# Two-dimensional bounded area defined parametrically

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.

• What kind of properties are you after for that surface? Sep 13 '15 at 12:35

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 $Q$.

• What, no MathJax ? :-(
– lhf
Sep 13 '15 at 14:24
• 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? Sep 13 '15 at 18:09
• I've added this question to the list of examples on meta to add to the case for adding MathJax to our site. Sep 14 '15 at 13:37
• "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". Sep 14 '15 at 13:45
• @ratchet, MathJax, yeah!
– lhf
Nov 3 '15 at 15:13