Scale and translate path from one coordinate system to another

Question

I have a path output as shown in the image below, in a coordinate system 1 wherein the start point and the end point are (40,40) and (10,20) respectively.

I want to scale this path to a new coordinate system (coordinate system 2) with a known start and end point, the path has to scale and adjust between the new points.

I believe Affine transforms might help / linear algebra. How do I achieve this? and will this be accurate or will it distort?

Answer 1

To find appropriate affine tranformation (there are many ways to transform two points into two another ones, but we choose the simplest way), you can apply these elementary steps:

Shift coordinates by (-startx, -starty) 
Scale along X-axis with coefficient (newendx-newstartx)/(endx-startx)  (here -80/3)
Scale along Y-axis with coefficient (newendy-newstarty)/(endy-starty)   (here -35)
Shift coordinates by (newstartx, newstarty) 

Resulting affine tranformation is product of these four matrices