Circle of a sphere the Equator the only great circle. By contrast, all meridians of longitude, paired with their opposite meridian in the other hemisphere, form great circles. The diameter of the sphere which passes through the center of the circle is called its axis and the endpoints of this diameter are called its poles. A circle of a sphere can also be defined as the set of points at a given angular distance from a given pole. When the intersection of a sphere and a plane is not empty or a single point, it is a circle. This can be seen as follows:
N-sphere space and is an example of an -manifold. The volume form of an -sphere of radius is given by where is the Hodge star operator; see for a discussion and proof of this formula in the case . As a result, The space enclosed by an -sphere is called an -ball. An -ball is closed if it includes the -sphere, and it is open if it does not include the -sphere. Specifically: Topologically, an -sphere can be constructed as a one-point compactification of -dimensional Euclidean space. Briefly, the -sphere can be described as , which is -dimensional Euclidean space plus