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In geometry, an intersection curve is a curve that is common to two geometric objects. In the simplest case, the intersection of two non-parallel planes in Euclidean 3-space is a line. In general, an intersection curve consists of the common points of two transversally intersecting surfaces, meaning that at any common point the surface normals are not parallel. This restriction excludes cases where the surfaces are touching or have surface parts in common.
The analytic determination of the intersection curve of two surfaces is easy only in simple cases; for example: a) the intersection of two planes, b) plane section of a quadric (sphere, cylinder, cone, etc.), c) intersection of two quadrics in special cases. For the general case, literature provides algorithms, in order to calculate points of the intersection curve of two surfaces. [1]
Given: two planes linearly independent, i.e. the planes are not parallel.
Wanted: A parametric representation of the intersection line.
The direction of the line one gets from the crossproduct of the normal vectors: .
A point of the intersection line can be determined by intersecting the given planes with the plane , which is perpendicular to and . Inserting the parametric representation of into the equations of und yields the parameters and .
Example:
The normal vectors are and the direction of the intersection line is . For point , one gets from the formula above Hence
is a parametric representation of the line of intersection.
Remarks:
In any case, the intersection curve of a plane and a quadric (sphere, cylinder, cone,...) is a conic section. For details, see. [2] An important application of plane sections of quadrics is contour lines of quadrics. In any case (parallel or central projection), the contour lines of quadrics are conic sections. See below and Umrisskonstruktion.
It is an easy task to determine the intersection points of a line with a quadric (i.e. line-sphere); one only has to solve a quadratic equation. So, any intersection curve of a cone or a cylinder (they are generated by lines) with a quadric consists of intersection points of lines and the quadric (see pictures).
The pictures show the possibilities which occur when intersecting a cylinder and a sphere:
In general, there are no special features to exploit. One possibility to determine a polygon of points of the intersection curve of two surfaces is the marching method (see section References). It consists of two essential parts:
For details of the marching algorithm, see. [3]
The marching method produces for any starting point a polygon on the intersection curve. If the intersection curve consists of two parts, the algorithm has to be performed using a second convenient starting point. The algorithm is rather robust. Usually, singular points are no problem, because the chance to meet exactly a singular point is very small (see picture: intersection of a cylinder and the surface ).
A point of the contour line of an implicit surface with equation and parallel projection with direction has to fulfill the condition , because has to be a tangent vector, which means any contour point is a point of the intersection curve of the two implicit surfaces
For quadrics, is always a linear function. Hence the contour line of a quadric is always a plane section (i.e. a conic section).
The contour line of the surface (see picture) was traced by the marching method.
Remark: The determination of a contour polygon of a parametric surface needs tracing an implicit curve in parameter plane. [4]
The intersection curve of two polyhedrons is a polygon (see intersection of three houses). The display of a parametrically defined surface is usually done by mapping a rectangular net into 3-space. The spatial quadrangles are nearly flat. So, for the intersection of two parametrically defined surfaces, the algorithm for the intersection of two polyhedrons can be used. [5] See picture of intersecting tori.
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