Four ways a fisheye maps angle to radius
A rectilinear lens puts a ray at r = f·tan θ, and past about 120° that number runs away. The four classical fisheye designs each choose a different function of θ, and this plots them together against a curve fitted to a real lens.
Try this
Push θ towards 90° and watch the rectilinear curve leave the plot while the equidistant one walks up in a straight line. That divergence is why a fisheye needs its own model.
Where this lab comes from
Fisheye Models: When Correcting the Lens Stops WorkingLesson 4 of the Image Formation unit. Past roughly 120 degrees the pinhole model is not inaccurate, it is the wrong shape, because tan θ runs to infinity while the sensor does not. Fitted to the same 15 photographs of one fisheye lens, pinhole plus Brown-Conrady reaches 7.326 px RMS and Kannala-Brandt reaches 0.644 px. At 80 degrees the first model predicts a radius of -35,572 px, a sign flip; the second predicts 458.8 px, which is on the sensor.
More in cameras and geometry
- Focus, aperture and depth of field — Move the subject and watch the focus plane and the blur follow.
- The scene, and what the camera sees — Orbit a 3-D scene on the left; the pinhole view updates on the right.
- Straight lines that bow — Brown–Conrady radial and tangential terms on a pixel grid.