how to make 2d shape deformations?

I'm trying to make an animation transition between 2 shapes, example: transform a square into a circle... or transform a triangle into 2 squares

pretty much like this: http://i.stack.imgur.com/hSxul.jpg (no rep for posting images)

my first idea is to interpolate 2 planes and give them 2 different values of Z, and then interpolate the shape of one plane into the other... but how?

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Parametrize both curves by normalized arc length. Now use the homotopy $H(s,t)=(1-t)A(s)+tB(s)$.