Self-balancing binary search tree - Revision history

5.65.63.85 at 00:34, 11 March 2014

2014-03-11T00:34:01Z

← Older revision		Revision as of 02:34, 11 March 2014
Line 1:		Line 1:
	~~Should you Google the title "Adam Short~~,~~" images of a young man~~ in ~~probably his late twenty's would pop up~~. ~~Alongside these pictures would be posts, training guides, and success stories about his triumphs as~~ an ~~Internet marketing guru. Most~~ of the ~~reviews would emphasize the fact that he only preaches what he himself has attempted, and has reportedly made thousands of dollars~~ in ~~earnings month-to-month~~.<br><br>~~Irritable Bowel Syndrome (IBS)~~ is a ~~syndrome that impacts the colon, or large bowel~~. ~~The bowel is component~~ of ~~the digestive tract that has the role~~ of ~~storing stool~~. ~~IBS~~ is ~~not a disease however it can effortlessly be mistaken~~ for ~~a illness~~. ~~It is essentially a functional condition which impacts~~ how ~~correctly the bowel functions. It has a selection~~ of ~~symptoms. These consist of abdominal discomfort or discomfort frequently observed by doctors as cramping~~, ~~gasoline, bloating, diarrhea and constipation. The real cause of~~ [http://~~www.Myfitnesspal~~.com~~/blog/DietingNewbie/view/-irritable-bowel-syndrome-foods-to-avoid-when-dealing-with-ibs-687335 diet for IBS~~] is not fully recognized by physicians. Nevertheless it appears the nerves and muscle tissues in the bowel have extra sensitivity. Muscle tissues may agreement to much when consuming creating cramping and diarrhea throughout or soon following a meal.<br><br>The previous yr has been 1 of the most demanding, ~~both bodily and mentally~~, ~~I have~~ at ~~any time experienced~~. ~~Last June~~ of ~~'08~~, ~~I was in an accident and smashed my neck, pretty poor. The rehab. has been the toughest thing I've at any time skilled. Just when I began to get a deal with on this thing~~, ~~I was diagnosed with malignant melanoma cancer on both my arm~~ and ~~chest~~.<br><br>~~If you live~~ with ~~IBS then you already know that not only can it lead to some very inconvenient times but it~~ can ~~be downright embarrassing as nicely~~. ~~There is no real cure for the disorder~~, ~~but you can maintain~~ it ~~nicely below manage if you take a small work.<br><br>The goal of holistic counselling~~ is ~~not~~ to ~~merely~~ help ~~you uncover what may be creating you psychological trauma or stress making~~ your ~~signs~~ and ~~symptoms and reason via why it~~ is ~~not so traumatic~~. ~~Instead, we examine your whole lifestyle~~ and ~~determine what modifications are needed~~ to ~~your entire atmosphere,~~ to ~~give you~~ a ~~happy, healthy life~~.<br><br>~~So what are hemorrhoids in any case? They are enlarged and instead inflamed blood vessels that are situated in the rectal area. They can also be interior as nicely. Some~~ of ~~the signs and symptoms may consist of:~~ a ~~bulging in the rectal region, bleeding,~~ and ~~sometimes a leakage from your stool. You~~ can ~~wind~~ up with ~~blood on~~ the ~~Foods~~ to ~~Avoid with Irritable Bowel Syndrome paper and whilst this can be a rather frightening experience, most of the time~~ it ~~is not serious or anything to worry about~~.<br><br>~~This kind of nutritional fiber dissolves effortlessly in drinking water and types a thick jelly like material. Therefore~~, it ~~can be safely used during constipation~~ as ~~well as diarrhea~~. ~~Foods that contain great amount of soluble fiber are rice~~, ~~oatmeal, pasta, corn tortillas, carrots, potatoes and so on~~. ~~These meals ought~~ to ~~be included~~ in ~~larger quantity~~ in ~~your foods and treats. You can take these foods even~~ on ~~an vacant abdomen~~.~~<br><br>Low carb diet programs, also known as low carbohydrates diet programs can~~ be ~~outlined as nutritional applications that restrict consumption of carbohydrates consumption for excess weight manage. These meals are generally changed with meals containing~~ a ~~higher proportion of proteins and fats~~.<br><br>~~If your high-protein meals are animal meals they are frequently also higher~~ in ~~animal fat~~. ~~Animal fat~~ are ~~mainly saturated fats~~ and ~~cholesterol and they are known to increase blood fat levels and improve the risk~~ of ~~coronary heart illness, and particular Irritable Bowel Syndrome Diet kinds~~ of ~~cancer.<br><br>Because, if you know how, you~~ can ~~get rid of most or all of your gerd, IBS~~, ~~allergic reactions and often other illnesses in a few times. The method addresses~~ the ~~basic cause. Your body is extremely low in~~ some thing that is a important metabolite for each cell. This material is at levels much as well low for you to be healthy. This material is normally occurring in our bodies. And inside particular limitations, the higher the ranges, the more healthy we are.<br><br>What is truly terrible about IBS for me, is I can't consume a great deal of eco-friendly leafy vegetables or ~~Irritable Bowel Syndrome Diet issues like tomatoes and onions I avoid totally unless~~ of ~~course somebody else has place~~ them ~~in a salad. (I will usually choose them out of the salad)~~.<br><br>~~Eating little meals is easier for your body to digest~~ the ~~foods you eat and is also best for your metabolic process~~. ~~It's also very essential to never consume fruits with your meal~~. ~~Fruits digest extremely quickly while other foods take two to three hours. You should usually consume fruit on your own and either 45 minutes prior to your food or two to three hrs following your meal~~. ~~Otherwise the fruit~~ will ~~ferment in your abdomen~~ and ~~possibly cause undesirable stomach pain and or gas.<br><br>Some individuals eat sugar out of melancholy or anxiousness~~. ~~To help cope with~~ these ~~circumstances exercise regularly~~ and ~~get plenty of sunlight~~. ~~This releases endorphins so that sugar is~~ no ~~lengthier a coping mechanism~~.		Gain access to it in excel, copy-paste this continued plan in accordance with corpuscle B1. An individual again access an bulk of time in abnormal located in corpuscle A1, the standard in treasures will surface in B1.<br><br>Getting to the higher level: it is essential when it comes to game, but when looking at Clash of Clans, you will have a lot more subtle simple steps. Despite making use of clash of clans hack tools, you furthermore acquire experience points as a result of matching on top linked to other players. Increased metabolism purpose of earning Pvp is to enable some other enhancements for your indigneous group. If you have any issues relating to where by and how to use clash of clans hack cydia, [http://circuspartypanama.com sneak a peek at this site],, you can get in touch with us at our own web-site. The restoration consists of better strive equipment, properties, troops and tribe people.<br><br>For anybody who is getting a online mission for your little one, look for one which enables numerous customers to do with each other. Video gaming can undoubtedly solitary action. Nevertheless, it is important to help motivate your youngster really social, and multi-player clash of clans hack is capable performing that. They set up sisters and brothers but buddies to all linked to take a moment or laugh and compete alongside one another.<br><br>All over Clash of Clans Hacks (a brilliant popular open architecture and arresting audacious by Supercell) participants can acceleration up accomplishments for [https://Www.Flickr.com/search/?q=instance instance] building, advance or exercise routine troops with gems which bought for absolute money. They're basically monetizing the valid player's impatience. Each single amusing architecture daring I actually apperceive of manages to do it.<br><br>Once the game is per mobile edition, it will do not lack substance as with many mobile games. So, defragging the routine registry will boost the system overall performance to a fantastic extent. I usually get everyplace from 4000 to 5000 m - Points in a day ($4 to $5 for Amazon. Apple mackintosh showed off the enormously anticipated i - Some of the 5 for the incredibly first time in San Francisco on Wednesday morning (September 12, 2012). Can be certainly a huge demand during some i - Telephone call 4 application not sole promoting business but likewise helps users to make extra money.<br><br>Precious metal and Elixir would function as main sources available in Clash of Clans. Each of these associated with are necessary and can be gathered by a level of ways. Frontrunners of individuals can use structures, recover the cash some other tribes or even clash of clans crack tools for acquiring them both.<br><br>Your leap into the pre-owned or operated xbox [http://www.wired.com/search?query=sport+marketplace sport marketplace]. Several ball players will get a Collide of Clans Hack and complete this game really really fast. Several shops let these betting games being dealt in and then also promote them at unquestionably the lessened cost. Might be by far probably the most cost-effective technique to request newer video games absolutely no higher cost.

50.43.74.47 at 02:11, 27 February 2014

2014-02-27T02:11:13Z

← Older revision		Revision as of 04:11, 27 February 2014
Line 1:		Line 1:
	~~{{About\|differential geometry\|\|}}~~		Should you Google the title "Adam Short," images of a young man in probably his late twenty's would pop up. Alongside these pictures would be posts, training guides, and success stories about his triumphs as an Internet marketing guru. Most of the reviews would emphasize the fact that he only preaches what he himself has attempted, and has reportedly made thousands of dollars in earnings month-to-month.<br><br>Irritable Bowel Syndrome (IBS) is a syndrome that impacts the colon, or large bowel. The bowel is component of the digestive tract that has the role of storing stool. IBS is not a disease however it can effortlessly be mistaken for a illness. It is essentially a functional condition which impacts how correctly the bowel functions. It has a selection of symptoms. These consist of abdominal discomfort or discomfort frequently observed by doctors as cramping, gasoline, bloating, diarrhea and constipation. The real cause of [http://www.Myfitnesspal.com/blog/DietingNewbie/view/-irritable-bowel-syndrome-foods-to-avoid-when-dealing-with-ibs-687335 diet for IBS] is not fully recognized by physicians. Nevertheless it appears the nerves and muscle tissues in the bowel have extra sensitivity. Muscle tissues may agreement to much when consuming creating cramping and diarrhea throughout or soon following a meal.<br><br>The previous yr has been 1 of the most demanding, both bodily and mentally, I have at any time experienced. Last June of '08, I was in an accident and smashed my neck, pretty poor. The rehab. has been the toughest thing I've at any time skilled. Just when I began to get a deal with on this thing, I was diagnosed with malignant melanoma cancer on both my arm and chest.<br><br>If you live with IBS then you already know that not only can it lead to some very inconvenient times but it can be downright embarrassing as nicely. There is no real cure for the disorder, but you can maintain it nicely below manage if you take a small work.<br><br>The goal of holistic counselling is not to merely help you uncover what may be creating you psychological trauma or stress making your signs and symptoms and reason via why it is not so traumatic. Instead, we examine your whole lifestyle and determine what modifications are needed to your entire atmosphere, to give you a happy, healthy life.<br><br>So what are hemorrhoids in any case? They are enlarged and instead inflamed blood vessels that are situated in the rectal area. They can also be interior as nicely. Some of the signs and symptoms may consist of: a bulging in the rectal region, bleeding, and sometimes a leakage from your stool. You can wind up with blood on the Foods to Avoid with Irritable Bowel Syndrome paper and whilst this can be a rather frightening experience, most of the time it is not serious or anything to worry about.<br><br>This kind of nutritional fiber dissolves effortlessly in drinking water and types a thick jelly like material. Therefore, it can be safely used during constipation as well as diarrhea. Foods that contain great amount of soluble fiber are rice, oatmeal, pasta, corn tortillas, carrots, potatoes and so on. These meals ought to be included in larger quantity in your foods and treats. You can take these foods even on an vacant abdomen.<br><br>Low carb diet programs, also known as low carbohydrates diet programs can be outlined as nutritional applications that restrict consumption of carbohydrates consumption for excess weight manage. These meals are generally changed with meals containing a higher proportion of proteins and fats.<br><br>If your high-protein meals are animal meals they are frequently also higher in animal fat. Animal fat are mainly saturated fats and cholesterol and they are known to increase blood fat levels and improve the risk of coronary heart illness, and particular Irritable Bowel Syndrome Diet kinds of cancer.<br><br>Because, if you know how, you can get rid of most or all of your gerd, IBS, allergic reactions and often other illnesses in a few times. The method addresses the basic cause. Your body is extremely low in some thing that is a important metabolite for each cell. This material is at levels much as well low for you to be healthy. This material is normally occurring in our bodies. And inside particular limitations, the higher the ranges, the more healthy we are.<br><br>What is truly terrible about IBS for me, is I can't consume a great deal of eco-friendly leafy vegetables or Irritable Bowel Syndrome Diet issues like tomatoes and onions I avoid totally unless of course somebody else has place them in a salad. (I will usually choose them out of the salad).<br><br>Eating little meals is easier for your body to digest the foods you eat and is also best for your metabolic process. It's also very essential to never consume fruits with your meal. Fruits digest extremely quickly while other foods take two to three hours. You should usually consume fruit on your own and either 45 minutes prior to your food or two to three hrs following your meal. Otherwise the fruit will ferment in your abdomen and possibly cause undesirable stomach pain and or gas.<br><br>Some individuals eat sugar out of melancholy or anxiousness. To help cope with these circumstances exercise regularly and get plenty of sunlight. This releases endorphins so that sugar is no lengthier a coping mechanism.
	~~{{no footnotes\|date=July 2011}}~~
	~~[[Image:Gauss map.svg\|thumb\|400px\|The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere]]~~
	~~In [[differential geometry]]~~, ~~the '''Gauss map''' (named after [[Carl Friedrich Gauss\|Carl F. Gauss]]) maps~~ a ~~[[surface]]~~ in ~~[[Euclidean space]]~~ '~~''R'''<sup>3</sup> to the unit [[sphere]] ''S''<sup>2</sup>~~. ~~Namely~~, ~~given a surface ''X'' lying in '''R'''<sup>3</sup>~~, the ~~Gauss map is a continuous map ''N'': ''X'' → ''S''~~<~~sup~~>2<~~/sup~~> ~~such that ''N''~~(~~''p''~~) is a ~~unit vector orthogonal to ''X'' at ''p''~~, ~~namely the normal vector to ''X'' at ''p''~~.

	The ~~Gauss map can be defined (globally) if and only if~~ the ~~surface is [[orientable]], in which case its degree is half~~ the ~~[[Euler characteristic]]~~. ~~The Gauss map~~ can ~~always~~ be ~~defined locally (i~~.e. on a ~~small piece~~ of ~~the surface)~~. ~~The [[Jacobian matrix and determinant\|Jacobian]] determinant~~ of ~~the Gauss map is equal to [[Gaussian curvature]]~~, and ~~the~~ [~~[differential (calculus)\|differential]~~] ~~of the Gauss map~~ is ~~called the [[shape operator]]~~.

	~~Gauss first wrote a draft on~~ the ~~topic in 1825~~ and ~~published~~ in ~~1827~~.

	~~There is also a Gauss map for~~ a ~~[[Link (knot theory)\|link]], which computes [[linking number]]~~.

	~~==Generalizations==~~
	~~The Gauss map can be defined for [[hypersurface]]s in '''R'''~~<~~sup~~>~~''n''~~<~~/sup~~> ~~as a map from a hypersurface to~~ the ~~unit sphere ''S~~''<~~sup~~>~~''n'' − 1~~<~~/sup~~> ~~∈ '''R'''~~<~~sup~~>~~''n''~~<~~/sup~~>.

	~~For~~ a ~~general oriented ''k''-[[submanifold]] of '''R'''~~<~~sup~~>~~''n''~~<~~/sup~~> the ~~Gauss map~~ can be also be ~~defined,~~ and ~~its target space is~~ the ~~''oriented'' [[Grassmannian]]~~
	~~<math>\tilde{G}_{k~~,~~n}</math>~~, ~~i.e~~. the ~~set of all oriented ''k''-planes in '''R'''<sup>''n''</sup>. In~~ this ~~case~~ a ~~point on~~ the ~~submanifold~~ is ~~mapped~~ to ~~its oriented tangent subspace~~. ~~One can also map to its oriented ''normal'' subspace; these are equivalent as~~ <~~math~~>~~\tilde{G}_{k,n} \cong \tilde{G}_{n-k,n}~~<~~/math~~> ~~via orthogonal complement~~.
	~~In [[Euclidean space\|Euclidean 3-space]]~~, ~~this says~~ that an ~~oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as~~ <~~math~~>~~\tilde{G}_{1,n} \cong S^{n-1}~~<~~/math~~>), ~~hence this is consistent with the definition above.~~

	~~Finally, the notion of Gauss map~~ can be ~~generalized to an oriented submanifold ''X''~~ of ~~dimension ''k'' in an oriented ambient [[Riemannian manifold]] ''M''~~ of ~~dimension ''n''~~. ~~In that case, the Gauss map then goes from ''X'' to the set of tangent ''k''~~-~~planes~~ in ~~the [[tangent bundle]] ''TM''~~. ~~The target space for~~ the ~~Gauss map ''N'' is a [[Grassmann bundle]] built on the tangent bundle ''TM''~~. ~~In the case where~~ <~~math~~>~~M=\mathbf{R}^n~~<~~/math~~>, ~~the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian)~~, and ~~we recover the previous definition~~.

	~~==Total curvature==~~
	The ~~area of~~ the ~~image of the Gauss map~~ is ~~called the '''total curvature''' and~~ is ~~equivalent~~ to ~~the [[surface integral]] of the [[Gaussian curvature]]~~. This is the ~~original interpretation given by Gauss~~. ~~The [[Gauss-Bonnet theorem]] links total curvature of a surface to its [[topology\|topological]] properties.~~
	:<~~math~~> ~~\iint_R \|N_u \times N_v\| \ du\, dv = \iint_R K\|X_u \times X_v\| \ du\, dv = \iint_R K \ dA~~<~~/math~~>

	~~==Cusps~~ of ~~the Gauss map==~~
	~~The Gauss map reflects many properties~~ of ~~the surface: when the surface~~ has ~~zero Gaussian curvature, (that is along~~ a ~~[[parabolic line]]) the Gauss map will have a [[Catastrophe theory#Fold catastrophe\|fold catastrophe]]~~. ~~This fold may contain [[cusp~~ (~~singularity)\|cusps]] and these cusps were studied in depth by [[Thomas Banchoff]], [[Terence Gaffney]] and [[Clint McCrory]]. Both parabolic lines and cusp are stable phenomena and~~ will ~~remain under slight deformations~~ of the ~~surface~~. ~~Cusps occur when:~~
	~~#The surface has a bi-tangent plane~~
	~~#A [[ridge (differential geometry)\|ridge]] crosses a parabolic line~~
	~~#at the closure of~~ the ~~set of inflection points of the [[asymptotic curve]]s of the surface.~~
	~~There are two types of cusp: ''elliptic cusp''~~ and ~~''hyperbolic cusps''.~~

	~~==References==~~
	*Gauss, K. F., '~~'Disquisitiones generales circa superficies curvas'' (1827)~~
	*Gauss, K. F.~~, ''General investigations of curved surfaces'', English translation. Hewlett, New York: Raven Press (1965).~~
	*Banchoff, T.~~, Gaffney T., McCrory C., ''Cusps of~~ the ~~Gauss Map'', (1982) Research Notes~~ in ~~Mathematics 55, Pitman, London~~. ~~[http://www~~.~~math~~.~~brown.edu/~dan/cgm/index~~.~~html online version]~~

	~~== External links ==~~
	* {{MathWorld \| urlname=GaussMap \| title=Gauss Map}}

	~~{{Chaos theory}}~~

	~~[[Category:Differential geometry]]~~
	~~[[Category:Differential geometry of surfaces]]~~
	~~[[Category:Riemannian geometry]]~~
	~~[[Category:Surfaces]]~~

en>Jochen Burghardt: /* Implementations */ added ref to 2-3 tree

2013-06-07T17:04:00Z

Implementations: added ref to 2-3 tree

← Older revision		Revision as of 19:04, 7 June 2013
Line 1:		Line 1:
	~~Hi there~~, ~~I am Aaron Tolan and I adore it~~. ~~My husband and I selected~~ to ~~reside~~ in ~~New Hampshire~~ and ~~I adore every day residing right here~~. ~~Managing people~~ is ~~how he tends~~ to ~~make cash~~. ~~Fish maintaining~~ is ~~something that he's been performing~~ for ~~years~~. ~~You~~ can ~~find my website right here: http:~~//~~www~~.~~Myfitnesspal.com~~/~~blog~~/~~DietingNewbie~~/~~view~~/-~~irritable~~-~~bowel~~-~~syndrome~~-~~foods~~-to-~~avoid-when-dealing-with-ibs-687335~~<br><br>~~my webpage~~ [http://www.~~Myfitnesspal~~.~~com~~/~~blog~~/~~DietingNewbie~~/~~view/-irritable-bowel-syndrome-foods-to-avoid-when-dealing-with-ibs-687335 diet for IBS~~]		{{About\|differential geometry\|\|}}
			{{no footnotes\|date=July 2011}}
			[[Image:Gauss map.svg\|thumb\|400px\|The Gauss map provides a mapping from every point on a curve or a surface to a corresponding point on a unit sphere]]
			In [[differential geometry]], the '''Gauss map''' (named after [[Carl Friedrich Gauss\|Carl F. Gauss]]) maps a [[surface]] in [[Euclidean space]] '''R'''<sup>3</sup> to the unit [[sphere]] ''S''<sup>2</sup>. Namely, given a surface ''X'' lying in '''R'''<sup>3</sup>, the Gauss map is a continuous map ''N'': ''X'' → ''S''<sup>2</sup> such that ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p'', namely the normal vector to ''X'' at ''p''.

			The Gauss map can be defined (globally) if and only if the surface is [[orientable]], in which case its degree is half the [[Euler characteristic]]. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The [[Jacobian matrix and determinant\|Jacobian]] determinant of the Gauss map is equal to [[Gaussian curvature]], and the [[differential (calculus)\|differential]] of the Gauss map is called the [[shape operator]].

			Gauss first wrote a draft on the topic in 1825 and published in 1827.

			There is also a Gauss map for a [[Link (knot theory)\|link]], which computes [[linking number]].

			==Generalizations==
			The Gauss map can be defined for [[hypersurface]]s in '''R'''<sup>''n''</sup> as a map from a hypersurface to the unit sphere ''S''<sup>''n'' − 1</sup> ∈ '''R'''<sup>''n''</sup>.

			For a general oriented ''k''-[[submanifold]] of '''R'''<sup>''n''</sup> the Gauss map can be also be defined, and its target space is the ''oriented'' [[Grassmannian]]
			<math>\tilde{G}_{k,n}</math>, i.e. the set of all oriented ''k''-planes in '''R'''<sup>''n''</sup>. In this case a point on the submanifold is mapped to its oriented tangent subspace. One can also map to its oriented ''normal'' subspace; these are equivalent as <math>\tilde{G}_{k,n} \cong \tilde{G}_{n-k,n}</math> via orthogonal complement.
			In [[Euclidean space\|Euclidean 3-space]], this says that an oriented 2-plane is characterized by an oriented 1-line, equivalently a unit normal vector (as <math>\tilde{G}_{1,n} \cong S^{n-1}</math>), hence this is consistent with the definition above.

			Finally, the notion of Gauss map can be generalized to an oriented submanifold ''X'' of dimension ''k'' in an oriented ambient [[Riemannian manifold]] ''M'' of dimension ''n''. In that case, the Gauss map then goes from ''X'' to the set of tangent ''k''-planes in the [[tangent bundle]] ''TM''. The target space for the Gauss map ''N'' is a [[Grassmann bundle]] built on the tangent bundle ''TM''. In the case where <math>M=\mathbf{R}^n</math>, the tangent bundle is trivialized (so the Grassmann bundle becomes a map to the Grassmannian), and we recover the previous definition.

			==Total curvature==
			The area of the image of the Gauss map is called the '''total curvature''' and is equivalent to the [[surface integral]] of the [[Gaussian curvature]]. This is the original interpretation given by Gauss. The [[Gauss-Bonnet theorem]] links total curvature of a surface to its [[topology\|topological]] properties.
			:<math> \iint_R \|N_u \times N_v\| \ du\, dv = \iint_R K\|X_u \times X_v\| \ du\, dv = \iint_R K \ dA</math>

			==Cusps of the Gauss map==
			The Gauss map reflects many properties of the surface: when the surface has zero Gaussian curvature, (that is along a [[parabolic line]]) the Gauss map will have a [[Catastrophe theory#Fold catastrophe\|fold catastrophe]]. This fold may contain [[cusp (singularity)\|cusps]] and these cusps were studied in depth by [[Thomas Banchoff]], [[Terence Gaffney]] and [[Clint McCrory]]. Both parabolic lines and cusp are stable phenomena and will remain under slight deformations of the surface. Cusps occur when:
			#The surface has a bi-tangent plane
			#A [[ridge (differential geometry)\|ridge]] crosses a parabolic line
			#at the closure of the set of inflection points of the [[asymptotic curve]]s of the surface.
			There are two types of cusp: ''elliptic cusp'' and ''hyperbolic cusps''.

			==References==
			*Gauss, K. F., ''Disquisitiones generales circa superficies curvas'' (1827)
			*Gauss, K. F., ''General investigations of curved surfaces'', English translation. Hewlett, New York: Raven Press (1965).
			*Banchoff, T., Gaffney T., McCrory C., ''Cusps of the Gauss Map'', (1982) Research Notes in Mathematics 55, Pitman, London. [http://www.math.brown.edu/~dan/cgm/index.html online version]

			== External links ==
			* {{MathWorld \| urlname=GaussMap \| title=Gauss Map}}

			{{Chaos theory}}

			[[Category:Differential geometry]]
			[[Category:Differential geometry of surfaces]]
			[[Category:Riemannian geometry]]
			[[Category:Surfaces]]

en>Qwertyus: /* See also */ lf

2012-07-03T12:50:24Z