Hi,
This seems inherently a simple task, but I'm having real problems with it!

I have a plane, let's call it P, with its direction defined by its normal vector n, and the camera distance, d, both of which I know:

P: \textbf{n}.\textbf{x} = d

Upon P, lie a set of coordinates:
<br /> C = \left(\begin{array}{ccc}<br /> x_1 & & x_n\\<br /> y_1 & ... & y_n\\<br /> z_1 & & z_n\\<br /> \end{array}\right)<br />

I need to rotate these coordinates onto the x-y plane to give a top-down view. I can't move the camera - it's fixed.

My initial attempt was to work out the rotation angles from the normal (x-rotation = phi, y-rotation = theta and z-rotation = psi):

<br /> \phi = cos^{-1}( n_z )<br /> \psi = sin^{-1}( \frac{-n_x}{sin(\phi}})<br />

I could then do:

<br /> C' = R_x(\phi) . R_z(\psi) . \left(\begin{array}{c}x_i\\y_i\\z_i\end{array}\rig ht)<br />

(where (xi,yi,zi) is each coordinate in C).

For some reason, this worked sometimes, but other times the angle was completely wrong!

So I tried something else. When the P is rotated to align with the x-y plane we have:

<br /> R .<br /> \left(\begin{array}{c}<br /> X\\<br /> Y\\<br /> Z<br /> \end{array}\right)<br /> = <br /> \left(\begin{array}{c}<br /> X'\\<br /> Y'\\<br /> 0<br /> \end{array}\right)<br />

And we know the combined x-y-z rotation matrix:


For ease of reading, hereafter we have:
<br /> r_3 = [ a, b, c ], where<br /> \begin{array}{l}<br /> a = \sin(\theta)\sin(\phi) \\<br /> b = -\sin( \theta )\cos( \phi ) \\<br /> c = \cos( \theta )<br /> \end{array}<br />

Since we don't know what X' and Y' will be, we only use the bottom row of the rotation
matrix. For three coordinates on the original plane, we then have 3 equations with which
to find phi and theta:

<br /> \begin{array}{l}<br /> a.x_1 + b.y_1 + c.z_1 = 0 \\<br /> a.x_2 + b.y_2 + c.z_2 = 0 \\<br /> a.x_3 + b.y_3 + c.z_3 = 0 \\<br /> \end{array}<br />

Which gives rise to the linear system
<br /> A = \left(<br /> \begin{array}{ccc}<br /> x_1 & y_1 & z_1 \\<br /> x_2 & y_2 & z_2 \\<br /> x_3 & y_3 & z_3 \\<br /> \end{array}<br /> \right)<br />
<br /> x = <br /> \left(<br /> \begin{array}{c}<br /> a \\<br /> b \\<br /> c \\<br /> \end{array}<br /> \right)<br />
<br /> Ax = \textbf{0}<br />

Which we solve using Singular Value Decomposition to get a,b and c (take the last column of the "V" matrix). We can then work out phi and theta:

<br /> \begin{array}{l}<br /> \theta = cos^{-1}( c )\\<br /> \phi = cos^{-1}( \frac{b}{sin(\theta)} )<br /> \end{array}<br />

These values were then subbed into the compound rotation matrix given above, and psi was set to 0 giving R. Finally I multiplied each coordinate by R:
<br /> C' = R(\phi, \theta, \psi) \,. \, C<br />

This again, didn't work. Although I'm convinced that in theory it should. Is anyone able to spot any errors I might have made? Or can anyone contribute a different idea? This has taken me the last week of work so any advice is most gratefully received!!!
Thanks!