Intersection curve

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.

linearly independent, i.e. the planes are not parallel.

The direction of the line one gets from the crossproduct of the normal vectors:

and the direction of the intersection line is

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.

[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.

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.

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

of the contour line of an implicit surface with equation

and parallel projection with direction

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,

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.

The display of a parametrically defined surface is usually done by mapping a rectangular net into 3-space.

[5] See picture of intersecting tori.