Snijcurves

Snijcurves are curves that arise as the intersection of two surfaces in three-dimensional space. If two surfaces are defined by implicit equations F1(x, y, z) = 0 and F2(x, y, z) = 0, their intersection consists of all points that satisfy both equations. Generically this set is a one-dimensional object—a curve. Degenerate cases can occur when the surfaces do not intersect, touch at a point, or coincide along a common portion, in which case the intersection may be empty, a single point, or a surface rather than a curve.

A standard example is the intersection of a plane z = 0 with a sphere x^2 + y^2 + z^2 = R^2, which yields a circle x^2 + y^2 = R^2 in the plane z = 0. More generally, if surfaces are given implicitly by F1 and F2, the intersection curve is the set of points where F1 = 0 and F2 = 0. When the surfaces are given parametrically as r1(u, v) and r2(s, t), the intersection is found where r1(u, v) = r2(s, t) for some parameters; solving this system yields the curve’s points and, by extension, its parameterization.

Tangent directions along a snijcurve are aligned with the cross product of the surface normals: t ∝ ∇F1 × ∇F2 at each point of the curve. In practice, snijcurves are computed numerically or symbolically using elimination, substitutions, or root-finding methods. They are central in applications such as computer-aided design and computer graphics for representing the boundaries created by the boolean intersection of surfaces, trimming operations, and the generation of accurate renderings of intersecting solids.