Monday, May 4, 2015

Unizor - Geometry3D - Elements - Spheres

Unizor - Creative Minds through Art of Mathematics - Math4Teens

A sphere is a three-dimensional surface that consists of all points in space located on the same distance R (called a radius of this sphere) from some fixed point O (called a center of this sphere).
This definition is a three-dimensional analogue of a two-dimensional circle.

Obviously, this definition is not perfect. Most importantly, we have not defined a concept of a distance in three-dimensional space. We also have to discuss the existence of such points for any real value of a radius R and a center O. These issues are very important and far from being clear. We will discuss them in further lectures. For now we would like to resort to intuitive understanding of these concept.

The surface of our planet is a rough approximation of a sphere and we will use some geography to exemplify the concepts we introduce in this lecture.

If a straight line passes through a center of a sphere, it intersects this sphere at two opposite points. A segment of this line between two opposite points of intersection with a sphere is a diameter.
If our planet is considered as an approximation of a sphere, its axis of rotation is such a line and two poles, North and South, are intersection of this line with a sphere.

Any segment that connects two points on a sphere along a straight line is called a chord (similarly to a two-dimensional case with a chord of a circle).
Diameter is the longest chord in a sphere.

If a plane cuts through a sphere, the intersection is a circle. If this plane passes through a center of a sphere, the intersection will be called a great circle, its radius will be equal to a radius of a sphere itself. All Earth's meridians and an equator are approximations of great circles.
A great circle has the largest radius among all circles formed by an intersection of a sphere and a plane.

Part of a great circle between any of its two points is a spherical arc (or simply arc if a context indicates that it's on a surface of a sphere).

A plane intersecting a sphere cuts it in two parts, each is called a spherical cap (or a spherical dome). This plane is called then a base plane for a spherical cap and a circle of its intersection with a sphere will be called a base circle of a spherical cap.
The radius of a base circle is considered a radius of a spherical cap. The altitude (or height) of a spherical cap is a perpendicular to a base plane from a center of a base circle to the surface of a sphere.

A spherical sector is an object bounded by a spherical cap and a conical surface with a circle of a spherical cap as a directrix and a center of a sphere as an apex.

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