How do I go about defining this area without using a piecewise function?

I think it has something to do with Bilinear Surfaces but I'm not sure how to get started.

Graph showing an area composed of a square region adjoining a triangular region

  • $\begingroup$ What kind of properties are you after for that surface? $\endgroup$
    – joojaa
    Sep 13, 2015 at 12:35

1 Answer 1


One solution is bilinear mapping, which works for every convex quadrilateral.

Let $A, B, C, D$ be the vertices of a quadrilateral $Q$ in the plane, in this order.

Then $f(u,v)=(1-u)(1-v)A + u(1-v)B + uvC + (1-u)vD$ maps the square $[0,1] \times [0,1]$ to $Q$.

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  • 3
    $\begingroup$ What, no MathJax ? :-( $\endgroup$
    – lhf
    Sep 13, 2015 at 14:24
  • $\begingroup$ Yep, looks like it's not working yet :(. Would you mind just using plain code formatting for time being, so all those symbols won't distract from the formulas? $\endgroup$
    – glampert
    Sep 13, 2015 at 18:09
  • $\begingroup$ I've added this question to the list of examples on meta to add to the case for adding MathJax to our site. $\endgroup$ Sep 14, 2015 at 13:37
  • $\begingroup$ "which works for every quadrilateral." Unless I've messed up my "back of the envelope sketch", I think you need to amend that to "every convex quadrilateral". $\endgroup$
    – Simon F
    Sep 14, 2015 at 13:45
  • $\begingroup$ @ratchet, MathJax, yeah! $\endgroup$
    – lhf
    Nov 3, 2015 at 15:13

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