Equations in R3
In analytic geometry, a sphere with centre (x0, y0, z0) and radius r is the locus of all points (x, y, z) such that
The points on the sphere with radius r can be parameterized via
(see also trigonometric functions and spherical coordinates).
A sphere of any radius centred at zero is an integral surface of the following differential form:
This equation reflects the fact that the position and velocity vectors of a point traveling on the sphere are always orthogonal to each other.
The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. For this reason, the sphere appears in nature: for instance bubbles and small water drops are roughly spherical, because the surface tension locally minimizes surface area. The surface area in relation to the mass of a sphere is called the specific surface area. From the above stated equations it can be expressed as follows:
A sphere can also be defined as the surface formed by rotating a circle about any diameter. If the circle is replaced by an ellipse, and rotated about the major axis, the shape becomes a prolate spheroid, rotated about the minor axis, an oblate spheroid.
Read more about this topic: Sphere