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I'm reviewing Projective Geometry. This is an exercise in 2D homogeneous points and lines. It is not a homework assignment - I'm way too old for that.

Given two points p1 (X1,Y1,W1) and p2 (X2,Y2,W2) find the equation of the line that passes through them (aX+bY+cW=0). (See http://vision.stanford.edu/~birch/projective/node4.html [Broken], "Similarly, given two points p1 and p2, the equation..." and http://vision.stanford.edu/~birch/projective/node16.html [Broken], Representing the Plucker Equations)

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 and the basic line equation aX + bY + cW = 0, but somewhere I take a wrong turn. Can someone provide the steps?

Given two points p1 (X1,Y1,W1) and p2 (X2,Y2,W2) find the equation of the line that passes through them (aX+bY+cW=0). (See http://vision.stanford.edu/~birch/projective/node4.html [Broken], "Similarly, given two points p1 and p2, the equation..." and http://vision.stanford.edu/~birch/projective/node16.html [Broken], Representing the Plucker Equations)

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 and the basic line equation aX + bY + cW = 0, but somewhere I take a wrong turn. Can someone provide the steps?

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