# 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∈x}}D{x}[m]W(m,v)} | {(1)} |

{W(m,v)} | {={{∏}a∈m[a],v[a])} | {(2)} |

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

{{{∫}x̂(v)dv} | {(3a)} |

... or defining inner products of variable values:

{x̂⋅ŷ} | {={{∫}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`}.