clamp a 2D coordinate to fit within an ellipse

Question

I need to clamp a 2D coordinate to fit within an ellipse.

Call of Duty: Modern Warfare 2 does something similar where capture points are translated from a 3D vector in the world to a 2D screen coordinate and then the 2D coordinates are clamped within an ellipse.

When the capture points are in view they're within the bounds of the ellipse.

When they're behind you they are clamped to be within the bounds of the ellipse.

Given a 2D coordinate that could be off screen, etc, what is the math behind clamping it within an ellipse?

Answer 1

Here's a general method to clamp a point \$A\$ to the ellipse of center \$C\$ and half-dimensions \$d\$ (i.e. the ellipse is \$2d_x\$ wide and \$2d_y\$ high):

• \$B = \frac{(A - C)}{d}\$ gives you the coordinates of \$A\$ in the frame in which the ellipse is the unit circle. The fraction is memberwise division (\$ \frac{a}{b} = (\frac{a_x}{b_x} ; \frac{a_y}{b_y})\$)

• \$B' = \frac{B}{|B|}\$ normalizes \$B\$: \$B'\$ is now on the unit circle.

• \$B'' = C + B' × d\$, where \$×\$ is the memberwise product. This brings our clamped point back into the original frame : \$B''\$ is \$A\$, clamped onto the ellipse.

This method can be greatly simplified if the ellipse is centered (\$C\$ disappears), and/or if your frame has the same aspect ratio as your ellipse (\$d_x = d_y\$, so the scaling isn't necessary).

Answer 2

The equation for an ellipse centered at the origin is just:

$$\bigg( \frac{x ^ 2}{Rx ^ 2}\bigg) + \bigg(\frac{y ^ 2}{Ry ^ 2} \bigg) = 1$$

where:

• Rx is the X-dimension radius; and

• Ry is the y-dimension radius

Combining this with a transformation that maps the relevant ellipse to the origin is straightforward.

Then just replace the = sign with a <= to get a test for (inclusive) containment in the ellipse.