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September 24th, 2012, 10:37 PM   #1
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Homogeneous Eqn of Line given 2 homogeneous points

I'm reviewing Projective Geometry. This is an exercise in 2D homogeneous points and lines.

Given two points p1 (X1,Y1,W1) and p2 (X2,Y2,W2), find u = (a, b, c) of the line (aX + bY + cW = 0). (See http://vision.stanford.edu/~birch/projective/node4.html, "Similarly, given two points p1 and p2, the equation..." and http://vision.stanford.edu/~birch/proje ... ode16.html, "Recall from the Section 2 that the coordinates...".)

The solution by means of linear algebra is u = p1 x p2 (cross product) = (Y1W2-Y2W1, W1X2-X1W2, X1Y2-Y1X2). I have worked out how to obtain that by calculating a determinant. However, I should be able to get the same result by using elementary algebra, but somewhere I take a wrong turn. Can someone provide at least the starting point?
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