> Our curves consist of one quadratic curve, with control points [...] per interpolated point p_i
So what happens is each segment between two knots (in the paper) is made of 2 pieces of parametric quadratic polynomials (i.e. parabolas).
Each knot is at the vertex of the same parabola to each side, which can be broken at the vertex into two parts, each of which belongs to one of the adjacent segments.
The solver in the paper makes sure that at the joint between two parabolas, the absolute magnitude of curvature matches.
This means there aren’t really any smooth inflection points and you always get a discrete jump in curvature at an inflection. This looks bad, so in practice Adobe implemented something different in their software.
So the name biquadratic is analogous to biarc. Although the usage here is slightly different to a biarc because it's the same parabola each side of the vertex.
The choice of the name biquadratic is possible unfortunate because it is already taken for an existing class of algebraic space curves [0]
>Biquadratics are algebraic curves of degree four that are the intersection between two quadrics that do not share a common line (in which case the curve would be of degree three).
then
>See also Cartesian ovals, Booth curves (including the lemniscate of Bernoulli), the Külp quartic, the Dürer shell curve, the bicorn, that are planar projections of biquadratics.
which was the source of my confusion as I had assumed that raphlinus's reference was to planar projections of these classical quartic curves.
So what happens is each segment between two knots (in the paper) is made of 2 pieces of parametric quadratic polynomials (i.e. parabolas).
Each knot is at the vertex of the same parabola to each side, which can be broken at the vertex into two parts, each of which belongs to one of the adjacent segments.
The solver in the paper makes sure that at the joint between two parabolas, the absolute magnitude of curvature matches.
This means there aren’t really any smooth inflection points and you always get a discrete jump in curvature at an inflection. This looks bad, so in practice Adobe implemented something different in their software.