Sphere (EntityTopic, 15)
From Hi.gher. Space
Geometry
A sphere refers to the surface of a perfectly symmetrical realmic object.
Equations
- Assumption: Sphere is centered at the origin.
- Variables:
r ⇒ radius of sphere
- All points (x, y, z) that lie on the surface of a sphere will satisfy the following equation:
x^{2} + y^{2} + z^{2} = r^{2}
- The hypervolumes of a sphere are given by:
total edge length = 0
surface area = 4πr^{2}
volume = 4πr^{3}3^{-1}
- The planar cross-sections (n) of a sphere are:
[!x,!y,!z] ⇒ circle of radius (rcos(πn/2))
Mapping
When the surface of a sphere is mapped onto a rectangle {(-1,-1),(1,-1),(1,1),(-1,1)}, the surface will be horizontally stretched such that the further away from the equator of the sphere a point is, the more it is stretched. The top and bottom edges of the rectangle will converge into a single point. Leaving the top edge of the rectangle will continue entering from the top edge at the position (x±1,1). Similarly, leaving the bottom edge of the rectangle will continue entering from the bottom edge at the position (x±1,-1).
Notable Trishapes | |
Regular: | tetrahedron • cube • octahedron • dodecahedron • icosahedron |
Direct truncates: | tetrahedral truncate • cubic truncate • octahedral truncate • dodecahedral truncate • icosahedral truncate |
Mesotruncates: | stauromesohedron • stauroperihedron • stauropantohedron • rhodomesohedron • rhodoperihedron • rhodopantohedron |
Snubs: | snub staurohedron • snub rhodohedron |
Curved: | sphere • torus • cylinder • cone • frustum • crind |