When Looking at a Parabola in Perspective
Projecting a 3D object onto a 2D plane often involves transforming geometric equations into a perspective view. Here’s a fun observation: when you view a parabolic curve drawn on the xz-plane from the xy-plane through a perspective projection, the curve can appear as an ellipse. The animation below shows the transition between a parabolic curve and an ellipse depending on the change in projection. Let’s see how that happens. This post shows how a parabola equation is represented and transformed in perspective projection, with a particular focus on the derivation of this transformation.
Problem definition
Let us imagine a parabolic curve drawn on a plane parallel to the xz-plane, defined by
\[\begin{align} z &= a x^2 + b x + c,\\\nonumber y &= d, \end{align}\]where \(a, b, c\), and \(d\) are constants, and \(x, y, z\) represent the Cartesian coordinates. When a perspective transformation is applied, the 3D coordinates are projected onto the 2D plane using the following equations.
\[\begin{align} u &= x/z, \\ v &= y/z, \nonumber \end{align}\]where \(u\) and \(v\) are the projected coordinates on the 2D plane. For simplicity, the focal length and principal point are assumed to be \(1\) and \(0\), respectively.
Problem solving
By substituting the parabola equation into perspective projection formula, we can rewrite the equation as:
\[\begin{align} u &= \frac{x}{a x^2 + b x + c}, \\ v &= \frac{d}{a x^2 + b x + c}. \nonumber \end{align}\]To eliminate the variable \(x\), we rearrange the expressions to obtain:
\[a u^2 + \frac{b}{d} u v + \frac{c}{d^2} v^2 = \frac{a x^2 + b x + c}{(a x^2 + b x + c)^2} = \frac{1}{a x^2 + b x + c} = \frac{1}{d} v\]Finally, we obtain:
\[a u^2 + \frac{b}{d} u + \frac{c}{d^2} v^2 - \frac{1}{d} v = 0\]This is the perspective-transformed equation relating \(u\) and \(v\). Although our original quadratic equation was a parabola in the xz‑plane, the perspective projection creates a conic equation, including circle, ellipse, hyperbola, that passes the origin.
Special case
When \(a = 1, b = 0, c = 1\) and \(d = 1\), the parabolic equation is:
\[z = x^2 + 1\]at \(y=1\) plane. When viewing this parabola through a perspective projection, it appears as a circle:
\[\begin{align} u^2 + v^2 - v &= 0, \\\nonumber u^2 + (v-\frac{1}{2})^2 &= \frac{1}{4} \end{align}\]The animation at the top describes the transition between the above parabola and circle.