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?

  • $\begingroup$ What is a "nonlinear transformation matrix"? $\endgroup$ – Travis Willse Feb 18 '15 at 0:15
  • $\begingroup$ such as eta = sqrt(x)+y, xi = x^2+y^2.....Sorry, that can not be written as a matrix, I described it incorrectly. $\endgroup$ – winterfly Feb 18 '15 at 0:21
  • $\begingroup$ Most transformations do not map straight lines into straight lines, and the second part of your question depends heavily on which of the numerous answers you take for the first part. As Robert points out, you might be looking for Mobius transformations, which map straight lines and circles to straight lines and circles (but not always straight lines to straight lines). $\endgroup$ – Travis Willse Feb 18 '15 at 0:26

I suspect you're looking for Möbius transformations

  • $\begingroup$ Here's a nice video illustrating them. $\endgroup$ – Blue Feb 18 '15 at 0:22
  • $\begingroup$ That is different. Mobius transformation has taken infty into account (tranform the infinity point to a finite point). But what I hope is to always tranform a finite point to a finite point. And the space we consider is a compact set in R^2 or R^3. $\endgroup$ – winterfly Feb 18 '15 at 0:25
  • $\begingroup$ @winterfly: You should include such details in your question. $\endgroup$ – Blue Feb 18 '15 at 0:30
  • $\begingroup$ Please see my updated question. $\endgroup$ – winterfly Feb 18 '15 at 0:43
  • $\begingroup$ If you transform a line to a circle, you're taking $\infty$ to a finite point (or at least points tending to $\infty$ along the line to points approaching a finite point on the circle). If you avoid infinity by restricting the domain of the transformation to a compact set, there's no objection to a Möbius transformation where the inverse image of $\infty$ is not in this set. $\endgroup$ – Robert Israel Feb 18 '15 at 1:48

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