Ellipsoidal coordinates - Revision history

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	~~{{Unreferenced\|date=March 2008}}~~		Ed is what individuals contact me and my wife doesn't like it at all. For many years she's been working as a travel agent. Mississippi is the only location I've been residing in but I will have to move in a yr or two. What I love doing is football but I don't have the time recently.<br><br>Here is my blog :: [http://isaworld.pe.kr/?document_srl=392088 online psychic readings]

	~~[[Image:Ellipse symmetry set.svg\|\|right\|thumb\|240px\|An [[ellipse]] (red), its [[evolute]] (blue),~~ and ~~its symmetry set (green and yellow)~~. ~~the [[medial axis]] is just the green portion of the symmetry set. One bi-tangent circle is shown.]]~~

	~~In [[geometry]], the~~ '~~''symmetry set''' is a method for representing the local symmetries of a curve, and can be used~~ as a ~~method for representing the [[shape]] of objects by finding the [[topological skeleton]]~~. ~~The [[medial axis]], a subset of the symmetry set~~ is ~~a set of curves which roughly run along~~ the ~~middle of an object.~~

	~~==The symmetry set~~ in ~~2 dimensions==~~
	~~Let <math>~~ I ~~\subseteq \mathbb{R} </math> be an open interval, and <math>\gamma : I \~~to ~~\mathbb{R}^2</math> be~~ a ~~parametrisation of a smooth plane curve~~.

	~~The symmetry set of <math> \gamma (~~I~~) \subset \mathbb{R}^2</math>~~ is ~~defined to be the closure of the set of centres of circles tangent to the curve at at least distinct two points ([[bitangent]] circles).~~

	~~The symmetry set will~~ have ~~endpoints corresponding to [[vertex (curve)\|vertices]] of the curve. Such points will lie at [[cusp (singularity)\|cusp]] of~~ the ~~[[evolute]]~~. ~~At such points the curve will have [[Contact (mathematics)\|4-point contact]] with the circle.~~

	~~==The symmetry set in ''n'' dimensions==~~
	~~For a smooth manifold of dimension <math>m</math> in~~ <~~math~~>~~\mathbb{R}^n~~<~~/math~~> ~~(clearly we need <math>m < n</math>). The symmetry set of the manifold~~ is ~~the closure of the centres of hyperspheres tangent to the manifold in at least two distinct places.~~

	~~==The symmetry set as a bifurcation set==~~
	~~Let <math> U \subseteq \mathbb{R}^m</math> be an open simply connected domain and <math>(u_1\ldots,u_m) := \underline{u} \in U</math>. Let <math>\underline{X}~~ : ~~U \to \R^n</math> be a parametrisation of a smooth piece of manifold.~~
	~~We may define a <math>n</math> parameter family of functions on the curve, namely~~
	:~~<math> F~~ : ~~\mathbb{R}^n \times U \to \mathbb{R} \ , \quad \mbox{where} \quad F(\underline{x},\underline{u}) = (\underline{x} - \underline{X}) \cdot (\underline{x} - \underline{X}) \ . <~~/~~math>~~
	~~This family is called the family of distance squared functions. This is because for a fixed <math>\underline{x}_0 \in \mathbb{R}^n</math> the value of <math>F(\underline{x}_0,\underline{u})<~~/~~math> is the square of the distance from <math>\underline{x}_0</math> to <math>\underline{X}</math> at <math>\underline{X}(u_1\ldots,u_m)~~.~~</math>~~

	~~The symmetry set is then the bifurcation set of the family of distance squared functions. I.e~~. ~~it is the set of <math>\underline{x} \in \R^n</math> such that <math>F(\underline{x},-)<~~/~~math> has a repeated singularity for some <math>\underline{u} \in U.</math>~~

	~~By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to <math>\mathcal{r} F = \underline{0}</math>.~~

	The symmetry set is then the set of <math>\underline{x} \in \mathbb{R}^n</math> such that there exist <math>(\underline{u}_1, \underline{u}_2) \in U \times U</math> with <math>\underline{u}_1 \neq \underline{u}_2</math>, and
	~~:<math> \mathcal{r} F(\underline{x},\underline{u}_1) = \mathcal{r} F(\underline{x},\underline{u}_2)~~ = ~~\underline{0} </math>~~
	~~together with the limiting points of this set.~~

	~~==References ==~~
	~~1. J. W. Bruce, P.J.Giblin and C. G. Gibson, Symmetry Sets. ''Proc. of the Royal Soc.of Edinburgh'' 101A (1985), 163-186.~~

	~~2. J. W. Bruce and P.J.Giblin, Curves and Singularities, Cambridge University Press (1993).~~

	~~[[Category:Differential geometry]~~]

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	~~Golda is what's created on my beginning real psychic readings -~~ [~~http~~:~~//conniecolin~~.~~com/xe/community/24580 http://conniecolin~~.~~com/xe/community/24580~~] ~~- certification although it~~ is ~~not~~ the ~~title on my beginning certificate~~. ~~Office supervising~~ is ~~where her main earnings arrives from but she~~'~~s currently utilized~~ for ~~another 1~~. ~~It's not~~ a ~~common thing but what~~ I ~~like doing~~ is to ~~climb but I don't~~ have the ~~time lately~~. ~~North Carolina~~ is the ~~place he enjoys most but now he is contemplating other options~~.<br><br>~~Here is my web blog free [http~~://~~www~~.~~prayerarmor.com~~/~~uncategorized/dont-know-which-kind-~~of~~-hobby-~~to-~~take-up-read~~-the~~-following-tips~~/ ~~cheap psychic readings] reading~~ (~~[http:~~//~~fashionlinked~~.~~com~~/~~index~~.~~php?do~~=/~~profile-13453~~/~~info~~/ ~~just click~~ the ~~following page~~])		{{Unreferenced\|date=March 2008}}

			[[Image:Ellipse symmetry set.svg\|\|right\|thumb\|240px\|An [[ellipse]] (red), its [[evolute]] (blue), and its symmetry set (green and yellow). the [[medial axis]] is just the green portion of the symmetry set. One bi-tangent circle is shown.]]

			In [[geometry]], the '''symmetry set''' is a method for representing the local symmetries of a curve, and can be used as a method for representing the [[shape]] of objects by finding the [[topological skeleton]]. The [[medial axis]], a subset of the symmetry set is a set of curves which roughly run along the middle of an object.

			==The symmetry set in 2 dimensions==
			Let <math> I \subseteq \mathbb{R} </math> be an open interval, and <math>\gamma : I \to \mathbb{R}^2</math> be a parametrisation of a smooth plane curve.

			The symmetry set of <math> \gamma (I) \subset \mathbb{R}^2</math> is defined to be the closure of the set of centres of circles tangent to the curve at at least distinct two points ([[bitangent]] circles).

			The symmetry set will have endpoints corresponding to [[vertex (curve)\|vertices]] of the curve. Such points will lie at [[cusp (singularity)\|cusp]] of the [[evolute]]. At such points the curve will have [[Contact (mathematics)\|4-point contact]] with the circle.

			==The symmetry set in ''n'' dimensions==
			For a smooth manifold of dimension <math>m</math> in <math>\mathbb{R}^n</math> (clearly we need <math>m < n</math>). The symmetry set of the manifold is the closure of the centres of hyperspheres tangent to the manifold in at least two distinct places.

			==The symmetry set as a bifurcation set==
			Let <math> U \subseteq \mathbb{R}^m</math> be an open simply connected domain and <math>(u_1\ldots,u_m) := \underline{u} \in U</math>. Let <math>\underline{X} : U \to \R^n</math> be a parametrisation of a smooth piece of manifold.
			We may define a <math>n</math> parameter family of functions on the curve, namely
			:<math> F : \mathbb{R}^n \times U \to \mathbb{R} \ , \quad \mbox{where} \quad F(\underline{x},\underline{u}) = (\underline{x} - \underline{X}) \cdot (\underline{x} - \underline{X}) \ . </math>
			This family is called the family of distance squared functions. This is because for a fixed <math>\underline{x}_0 \in \mathbb{R}^n</math> the value of <math>F(\underline{x}_0,\underline{u})</math> is the square of the distance from <math>\underline{x}_0</math> to <math>\underline{X}</math> at <math>\underline{X}(u_1\ldots,u_m).</math>

			The symmetry set is then the bifurcation set of the family of distance squared functions. I.e. it is the set of <math>\underline{x} \in \R^n</math> such that <math>F(\underline{x},-)</math> has a repeated singularity for some <math>\underline{u} \in U.</math>

			By a repeated singularity, we mean that the jacobian matrix is singular. Since we have a family of functions, this is equivalent to <math>\mathcal{r} F = \underline{0}</math>.

			The symmetry set is then the set of <math>\underline{x} \in \mathbb{R}^n</math> such that there exist <math>(\underline{u}_1, \underline{u}_2) \in U \times U</math> with <math>\underline{u}_1 \neq \underline{u}_2</math>, and
			:<math> \mathcal{r} F(\underline{x},\underline{u}_1) = \mathcal{r} F(\underline{x},\underline{u}_2) = \underline{0} </math>
			together with the limiting points of this set.

			==References ==
			1. J. W. Bruce, P.J.Giblin and C. G. Gibson, Symmetry Sets. ''Proc. of the Royal Soc.of Edinburgh'' 101A (1985), 163-186.

			2. J. W. Bruce and P.J.Giblin, Curves and Singularities, Cambridge University Press (1993).

			[[Category:Differential geometry]]

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