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copied!<p>First of all, unless in your plane equation, d=0, there is no <a href="http://en.wikipedia.org/wiki/Linear_transformation" rel="noreferrer">linear transformation</a> you can apply. You need to instead perform an <a href="http://en.wikipedia.org/wiki/Affine_transformation" rel="noreferrer">affine transformation</a>.</p> <p>One way to do this is to determine an angle and vector about which to rotate to make your pointset lie in a plane parallel to the XY plane (ie. the Z component of your transformed pointset to all have the same values). Then you simply drop the Z component.</p> <p>For this, let <B>V</B> be the normalized plane normal for the plane containing your points. For convenience define from your plane equation above Ax+By+Cz+d=0:</p> <pre><B>V</b> = (A, B, C) <B>V'</b> = <B>V</b> / ||<B>V</b>|| = (A', B', C') <B>Z</b> = (0, 0, 1)</pre> <p>where</p> <pre>A' = A / ||<B>V</B>|| B' = B / ||<B>V</B>|| C' = C / ||<B>V</B>|| ||<B>V</B>|| = (A<sup>2</sup>+B<sup>2</sup>+C<sup>2</sup>)<sup>1/2</sup></pre> <p>The angle will simply be:</p> <pre>θ = cos<sup>-1</sup>(<B>Z</B>∙<B>V</B> / ||<B>V</B>||) = cos<sup>-1</sup>(<B>Z</B>∙<B>V'</B>) = cos<sup>-1</sup>(C')</pre> <p>The axis <b>R</b> about which to rotate is just the cross product of the normalized plane normal <b>V'</b> and <B>Z</B>. That is</p> <pre><b>R</b> = <b>V'</b>×<b>Z</b> = (B', -A', 0)</pre> <p>You can now use this angle / axis pair to build the <a href="http://en.wikipedia.org/wiki/Quaternion_rotation" rel="noreferrer">quaternion rotation</a> needed to rotate all of the points in your dataset to a plane parallel to the XY plane. Then, a I said earlier, just drop the Z component to perform an orthogonal projection onto the XY plane.</p> <p><i>Update: </i> antonakos makes a good point about normalizing the <b>R</b> before using an API taking axis / angle pairs.</p>

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