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    <p>Let me say this up front: this problem is hard. There is a reason Dan Story's linked question has not been answered. Let provide an explanation for people who want to take a stab at it. I hope I'm wrong about how hard it is, though.</p> <p>I will assume that the 2D screen coordinates and projection/perspective matrix is known to you. You need to know at least this much (if you don't know the projection matrix, essentially you are using a different camera to look at the world). Let's call each pair of 2D screen coordinates <code>(a_i, b_i)</code>, and I will assume the projection matrix is of the form</p> <pre><code>P = [ px 0 0 0 ] [ 0 py 0 0 ] [ 0 0 pz pw] [ 0 0 s 0 ], s = +/-1 </code></pre> <p>Almost any reasonable projection has this form. Working through the rendering pipeline, you find that</p> <pre><code>a_i = px x_i / (s z_i) b_i = py y_i / (s z_i) </code></pre> <p>where <code>(x_i, y_i, z_i)</code> are the original 3D coordinates of the point.</p> <p>Now, let's assume you know your shape in a set of canonical coordinates (whatever you want), so that the vertices is <code>(x0_i, y0_i, z0_i)</code>. We can arrange these as columns of a matrix <code>C</code>. The actual coordinates of the shape are a rigid transformation of these coordinates. Let's similarly organize the actual coordinates as columns of a matrix <code>V</code>. Then these are related by</p> <pre><code>V = R C + v 1^T (*) </code></pre> <p>where <code>1^T</code> is a row vector of ones with the right length, <code>R</code> is an orthogonal rotation matrix of the rigid transformation, and <code>v</code> is the offset vector of the transformation.</p> <p>Now, you have an expression for each column of <code>V</code> from above: the first column is <code>{ s a_1 z_1 / px, s b_1 z_1 / py, z_1 }</code> and so on.</p> <p>You must solve the set of equations <code>(*)</code> for the set of scalars <code>z_i</code>, and the rigid transformation defined <code>R</code> and <code>v</code>.</p> <p><strong>Difficulties</strong></p> <ul> <li>The equation is nonlinear in the unknowns, involving quotients of <code>R</code> and <code>z_i</code></li> <li>We have assumed up to now that you know which 2D coordinates correspond to which vertices of the original shape (if your shape is a square, this is slightly less of a problem).</li> <li>We assume there is even a solution at all; if there are errors in the 2D data, then it's hard to say how well equation <code>(*)</code> will be satisfied; the transformation will be nonrigid or nonlinear.</li> </ul>
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