Is there any nonlinear tranformation that can transfrom a line segment to a part of a circle or any other shapes? If yes, how is the plane/space transformed and does the underlying function after tranformation have similar property to the original one?
Suppose that we have a square as the domain and there is a straight line dividing the square into halves. The function over this domain is xy at the left half the square and x^2+y^2 at the right half. What if I want to tranform the straight line into a quadratic or circular curve, and then what is the function after transformation?