geometry - 3d to 2d Projection Matrix

Question

I have 3 points in a 3D space of which I know the exact locations. Suppose they are: (x0,y0,z0), (x1,y1,z1) and (x2,y2,z2).

Also I have a camera that is looking at these 3 points and I know the 2D locations of those three points on camera view plane. So for example (x0,y0,z0) will be (x0',y0'), and (x1,y1,z1) will be (x1',y1') and (x2,y2,z2) will be (x2',y2') from the camera's point of view.

What is the easiest way to find the projection matrix that will project those 3D points into 2D points on camera view plane. We don't know anything about the camera location.

Answer 1

This gives you two sets, each of three equations in 3 variables:

a*x0+b*y0+c*z0 = x0'
a*x1+b*y1+c*z1 = x1'
a*x2+b*y2+c*z2 = x2'

d*x0+e*y0+f*z0 = y0'
d*x1+e*y1+f*z1 = y1'
d*x2+e*y2+f*z2 = y2'

Just use whatever method of solving simultaneous equations is easiest in your situation (it isn't even hard to solve these "by hand"). Then your transformation matrix is just ((a,b,c)(d,e,f)).

...

Actually, that is over-simplified and assumes a camera pointed at the origin of your 3D coordinate system and no perspective.

For perspective, the transformation matrix works more like:

               ( a, b, c, d )   ( xt )
( x, y, z, 1 ) ( e, f, g, h ) = ( yt )
               ( i, j, k, l )   ( zt )

( xv, yv ) = ( xc+s*xt/zt, yc+s*yt/zt ) if md < zt;

but the 4x3 matrix is more constrained than 12 degrees of freedom since we should have

a*a+b*b+c*c = e*e+f*f+g*g = i*i+j*j+k*k = 1
a*a+e*e+i*i = b*b+f*f+j*j = c*c+g*g+k*k = 1

So you should probably have 4 points to get 8 equations to cover the 6 variables for camera position and angle and 1 more for scaling of the 2-D view points since we'll be able to eliminate the "center" coordinates (xc,yc).

So if you have 4 points and transform your 2-D view points to be relative to the center of your display, then you can get 14 simultaneous equations in 13 variables and solve.

Unfortunately, six of the equations are not linear equations. Fortunately, all of the variables in those equations are restricted to the values between -1 and 1 so it is still probably feasible to solve the equations.