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I have multiple points in a circle and I need to project them onto a finite plane that is angled in respect to the circle. This is actually a light cone, that originates some millimetres behind the circle and is projecting these point on the plane (we get the intersection of a light cone with the plane so we get points in the ellipse). Can somebody help me with the proper mathematical equations, because I couldn't find any for my specific problem?

Thank you.

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A way to compute this projection is to parameterize the circle, and then compute the intersection of a ray from the center of the projection (the view point) through a general point on the circle with the plane. This is a matter of solving a system of linear equations.

For a direct computation, you can use the following formula for the projection matrix: Working in homogeneous coordinates, if $\mathbf V$ is the view point (center of projection) and $\mathbf P$ the projection plane, then the matrix $$\mathtt M = \mathbf V\mathbf P^T-(\mathbf V^T\mathbf P)\mathtt I_4$$ computes the associated projection. Note that the first term is the outer (a.k.a. tensor) product of the two vectors, while the second term is identity matrix times the dot product of the two vectors. You can apply this matrix directly to a parameterization of the circle to get a parameterization of its projection.

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