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:

{x̂(v)}	{=x̄+{{∑}m∈𝔐{x}}D{x}[m]W(m,v)}	{(1)}
{W(m,v)}	{={{∏}a∈𝔄}Λ(m[a],v[a])}	{(2)}

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

{{{∫}𝔙}x̂(v)dv}

{(3a)}

... or defining inner products of variable values:

{x̂⋅ŷ}

{={{∫}𝔙}x̂(v)ŷ(v)dv}

{(3b)}

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 Ã(t). We could measure the overall conversion error as:

{{{∫}t}|A(t)−Ã(t)|}

{(4)}

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