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make a matrix not modify a coordinate

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Don't worry. You won't have to debug my code. But in case you're curious...
// compute a homography matrix
matrix math::matrix_homography(	float src_x0, float src_y0, float src_x1, float src_y1, float src_x2, float src_y2, float src_x3, float src_y3,
										float dst_x0, float dst_y0, float dst_x1, float dst_y1, float dst_x2, float dst_y2, float dst_x3, float dst_y3)
	float dx1, dy1, dx2, dy2, sx, sy, g, h, a, b, c, d, e, f;

	// quad to square
	dx1 = src_x1-src_x2;
	dy1 = src_y1-src_y2;
	dx2 = src_x3-src_x2;
	dy2 = src_y3-src_y2;
	sx = src_x0-src_x1+src_x2-src_x3;
	sy = src_y0-src_y1+src_y2-src_y3;
	g = (sx*dy2-dx2*sy)/(dx1*dy2-dx2*dy1);
	h = (dx1*sy-sx*dy1)/(dx1*dy2-dx2*dy1);
	a = src_x1-src_x0+g*src_x1;
	b = src_x3-src_x0+h*src_x3;
	c = src_x0;
	d = src_y1-src_y0+g*src_y1;
	e = src_y3-src_y0+h*src_y3;
	f = src_y0;

	matrix quad_to_square_inv;
	quad_to_square_inv.m[0][0] = a;		quad_to_square_inv.m[0][1] = d;		quad_to_square_inv.m[0][2] = 0.0f;	quad_to_square_inv.m[0][3] = g;
	quad_to_square_inv.m[1][0] = b;		quad_to_square_inv.m[1][1] = e;		quad_to_square_inv.m[1][2] = 0.0f;	quad_to_square_inv.m[1][3] = h;
	quad_to_square_inv.m[2][0] = 0.0f;	quad_to_square_inv.m[2][1] = 0.0f;	quad_to_square_inv.m[2][2] = 1.0f;	quad_to_square_inv.m[2][3] = 0.0f;
	quad_to_square_inv.m[3][0] = c;		quad_to_square_inv.m[3][1] = f;		quad_to_square_inv.m[3][2] = 0.0f;	quad_to_square_inv.m[3][3] = 1.0f;
	matrix quad_to_square = matrix_invert(&quad_to_square_inv);

	// square to quad
	dx1 = dst_x1-dst_x2;
	dy1 = dst_y1-dst_y2;
	dx2 = dst_x3-dst_x2;
	dy2 = dst_y3-dst_y2;
	sx = dst_x0-dst_x1+dst_x2-dst_x3;
	sy = dst_y0-dst_y1+dst_y2-dst_y3;
	g = (sx*dy2-dx2*sy)/(dx1*dy2-dx2*dy1);
	h = (dx1*sy-sx*dy1)/(dx1*dy2-dx2*dy1);
	a = dst_x1-dst_x0+g*dst_x1;
	b = dst_x3-dst_x0+h*dst_x3;
	c = dst_x0;
	d = dst_y1-dst_y0+g*dst_y1;
	e = dst_y3-dst_y0+h*dst_y3;
	f = dst_y0;

	matrix square_to_quad;
	square_to_quad.m[0][0] = a;		square_to_quad.m[0][1] = d;		square_to_quad.m[0][2] = 0.0f;	square_to_quad.m[0][3] = g;
	square_to_quad.m[1][0] = b;		square_to_quad.m[1][1] = e;		square_to_quad.m[1][2] = 0.0f;	square_to_quad.m[1][3] = h;
	square_to_quad.m[2][0] = 0.0f;	square_to_quad.m[2][1] = 0.0f;	square_to_quad.m[2][2] = 1.0f;	square_to_quad.m[2][3] = 0.0f;
	square_to_quad.m[3][0] = c;		square_to_quad.m[3][1] = f;		square_to_quad.m[3][2] = 0.0f;	square_to_quad.m[3][3] = 1.0f;

	return quad_to_square*square_to_quad;
...and I'm applying this matrix to the vertices (x, y, z, w) in screen space. The problem is that this matrix modifies the w component of the vertex, which normally isn't a problem because we only really care about x and y (and z, because it is used for perspective projections). BUT OpenGL does care. It uses the w component to compute depth and, therefore, clipping. When I apply a homography matrix to the screen-space vertices, it causes triangles to be unnecessarily clipped (depending on my view angle etc.). Is there any way to take this matrix, and make it leave the w coordinate alone?

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Well, you know how matrix multiplication works. The new "w" component is just the dot product the bottom row of the matrix with the old vector. What does that row need to be to just pick out the old 'w' value? Hint: It'll be mostly zeros, but with a '1' in one place...

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Indeed, I know how matrix multiplication works. But I can't just make that row (0, 0, 0, 1). That would affect the other coordinates, wouldn't it? I'm pretty sure it would.

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I'm not sure what you mean. You mean like expanding it out to see all the dot products?

I have looked at an expansion of matrix multiplication before. I can't seem to figure it out.

I'm tempted to just write a vertex shader that sets the w component back to what it was before the transformation. Is this a good solution?

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Here's all I was saying:

Say you've got

y = A x

where x and y are nx1 matrices (i.e., column vectors), and A is an nxn matrix. Then, expanding this a bit,

[ y1 ] [ ---- Row 1 of A ---- ] [ x1 ] [ dot((Row 1 of A), (x1,x2,...,xn)) ]
[ y2 ] = [ ---- Row 2 of A ---- ] [ x2 ] = [ dot((Row 2 of A), (x1,x2,...,xn)) ]
[ ... ] [ ....... ] [ ... ] [ ....... ]
[ yn ] [ ---- Row n of A ---- ] [ xn ] [ dot((Row n of A), (x1,x2,...,xn)) ]

Now, look at this and pay attention to what depends on what...

In particular, notice that

yi = dot(Row i of A, x)

for each i from 1 to n.

The upshot: changing the i-th row of A changes only the i-th component of y.

Do you understand what to do with your transformation matrix now?

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