Curve families - Linebender

submited by

Style Pass

2024-11-18 12:30:01

This graph has an arrow when one curve family fully includes the parameter space of another. A dotted line represents an approximation which is better than can be expected from scaling based on the number of parameters.

We use the same parameter counting policy as Raph's thesis (section 2.8). The endpoints are considered to be fixed, so to use in an actual application, there is an additional transform consisting of translation, rotation, and scaling. This additional transformation is the same for all curves.

The number of parameters is a tradeoff. To represent an arbitrary curve, with a small number of parameters more subdivisions are needed. Increasing the number of parameters has a cost, for representing the parameter values, presenting them in a UI for humans to manipulate, and for doing computations. Generally, higher degrees of continuity requires more parameters.

In general, the accuracy of curve fitting scales with an exponent equal to the number of parameters plus 2. For example, fitting with a quadratic Bézier or Euler spiral has O(n^4) scaling, meaning that subdivision in half improves accuracy by a factor of 16. That said, actually achieving optimal curve fitting may be challenging; I'm