2-sphere in Euclidean space

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Definition

This is the special case of the sphere in Euclidean space where the sphere has dimension 2 and the Euclidean space it is embedded in has dimension 3.

2sphere.png

Implicit and parametric descriptions

Degree of generality Implicit description What the parameters mean Parametric description What the additional parameters mean
Arbitrary (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2 (x_0,y_0,z_0) are the coordinates of the center and r \ge 0 is the radius of the sphere x = x_0 + r \cos \theta \sin \phi, y = y_0 + r \sin \theta \sin \phi, z = z_0 + r \cos \phi \theta and \phi play roles analogous to the azimuthal and polar angles. See spherical polar coordinates.
Up to translations, i.e., given any 2-sphere in Euclidean space, we can do a translation and bring it into this form x^2 + y^2 + z^2 = r^2 r \ge 0 is the radius. We have used a translation to move the center of the sphere to the origin. x = r \cos \theta \sin \phi, y = r \sin \theta \sin \phi, z = r \cos \phi. \theta and \phi play roles analogous to the azimuthal and polar angles. See spherical polar coordinates.
Up to all rigid motions (translations, rotations, reflections) x^2 + y^2 + z^2 = r^2 r \ge 0 is the radius. We have used a translation to move the center of the sphere to the origin. Since the sphere has rotational and reflection symmetry, allowing freedom of rotation does not result in any simplification of the equation. x = r \cos \theta \sin \phi, y = r \sin \theta \sin \phi, z = r \cos \phi. \theta and \phi play roles analogous to the azimuthal and polar angles. See spherical polar coordinates.
Up to all similarity transformations (transformations, rotations, reflections, scaling) x^2 + y^2 + z^2 = 1 no parameters any more. This is the unit 2-sphere centered at the origin. x = \cos \theta \sin \phi, y = \sin \theta \sin \phi, z = \cos \phi. \theta and \phi play roles analogous to the azimuthal and polar angles. See spherical polar coordinates.