each coordinate in the domain, a nonlinear transformation of 2D space is performed.
To play with this transformation idea, I made this desmos graph, where the 'basis' points can be dragged around. From this experiment it appears that the operation is similar, in convex cases, to planar homography/ perspective projection. The concept also seems to bear similarity to barycentric coordinate systems.
