Spherical cone
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In geometry, a spherical cone is the figure in the 4-dimensional Euclidean space represented by the equation
- x2 + y2 + z2 − w2 = 0.
It is a quadric surface, and is one of the possible 3-manifolds which are 4-dimensional equivalents of the conical surface in 3 dimensions. It is so named because its intersections with hyperplanes perpendicular to the w-axis are spheres.
[edit] Geometrical interpretation
The spherical cone consists of two unbounded nappes, which meet at the origin and are the analogues of the nappes of the 3-dimensional conical surface. The upper nappe corresponds with the half with positive w-coordinates, and the lower nappe corresponds with the half with negative w-coordinates.
If it is restricted between the hyperplanes w = 0 and w = r for some non-zero r, then it may be closed by a 3-ball of radius r, centered at (0,0,0,r), so that it bounds a finite 4-dimensional volume. This volume is given by the formula πr4 / 3, and is the 4-dimensional equivalent of the solid cone. The ball may be thought of as the 'lid' at the base of the 4-dimensional cone's nappe, and the origin becomes its 'apex'.
This shape may be projected into 3-dimensional space in various ways. If projected onto the XYZ hyperplane, its image is a ball. If projected onto the XYW, XZW, or YZW hyperplanes, its image is a solid cone. If projected onto an oblique hyperplane, its image is either an ellipsoid or a solid cone with an ellipsoidal base (resembling an ice cream cone). These images are the analogues of the possible images of the solid cone projected to 2 dimensions.
[edit] Temporal interpretation
If the w-coordinate of the equation of the spherical cone is interpreted as time, then it is the shape of the light cone in special relativity. In this case, the equation is usually written as:
- x2 + y2 + z2 − t2 = 0.
The upper nappe is then the future light cone and the lower nappe is the past light cone.
