Intersections

We do support all sorts of intersections between curves and surfaces.

Curve-Curve Intersections

Curve Curve intersections result in one of three cases:

• Intersection is a curve. This can only happen with two equal curves.
• Intersection is a finite set of discrete points
• Intersection is a infinite set of discrete points (this can happen between a helix and a line). This is just models as a PointArray, a set of equidistant points.
• No intersection

#![allow(unused)]
fn main() {
pub enum CurveCurveIntersection {
    None,
    FinitePoints(Vec<Point>),
    InfiniteDiscretePoints(PointArray),
    Curve(Curve),
}
}

One very imporant thing to understand is that curve-curve intersections cannot result in something like a short edge and a point. The intersection is EITHER the same curve, OR a set of points. This follows from one of the fundamental properties of curves. The proof looks something along the lines of:

1. Let C1 and C2 be two curves.
2. Let P be an intersection point on C1 and C2.
3. Now do the taylor expansion of C1 and C2 around P.
4. Either the taylor expansions are equal, or they are not.
   1. If they are equal, then C1 and C2 are the same curve.
   2. If they are not equal, then there is an infinitly small neighborhood around P where C1 and C2 are not equal, hence P is a discrete intersection point.

Similar proofs can be made for surfaces etc. You can look it up under Bézout's theorem.

Curve-Surface Intersections

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