# 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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