Stereographic projection preserves circles - Revision history

Vipul: 2 revisions

2008-05-18T19:19:01Z

2 revisions

← Older revision	Revision as of 19:19, 18 May 2008
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Vipul at 14:37, 27 April 2008

2008-04-27T14:37:30Z

← Older revision		Revision as of 14:37, 27 April 2008
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	* Circles in <math>S^2 \setminus \{ N \}</math> get mapped to circles in <math>\mathbb{C}</math>		* Circles in <math>S^2 \setminus \{ N \}</math> get mapped to circles in <math>\mathbb{C}</math>
	* A circle in <math>S^2</math> that passes through <math>N</math>, minus the point <math>N</math>, gets mapped to a straight line in <math>\mathbb{C}</math>		* A circle in <math>S^2</math> that passes through <math>N</math>, minus the point <math>N</math>, gets mapped to a straight line in <math>\mathbb{C}</math>

			==Facts used==

			* [[Distance formula for inverse stereographic projection]]
			==Proof==

			===Computational proof===

			{{fillin}}

Vipul: New page: ==Statement== ===Setup=== {{further|Stereographic projection}} Consider $\R^3$ , three-dimensional Euclidean space, with coordinates $x,y,z$ . Denote: $S^2 :...$

2008-04-27T14:15:00Z

New page: ==Statement== ===Setup=== {{further|Stereographic projection}} Consider <math>\R^3</math>, three-dimensional Euclidean space, with coordinates <math>x,y,z</math>. Denote: <math>S^2 :...

New page

==Statement==

===Setup===
{{further|[[Stereographic projection]]}}
Consider <math>\R^3</math>, three-dimensional Euclidean space, with coordinates <math>x,y,z</math>. Denote:

<math>S^2 := \{ (x,y,z) \in \R^3 \mid x^2 + y^2 + z^2 = 1 \}</math>

Identify <math>\mathbb{C}</math> with the <math>xy</math>-plane under the map:

<math>(x,y,0) \mapsto x + iy</math>

Let <math>N = (0,0,1)</math> denote the north pole in <math>S^2</math>, and define the stereographic projection as a bijective map:

<math>S^2 \setminus \{ N \} \to \mathbb{C}</math>

which sends a point <math>P \in S^2 \setminus \{ N \}</math> to the unique point in <math>\mathbb{C}</math> that is collinear with <math>N</math> and <math>P</math>.

===Actual statement===

The following two facts are true:

* Circles in <math>S^2 \setminus \{ N \}</math> get mapped to circles in <math>\mathbb{C}</math>
* A circle in <math>S^2</math> that passes through <math>N</math>, minus the point <math>N</math>, gets mapped to a straight line in <math>\mathbb{C}</math>

Stereographic projection preserves circles - Revision history

Vipul: 2 revisions

Vipul at 14:37, 27 April 2008

Vipul: New page: ==Statement== ===Setup=== {{further|Stereographic projection}} Consider \R^3, three-dimensional Euclidean space, with coordinates x,y,z. Denote: S^2 :...

Vipul: New page: ==Statement== ===Setup=== {{further|Stereographic projection}} Consider $\R^3$ , three-dimensional Euclidean space, with coordinates $x,y,z$ . Denote: $S^2 :...$