# Difference between revisions of "2-sphere in Euclidean space"

From Diffgeom

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− | ! Degree of generality !! Implicit description !! What the parameters mean !! Parametric description !! What the additional parameters mean | + | ! Degree of generality !! Implicit description !! What the parameters mean !! Parametric description !! What the additional parameters mean !! Comment |

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− | | Arbitrary || <math>(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2</math> || <math>(x_0,y_0,z_0)</math> are the coordinates of the center and <math>r \ge 0</math> is the radius of the sphere || <math>x = x_0 + r \cos \theta \sin \phi, y = y_0 + r \sin \theta \sin \phi, z = z_0 + r \cos \phi</math> || <math>\theta</math> and <math>\phi</math> play roles analogous to the azimuthal and polar angles. See [[spherical polar coordinates]]. | + | | Arbitrary || <math>(x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2</math> || <math>(x_0,y_0,z_0)</math> are the coordinates of the center and <math>r \ge 0</math> is the radius of the sphere || <math>x = x_0 + r \cos \theta \sin \phi, y = y_0 + r \sin \theta \sin \phi, z = z_0 + r \cos \phi</math> || <math>\theta</math> and <math>\phi</math> play roles analogous to the azimuthal and polar angles. See [[spherical polar coordinates]]. || |

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− | | Up to translations, i.e., given any 2-sphere in Euclidean space, we can do a translation and bring it into this form || <math>x^2 + y^2 + z^2 = r^2</math> || <math>r \ge 0</math> is the radius. | + | | Up to translations, i.e., given any 2-sphere in Euclidean space, we can do a translation and bring it into this form || <math>x^2 + y^2 + z^2 = r^2</math> || <math>r \ge 0</math> is the radius. || <math>x = r \cos \theta \sin \phi, y = r \sin \theta \sin \phi, z = r \cos \phi</math>. || <math>\theta</math> and <math>\phi</math> play roles analogous to the azimuthal and polar angles. See [[spherical polar coordinates]]. || We have used a translation to move the center of the sphere to the origin. |

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− | | Up to all rigid motions (translations, rotations, reflections) || <math>x^2 + y^2 + z^2 = r^2</math> || <math>r \ge 0</math> is the radius. We have used a translation to move the center of the sphere to the origin | + | | Up to all rigid motions (translations, rotations, reflections) || <math>x^2 + y^2 + z^2 = r^2</math> || <math>r \ge 0</math> is the radius. We have used a translation to move the center of the sphere to the origin. || <math>x = r \cos \theta \sin \phi, y = r \sin \theta \sin \phi, z = r \cos \phi</math>. || <math>\theta</math> and <math>\phi</math> play roles analogous to the azimuthal and polar angles. See [[spherical polar coordinates]]. || Since the sphere has rotational and reflection symmetry, allowing freedom of rotation does not result in any simplification of the equation. |

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− | | Up to all similarity transformations (transformations, rotations, reflections, scaling) || <math>x^2 + y^2 + z^2 = 1</math> || no parameters any more. This is the unit 2-sphere centered at the origin. || <math>x = \cos \theta \sin \phi, y = \sin \theta \sin \phi, z = \cos \phi</math>. ||<math>\theta</math> and <math>\phi</math> play roles analogous to the azimuthal and polar angles. See [[spherical polar coordinates]]. | + | | Up to all similarity transformations (transformations, rotations, reflections, scaling) || <math>x^2 + y^2 + z^2 = 1</math> || no parameters any more. This is the unit 2-sphere centered at the origin. || <math>x = \cos \theta \sin \phi, y = \sin \theta \sin \phi, z = \cos \phi</math>. ||<math>\theta</math> and <math>\phi</math> play roles analogous to the azimuthal and polar angles. See [[spherical polar coordinates]]. || We've scaled the sphere to unit radius. |

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## Latest revision as of 15:05, 5 August 2011

## 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.

### Implicit and parametric descriptions

Degree of generality | Implicit description | What the parameters mean | Parametric description | What the additional parameters mean | Comment |
---|---|---|---|---|---|

Arbitrary | are the coordinates of the center and is the radius of the sphere | and 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 | is the radius. | . | and play roles analogous to the azimuthal and polar angles. See spherical polar coordinates. | We have used a translation to move the center of the sphere to the origin. | |

Up to all rigid motions (translations, rotations, reflections) | is the radius. We have used a translation to move the center of the sphere to the origin. | . | and play roles analogous to the azimuthal and polar angles. See spherical polar coordinates. | Since the sphere has rotational and reflection symmetry, allowing freedom of rotation does not result in any simplification of the equation. | |

Up to all similarity transformations (transformations, rotations, reflections, scaling) | no parameters any more. This is the unit 2-sphere centered at the origin. | . | and play roles analogous to the azimuthal and polar angles. See spherical polar coordinates. | We've scaled the sphere to unit radius. |