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Intersection in two points. Methods for distinguishing these cases, and determining the coordinates for the points in the latter cases, are useful in a number of circumstances. For example, it is a common calculation to perform during ray tracing .
The intersection points are: (−0.8587, 0.7374, −0.6332), (0.8587, 0.7374, 0.6332). A line–sphere intersection is a simple special case. Like the case of a line and a plane, the intersection of a curve and a surface in general position consists of discrete points, but a curve may be partly or totally contained in a surface.
Subtracting the two equations given above gives + (+) =. Since is a quadratic function of , the projection of the intersection onto the xz-plane is the section of an orthogonal parabola; it is only a section due to the fact that < <.
The intersection of two planes. 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 ...
This is more than an analogy; spherical and plane geometry and others can all be unified under the umbrella of geometry built from distance measurement, where "lines" are defined to mean shortest paths (geodesics). Many statements about the geometry of points and such "lines" are equally true in all those geometries provided lines are defined ...
The set of all spheres satisfying this equation is called a pencil of spheres determined by the original two spheres. In this definition a sphere is allowed to be a plane (infinite radius, center at infinity) and if both the original spheres are planes then all the spheres of the pencil are planes, otherwise there is only one plane (the radical ...
The disk bounded by a great circle is called a great disk: it is the intersection of a ball and a plane passing through its center. In higher dimensions, the great circles on the n-sphere are the intersection of the n-sphere with 2-planes that pass through the origin in the Euclidean space R n + 1.
This great circle is defined by the intersection of a diametral plane with the surface. Draw the normal to that plane at the centre: it intersects the surface at two points and the point that is on the same side of the plane as A is (conventionally) termed the pole of A and it is denoted by A'. The points B' and C' are defined similarly.