San José State University |
---|
applet-magic.com Thayer Watkins Silicon Valley & Tornado Alley USA |
---|
Subsphere in Terms of the Arc Distance in the n-Dimensional Sphere |
An n-dimensional sphere (hereafter an n-sphere) centered at the origin of a coordinate system is the set of points (x_{1}, x_{2}, …, x_{n-1}, x_{n}) such that
where R is a constant called the radius.
Consider the point (0, 0, …, 0, R). As x_{n} decreases from R a set of (n-1)-spheres are generated. The radius r of the (n-1)-sphere corresponding to x_{n} is given by
The set of points on the n-sphere such that x_{i}=0 for all i from 1 to (n-2) is a 2-sphere (circle) of radius R. The set of points from (0, 0, …, 0, R) to (0, 0, …, r, (R²-r²)^{½}) is the arc of a circle of radius R. Let s be the length of this arc and θ be the angle subtended by the arc. Then
The (n-1)-sphere on the n-sphere has a radius of r where
and the (n-1)-sphere is centered at (0, 0, …, 0, Rcos(θ)).
The volume of the (n-1)-sphere is a function of its radius r, but r=Rsin(θ) so its volume is a function of Rsin(s/R) where s is the arc distance from (0, 0, …, 0, R).
For example for n=2 the 1-sphere is a circle on the sphere, as shown below, and its volume is its circumference of 2πr. In terms of the arc distance s this circumference is 2πRsin(s/R). For s small compared to R this reduces to 2πs.