2-sphere in Euclidean space - Revision history

Vipul: /* Implicit and parametric descriptions */

2011-08-05T15:05:42Z

Implicit and parametric descriptions

← Older revision		Revision as of 15:05, 5 August 2011
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! 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
	\|-		\|-
	\| 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]]. \|\|
	\|-		\|-
	\| 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. ~~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]].		\| 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.
	\|-		\|-
	\| 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~~. \|\| Since the sphere has rotational and reflection symmetry, allowing freedom of rotation does not result in any simplification of the equation~~. \|\| <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]].		\| 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.
	\|-		\|-
	\| 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.
	\|}		\|}

Vipul at 15:04, 5 August 2011

2011-08-05T15:04:36Z

← Older revision		Revision as of 15:04, 5 August 2011
Line 4:		Line 4:

	[[File:2sphere.png\|400px]]		[[File:2sphere.png\|400px]]

			===Implicit and parametric descriptions===

			{\| class="sortable" border="1"
			! Degree of generality !! Implicit description !! What the parameters mean !! Parametric description !! What the additional parameters mean
			\|-
			\| 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]].
			\|-
			\| 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. 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]].
			\|-
			\| 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. \|\| Since the sphere has rotational and reflection symmetry, allowing freedom of rotation does not result in any simplification of the equation. \|\| <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]].
			\|-
			\| 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]].
			\|}

Vipul: Created page with "==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. [[File:..."

2011-07-29T23:41:50Z

Created page with "==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. [[File:..."

New page

==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.

[[File:2sphere.png|400px]]