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matlablinear-algebrahomogenous-transformation

Forcing homogeneous output in matlab

As part of an assignment related to graphics, I have to solve the simple equation Ax=b. In this equation, A is a known 2x3 matrix, b is a known 2x1 vector, and x is the unknown 3x1 vector, which has to be homogeneous.

Now, I know of the standard MATLAB solution x = A\b; However, this does not force x to be homogeneous. Is there a way I can force the third element of x to be 1?

Solution

  • To force the third element of x to be 1, add an equation that says "x3=1". That is, add row [0 0 1] to the matrix and corresponding entry 1 to the vector b. Like this: 

    x = [A; 0 0 1] \ [b; 1]

    Example

    A = [1 2 4; 3 4 5];
b = [6; 7];

    Simply entering A\b returns [-0.2857; 0; 1.5714].

    But [A; 0 0 1] \ [b; 1] returns [-2; 2; 1].