Integrals of variable values

This is an open problem about “variable values” in variable fonts.

Consider the variable value model we discussed in previous posts:


We could consider the integral of a variable value over the design space:


... or defining inner products of variable values:


Computing such integrals’ exact value might be non-trivial, though, acquiring an numeric result is not hard.

Being able to define inner products is helpful for certain font processing processes, for example, cubic-to-quadratic conversion. In this process, we are converting an arc A(t) to its approximation (t). We could measure the overall conversion error as:


Where normals |x| could be defined as {xx}.