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	Surface normal - Wikipedia, the free encyclopedia
		A normal to a non-flat surface at a point p on the surface is a vector which is perpendicular to the tangent plane to that surface at p.
		For a plane given by the equation ax + by + cz = d, the vector (a,b,c) is a normal.
		Surface normal at a point to a surface does not have a unique direction; the vector pointing in the opposite direction of a surface normal is also a surface normal.
	en.wikipedia.org /wiki/Normal_vector (341 words)

	Normal mapping - Wikipedia, the free encyclopedia
		But where a bump map is usually calculated based on a single-channel (interpreted as grayscale) image, the source for the normals in normal mapping is usually a multichannel image (that is, channels for "red", "green" and "blue" as opposed to just a single color) derived from a set of more detailed versions of the objects.
		To calculate the lambertian (diffuse) lighting of a surface, the unit vector from the shading point to the light source is dotted with the unit vector normal to that surface, and the result is the intensity of the light on that surface.
		Normal mapping's increasing popularity amongst video-game designers is due to its combination of excellent graphical quality and decreased processing requirements versus other methods of producing similar effects.
	en.wikipedia.org /wiki/Normal_mapping (798 words)

	Normal Vector and Curvature
		Thus, the binormal vector b(u) is perpendicular to both f'(u) and f''(u) and hence perpendicular to the osculating plane.
		The binormal vector is always perpendicular to the xy-plane while both the tangent and normal vectors lie on the xy-plane.
		Therefore, you are moving in the direction of the tangent vector, your "up" vector is in the direction of the binormal vector and the rate of turning and turning direction are given by the curvature and the direction of the normal vector, respectively.
	www.cs.mtu.edu /~shene/COURSES/cs3621/NOTES/curves/normal.html (890 words)

	Normal Vector and Curvature
		vector b is the unit-length vector of the cross-product of f'(u) and f''(u):
		The tangent line, binormal line and normal line are the three coordinate axes with positive directions given by the tangent vector, binormal vector and normal vector.
		Thus, the unit-length tangent vector is (-sin(u), cos(u), 0), the binormal vector is (0, 0, 1), and the normal vector is (-cos(u), sin(u), 0).
	www.css.tayloru.edu /~btoll/s99/424/res/mtu/Notes/curves/normal.htm (917 words)

	VECTOR ANALYSIS - Online Information article about VECTOR ANALYSIS
		As already explained, two vectors which are represented by equal and parallel straight lines drawn in the same sense are regarded as identical.
		When the sum (or difference) of two vectors is to be further dealt with as a single vector, this may be indicated by the use of curved brackets, e.g.
		The members of the second class, that of axial vectors, are primarily not vectors at all.
	encyclopedia.jrank.org /VAN_VIR/VECTOR_ANALYSIS.html (2673 words)

	Normal vector computation method - Patent 4905158
		A normal vector computation method according to claim 1, wherein said inputting in step (a) of the data specifying the three-dimensional curved surface includes inputting curve data defining four section curves for specifying an external form of the three-dimensional curved surface.
		However, since this conventional method of calculating the normal vector at the cutting point uses the coordinates of a total of five points, it takes time to obtain the normal vector at each cutting point and, as a result, a considerable period of time is required until NC data are obtained.
		This makes it possible to compute a normal vector at a cutting point using the coordinates of a fewer number of points, and enables the normal vector to be computed simply in a short period of time and with sufficient accuracy.
	www.freepatentsonline.com /4905158.html (1834 words)

	Surface Normal \| Normal Vector
		In computer graphics, manipulations of the normal vector are often used as a way to simulate geometrical detail on otherwise planar surfaces.
		In this case, a function will determine small aberrations of the true direction of the normal vector on every point of the surface, in order special create highlight or shadow effects.
		the vector is slightly shifted in accordance to a sinus function, then the surface will appear in a rendered image, as if it were made of corrugated material (except for the edges).
	www.schorsch.com /kbase/glossary/normal_vector.html (126 words)

	Normal vector shading for 3-D graphics display system - Patent 4901064
		N is the vector normal to the surface of a displayed object 15, V is the viewpoint normal vector, L is the light source vector, and R is the reflectance of the viewpoint vector through the surface normal vector.
		The interpolation components of viewpoint vector interpolator are determined by the field of view of the image, the aspect ratio of the image, the aspect ratio of the pixels, the number of pixels in each of X and Y directions, and the direction of view.
		The xinc v vector register is repeated added to the span x vector register when processing an image, thereby interpolating the span x from its initial value of left x to finally point to the last pixel on the right of the scan line.
	www.freepatentsonline.com /4901064.html (7692 words)

	Normal Map (Site not responding. Last check: 2007-11-06)
		Thus, each pixel in a normal map encodes which direction that particular point is facing - the "normal vector" of the surface.
		The red channel is used to encode normal vectors in the X direction.
		Normal maps don't contain values below 50% in the blue channel since these would be pointing behind the surface.
	members.shaw.ca /jimht03/normal.html (1258 words)

	3D Glossary-- Normal - webreference.com
		As there will always be two normals, one on each side of the surface, and pointing in opposing directions, the choice of the side from which the normal projects defines the front or "face" of the polygon.
		Normals can be associated not only with the flat surfaces of the polygons, but also with the individual points that make up the vertices where polygons meet on the surface of a model.
		Such vertex normals can be directly assigned in the model file, but are usually computed during rendering by averaging the normals of the adjacent polygons.
	www.webreference.com /3d/glossary/normal.html (377 words)

	Normal vector - DmWiki
		Normal vectors are very important for lighting calculations, as well as many other special effects, and for physics calculations.
		You can calculate the normal of a flat surface by taking any three points on the surface (for a triangle, these could be its three vertices) and then compute:
		The resulting vector will be the normal vector that is perpendicular to the surface that is formed by the three points.
	www.devmaster.net /wiki/Normal_vector (139 words)

	CSPICE Routines: PL2NVC_C
		normal, constant O A normal vector and constant defining the geometric plane represented by plane.
		normal, constant are, respectively, a unit normal vector and constant that define the geometric plane represented by plane.
		nvp2pl_c (normal, point, andplane); pl2nvc_c (andplane, normal, andconstant); 2) Apply a linear transformation represented by the matrix m to a plane represented by the normal vector n and the constant c.
	www.gps.caltech.edu /~marsdata/cspice/pl2nvc_c.html (495 words)

	Normal vector
		Vector which is perpendicular to said surface or manifold.
		In a inner product space, the inner product of the normal vector with all vectors which comprise the surface is 0.
		The text of this article is licensed under the GFDL.
	www.ebroadcast.com.au /lookup/encyclopedia/no/Normal_vector.html (50 words)

	Multivariate Normal Distribution
		During this period, there was a popular misperception that a sum of normal random variables is itself normal.
		Perhaps the simplest is this: A random vector has a joint-normal distribution if every non-trivial liner polynomial of the random vector is itself normal.
		Both of its components have marginal distributions that are normal, but the random vector is not joint-normal.
	www.riskglossary.com /articles/joint_normal_distribution.htm (525 words)

	THE CONSTANCY OF STRESS FIELDS DURING FRACTURING INDICATED BY JOINT SET NORMAL VECTOR DISTRIBUTIONS (Site not responding. Last check: 2007-11-06)
		Joint normal vectors for a set formed in a homogeneous stress field, hosted by an isotropic material and measured with a perfectly precise compass would plot as a point on a stereographic projection.
		Joint vector distributions from fold belt and basin settings are quantified with an eigenvalue method, and used to determine the relative constancy of the horizontal tectonic and vertical gravitational stresses in the rock volume over the jointing interval.
		Joint vector distributions in horizontal beds, and especially in rock sequences with low strength anisotropy, demonstrate that the vertical dimension of joints, while not constant, is better constrained in the joint set than the strike dimension in both passive and active tectonic settings.
	gsa.confex.com /gsa/2004AM/finalprogram/abstract_77505.htm (476 words)

	ipedia.com: Outer product Article (Site not responding. Last check: 2007-11-06)
		By virtue of properties (1) and (2), the vector space becomes an algebra, and by property (4) is also associative.
		A vector can be seen as a "piece" of a straight line with an orientation; a bivector is a piece of a plane with an orientation.
		If we took our vectors from an n-dimensional vector space, then we cannot get more than n LI vectors; thus, the outer product of more than n vectors is always 0, and the n-vector is the "highest order" k-vector that can be generated.
	www.ipedia.com /outer_product.html (455 words)

	untitled
		Draw the vector v in the plane, and a filament surrounding him with a given twist and in a certain interval.
		Draw the vector v in the plane, and a filament surrounding him with a width which decrements as a negative exponential, with a given twist and in a certain interval.
		Used to create a list containing the unit normal vectors, the curvatures, the kappa1 and the kappa2 in precise points of a discretisized curve.
	cso.ulb.ac.be /cso/povweb/reference.html (4159 words)

	Calculus II (Math 2414) - 3-Dimensional Space - Tangent, Normal and Binormal Vectors (Site not responding. Last check: 2007-11-06)
		While, the components of the unit tangent vector can be somewhat messy on occasion there are times when we will need to use the unit tangent vector instead of the tangent vector.
		The unit normal is orthogonal (or normal) to the unit tangent vector and hence to the curve as well.
		The binormal vector is orthogonal to both the tangent vector and the normal vector.
	tutorial.math.lamar.edu /AllBrowsers/2414/TangentNormalVectors.asp (528 words)

	MegaCads Help: Dialogbox `Create Normal Vector' (Site not responding. Last check: 2007-11-06)
		The normal vector can be attached to the start or the end point of a spline curve.
		The normal vector is computed with respect to the current working surface.
		The vector can also be normal to the spline curve and the working surface.
	www.megacads.dlr.de /docus/FctNormalVector.html (180 words)

	Equations of Lines and Planes
		If we had taken the point to be (2,2,-4) and the vector to be <-6,0,2> in the previous example we would have found the parametric equations of the line to be x=2-6t, y=2, and z=-4+2t.
		since each vector in the plane must be orthogonal to the normal vector n and the vector r-r_0 is a vector in the plane.
		By taking the cross product of the vector a from P to Q and the vector b from Q to R, we obtain a vector which is orthogonal to each of the original vectors (and thus orthogonal to the plane).
	www.math.oregonstate.edu /home/programs/undergrad/CalculusQuestStudyGuides/vcalc/lineplane/lineplane.html (768 words)

	Calculus III (Math 2415) - Surface Integrals - Surface Integrals of Vector Fields (Site not responding. Last check: 2007-11-06)
		E while the negative orientation will be the set of unit normal vectors that point in towards the region E.
		In order to work with surface integrals of vector fields we will need to be able to write down a formula for the unit normal vector corresponding to the orientation that we’ve chosen to work with. We have two ways of doing this depending on how the surface has been given to us.
		So, as with the previous problem we have a closed surface and since we are also told that the surface has a positive orientation all the unit normal vectors must point away from the enclosed region. To help us visualize this here is a sketch of the surface.
	tutorial.math.lamar.edu /AllBrowsers/2415/SurfIntVectorField.asp (1633 words)

	Note on FORTRAN and C Versions
		In the case of the routine PL2NVP (Plane to normal vector and point), the output normal vector is always a unit vector, and the output point is always the closest point in the plane to the origin.
		Suppose a plane is defined by the point P and the normal vector N, and you wish to translate it by the vector X. That is, you wish to find data defining the plane that results from adding X to every vector in the original plane.
		Suppose we have a normal vector N and constant C defining a plane, and we wish to apply a non-singular linear transformation T to the plane.
	www.gps.caltech.edu /~marsdata/req/planes.html (3251 words)

	Normal Vector
		The first step in this process is determining an equation for the normal vector
		An important realization is that the vector marked in red in Figure 3 may be represented as the quantity
		may be represented as some combination of the two vectors on either side of it.
	users.stargate.net /~zrm/usma/corneal/normal.html (91 words)

	APPLICATIONS TO ANALYTICGEOMETRY
		A normal vector for a plane is said to be perpendicular to the plane.
		In three-space, the normal vector serves the function of the slope in two-space.
		The use of vectors in the study of analytic geometry is a very effective method.
	distance-ed.math.tamu.edu /Math640/chapter2/node8.html (584 words)

	Curvature Article, Curvature Information (Site not responding. Last check: 2007-11-06)
		For a plane curve C, the curvature at a given point P has magnitude equal to the reciprocal of the radius of an osculating circle (a circle that "kisses" or closely touches the curve atthe given point), and is a vector pointing in the direction of that circle's center.
		, normal vectors, external planes etc. Gaussian curvature ishowever in fact an intrinsic property of the surface, meaning it does not depend on the particular embedding of the surface; intuitively, this means that ants living on the surface coulddetermine the Gaussian curvature.
		Curvature vector and geodesic curvature for appropriate notions of curvature of curves in Riemannian manifolds, ofany dimension.
	www.anoca.org /surface/plane/curvature.html (805 words)

	The Unit Tangent and the Unit Normal Vectors
		The derivative of a vector valued function gives a new vector valued function that is tangent to the defined curve.
		Comparing this with the formula for the unit tangent vector, if we think of the unit tangent vector as a vector valued function, then the principal unit normal vector is the unit tangent vector of the unit tangent vector function.
		As you may guess the tangential component of acceleration is in the direction of the unit tangent vector and the normal component of acceleration is in the direction of the principal unit normal vector.
	ltcconline.net /greenl/courses/202/vectorFunctions/tannorm.htm (538 words)

	Normal Vector, Osculating Plane
		This is a unit vector in n space that indicates the direction of travel, without regard to speed.
		Let the normal vector n(t) be the derivative of d(t), recording the instantaneous change in direction at each point in time.
		Subtract the two velocity vectors determined by the three points, the two connecting line segments, and we have a scaled version of the difference between direction vectors near t.
	www.mathreference.com /ca-path,norm.html (636 words)

	Documentation for XFree86 - glNormal3sv - set the current normal vector (Site not responding. Last check: 2007-11-06)
		The initial value of the current normal is the unit vector, (0, 0, 1).
		Specifies a pointer to an array of three elements: the $x$, $y$, and $z$ coordinates of the new current normal.
		The current normal is set to the given coordinates whenever glNormal is issued.
	xfree86.activeventure.org /gl/glNormal3s.3.html (323 words)

	POV-Ray: Newsgroups: povray.advanced-users: Normal vector deformation: Re: Normal vector deformation
		The normal to the -x side of a box is -x.
		The top of the box is then sheared +x so the -x and +x faces are at a 45 degree angle...but points still remain in the same xz plane as they were before, the -y and +y faces remain perpendicular to the y axis.
		The normal should be at a 45 degree angle now, <-sqrt(2)/2, sqrt(2)/2, 0> to be precise, but since the two sample points were in the same xz plane, their position relative to each other is the same, and the normal is still -x.
	news.povray.org /chrishuff-EE1B2C.22440613012001%40news.povray.org (167 words)

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