In Perspective

Geometry Level 4

In perspectival drawings, a 3D situation is projected onto the 2D plane. This involves a projection function \((x,y,z) \mapsto (\xi, \eta)\). A simple projection function is of the form \[\xi = \frac x z;\ \ \ \ \eta = \frac y z.\] Now consider in 3D space a circle in a horizontal plane, centered at \((0, b, c)\). Its points satisfy the equations \[x^2 + (z - c)^2 = r^2;\ \ \ y = b.\] When projected according to the formula above, this circle becomes a curve, which looks like an ellipse. It is obviously symmetric around the \(\xi\)-axis.

Is this curve actually an ellipse? I.e. does it satisfy \[\left(\frac \xi \rho\right)^2 + \left(\frac{\eta-a} \sigma\right)^2 = 1\] for some constants \(\rho, \sigma, a > 0\)?

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