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Topic: Tangent space

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	PlanetMath: tangent plane (elementary)
		The notion of tangent plane is a generalization of the notion of tangent vector to surfaces.
		Just as the tangent line is a special line which is associated to a point of a smooth curve, so too a tangent plane is a special plane which is associated to a point on a smooth surface.
		Just as the tangent to a line may be understood as the limit of a line connecting two nearby points in the limit where the points coalesce, so too the tangent plane can be regarded as a limit.
	planetmath.org /encyclopedia/TangentPlane.html (1872 words)

	Tangent space
		For example, if the given manifold is a 2-sphere, one can picture the tangent space at a point as the plane which touches the sphere at that point and is perpendicular to the sphere's radius through the point.
		Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold.
		All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the tangent bundle of the manifold.
	www.ebroadcast.com.au /lookup/encyclopedia/ta/Tangent_vector.html (946 words)

	PlanetMath: tangent space
		The notion of tangent space derives from the observation that there is no natural way to relate and compare velocities at different points of a manifold.
		This is already evident when we consider objects moving on a surface in 3-space, where the velocities take their value in the tangent planes of the surface.
		This is version 2 of tangent space, born on 2002-02-16, modified 2003-02-16.
	planetmath.org /encyclopedia/TangentVector.html (374 words)

	Differential geometry and topology - Gurupedia
		At every point of the manifold, there is the tangent space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point.
		One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point.
		A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point.
	www.gurupedia.com /d/di/differential_geometry.htm (938 words)

	Tangent space - Wikipedia, the free encyclopedia
		The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.
		All the tangent spaces have the same dimension, equal to the dimension of the manifold.
		This defines an equivalence relation on such curves, and the equivalence classes are known as the tangent vectors of M at p.
	en.wikipedia.org /wiki/Tangent_space (1221 words)

	[No title]
		The tangent bundle of LV is thus an (infinite-dimensional) c* *omplex T-equivariant vector bundle; in fact it is the free loopspace of T V.
		The space ToeLV [0,1)spanned by eigen* vectors of -iDoewith eigenvalues in [0, 1) is thus isomorphic to the tangent* space Toe(* *0)V of V at the basepoint oe(0), by the map which assigns to an initial vector, its continuation by parallel transport around the loop.
		The space of conjugacy classes in a c* on- nected Lie group is just the quotient of a maximal torus by the Weyl group, whi ch in this case is the space* Tn= n of unordered n-tuples of points on the circle.
	www.math.purdue.edu /research/atopology/Morava/Looptangent.txt (2625 words)

	Differentiable Manifolds
		For flat spaces this works well, but for mapped regions there are two coordinate systems available, one in the base space (which becomes a basis for the tangent space), and one in the embedding space.
		A "differential structure" on a manifold is a local description of the tangent space and normal space.
		On manifolds where the basis for the tangent space is not constant, derivatives of a point which is described by it's coordinates must take into account the way that the basis vectors change, in addition to how the coordinates change.
	www.research.ibm.com /nao/Primer/DifferentialManifoldPrimer.html (806 words)

	Calculating mesh tangent space vectors - DevMaster.net Forums
		The u and v axes of the tangent space are defined by the texture coordinates, and if the texture mapping is not orthogonal, then the tangent space mapping will not be either.
		Averaging the tangent bases at each vertex is completely correct and results in an approximation to the continuous parameterization of the curved surface that is represented by the triangle mesh, in just the same way that averaging normals at vertices approximates the continuous normal map of a curved surface.
		The tangent space vectors should be the partial derivatives of our atlas of smooth local diffeomorphisms from R^2 (these diffeomorphisms are defined by the 'inverse' texture mapping btw).
	www.devmaster.net /forums/showthread.php?p=28170 (2971 words)

	shape space
		This projection results in a (k-1)m vector of tangent space shape coordinates with respect to the mean for each specimen.
		Principal components analysis can be carried out using tangent space coordinates to extract km-m-m(m-1)-1 eigenvectors; which are the principal components of variation of shape.
		Equilateral triangles lie at the poles, the southern hemisphere is a reflection of the northern.
	www.york.ac.uk /res/fme/resources/morphologika/helpfiles/shapespace.html (604 words)

	The affine connection (Site not responding. Last check: 2007-10-04)
		This is called the tangent space at the point for one may imagine the manifold embedded in a larger flat space, where there would be a flat subspace tangent to each point of the manifold.
		Even when there is no known flat space in which to embed the manifold we speak of the tangent space, merely referring to those vectors found at the point of the manifold in question.
		If it is very near we are tempted to identify vectors in the two tangent spaces just as we are tempted to compare wind velocity in two nearby cities, but when the cities are far apart it is not clear what it means to say the wind velocity is the same.
	www.cap-lore.com /MathPhys/Affine.html (479 words)

	Oregon Business Park - Tangent Business Park
		Tangent Business Park is a 100 acre mixed use site providing office, research and development, light manufacturing, and warehousing space.
		Tangent Business Park provides active on-site management and strives for a high quality faciility with a professional setting and superior customer service.
		Tangent Business Park is located in the heart of the Willamette Valley, is two miles west of the I-5 corridor, and is served by the Southern Pacific rail system.
	www.tangentbusinesspark.com (106 words)

	another object space vs. tangent space question - GameDev.Net Discussion Forums
		With tangent-space normalmapping the sides could share the same section of a dot3 texture because their normals in tangent space are the same.
		So you have L and E renormalized in tangent space and compute diffuse by N.L. How do you compute specular from E? May be I would reflect E by N to get R and use R.L then but you said that you did't do that.
		Additionally, with tangent space, you can store the x,y of the normal in the r,g channels of the texture, Ks in the z, and the power in w component.
	www.gamedev.net /community/forums/viewreply.asp?ID=1868589 (4481 words)

	Cotangent space - Wikipedia, the free encyclopedia
		All cotangent spaces on a manifold have the same dimension, equal to the dimension of the manifold.
		All the cotangent spaces of a manifold can be "glued together" to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold.
		The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other.
	en.wikipedia.org /wiki/Cotangent_space (731 words)

	Gamasutra - Feature - "Let There Be Light!: A Unified Lighting Technique for a New Generation of Games" ...
		Next, when processing the vertices of the surface to be normal mapped on the vertex shader, a tangent matrix must be created to transform all positional lighting information into tangent space (all lighting equations must be performed in the same coordinate space).
		The tangent space matrix is a 3x3 matrix made up of the vertex's tangent, binormal and normal vectors.
		This stems from the fact that we are re-creating a tangent space matrix from interpolated vertex data, as opposed to the actual texel-based tangent space matrix that is computed during normal map generation and used to transform the normal.
	www.gamasutra.com /features/20050729/lacroix_pfv.htm (2937 words)

	BioWare Forums: View Post For Code Display
		Now, getting "a" binormal and tangent vector that are perpendicular to the normal and to each other is easy - but getting the "same" binormal and tangents as the models use has been more difficult.
		Are the tangent and binormal vectors only used to represent a plane that is perpendicular to the normal (and then the difference does not matter, as they represent the same plane), or does the orientation of the vectors within the plane matter, for example with regard to shading or texture layout?
		The proper tangent space basis for each vertex is built from the weighted average of all the faces that share a particular vertex.
	nwn2forums.bioware.com /viewcodepost.html?post=4569982 (679 words)

	More on Differential Topology
		Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions).
		For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rn.
		An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V* of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point.
	www.artilifes.com /differential-topology.htm (1008 words)

	Paul's Projects - Simple Bump Mapping Tutorial
		After clip space, your coordinates undergo "perspective divide" and the viewport transformation and are then in "window coordinates" ready to be drawn on the screen.
		This is the vector from the vertex to the light, in tangent space.
		We have the normal in texture 0, and the normalised tangent space light vector in unit 1.
	www.paulsprojects.net /tutorials/simplebump/simplebump.html (3536 words)

	10.6.0.3 Tangent Space
		The light source is transformed into tangent space at each vertex of the polygon.
		This transformation brings coordinates into tangent space, where the plane tangent to the surface lies in the X-Y plane, and the normal to the surface coincides with the Z axis.
		Note that the tangent space transformation varies for vertices representing a curved surface, and so this technique makes the approximation that curved surfaces are flat and the tangent space transformation is interpolated from vertex to vertex.
	www.opengl.org /resources/code/samples/sig99/advanced99/notes/node140.html (308 words)

	Doom 3 normal mapping, object or tangent? - Beyond3D Forum
		On dynamic geometry, you have to find the transform from the old object space to the new object space, which I suppose can be done in pretty much the same way as tangent-space.
		CPU skinning, shadow generation, and tangent space reconstruction are the parts of Doom that take up significant time in timedemos, but there is a lot of driver overhead as well.
		If part of your world is known to exist in tangent space it's a reasonable assumption that the rest would be stored in tangent space also where possible to simplify processing.
	www.beyond3d.com /forum/showthread.php?t=12991 (786 words)

	Surface/Curve Notes
		In Calc I, the tangent of y=f(x) was specified
		parlance, we specify the tangent as being the span of
		For a curve, this is a 1-dimensional vector space whose
	coweb.math.gatech.edu:8888 /model/827 (393 words)

	Object Space Normal Mapping Tutorial
		The main idea behind using tangent space is going through extra steps to allow the reuse of a normal map texture across multiple parts of the model (storing normals/tangents/binormals at the vertices, and computing the normals of the normal map relative to them, then converting back when rendering.
		Another main disadvantage to not using tangent space is that you can't use a detail normal map for fine close-up detail in addition to the one that approximates the high resolution mesh.
		A limitation of Object Space normal mapping, as opposed to using Tangent Space, is that every point on the skin must have its own distinct UV coordinates.
	www.3dkingdoms.com /tutorial.htm (3733 words)

	P.P. Cook's Tangent Space
		The end of the talk was dedicated to the parametric space which is a new approach to noncommuative field theory described by Gurau and Rivasseau in their paper.
		In particular the conjugation properties of the relevant spinors and the necessity of removing his double fermions leads to picking the KO-dimension of the required space F, which Lubos has taken to calling the Connes manifold, to be 6mod8.
		At the end of the talk Connes told the audience that the finiteness of the space F is really tantamount to there existing a basic unit of length, and it was revealed during the questions that it was really the Euclidean version of the standard model that had been constructed.
	ppcook.blogspot.com (6197 words)

	Tangent Space Representation
		A digital curve C is represented in the tangent space by the graph of a step function, where the x-axis represents the arclength coordinates of points in C and the y-axis represents the direction of the line segments in the decomposition of C.
		We will display the tangent space as a rectangle with the parallel sides identified in the standard way to obtain a topological torus.
		[21] uses the transformation of polygonal arcs to the tangent space to determine a circular approximation of polygonal arcs.
	www.math.uni-hamburg.de /projekte/shape/tangent_space.html (622 words)

	Zariski tangent space - Wikipedia, the free encyclopedia
		In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space, at a point P on an algebraic variety V (and more generally).
		In the first case the (Zariski) tangent space to C at (0,0) is the whole plane, considered as a two-dimensional affine space.
		In the second case, the tangent space is that line, considered as affine space.
	en.wikipedia.org /wiki/Zariski_tangent_space (467 words)

	3D Programming - Weekly : Tangent Space
		Since I do use object space for many of my complex meshes, I may not be as famaliar with tangent space issues as I would be otherwise.
		Their direction is determined by the UV coordinates, one points in the direction of U-axis in 3d space, the other in the direction of the V-axis.
		Note that the 3 vectors (normal, tangent, bitangent) may not form an orthogonal basis since the tangent and bitangent directions are based on texture uv coordinates and possibly averaged from multiple vectors.
	www.3dkingdoms.com /weekly/weekly.php?a=37 (601 words)

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