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	Projective transformation - Wikipedia, the free encyclopedia
		A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections.
		Projective transformations do not preserve sizes or angles but do preserve incidence and cross-ratio: two properties which are important in projective geometry.
		Composition of quadrilinear transformations is associative, therefore the set of all quadrilinear transformations, together with the operation of composition, form a group.
	en.wikipedia.org /wiki/Projective_transformation (2853 words)

	Projective geometry - Wikipedia, the free encyclopedia
		Projective geometry is a non-metrical form of geometry that emerged in the early 19th century.
		Projective geometry can be formulated as an axiomatic first order theory (with identity), whose universe contains "points" and "lines." Hence there are two primitive sets, one whose members are the points and the other whose members are the lines.
		Projective geometry also includes a full theory of conic sections, a subject already very well developed in Euclidian geometry (and mainly useful as a source of examination questions).
	en.wikipedia.org /wiki/Projective_geometry (1199 words)

	Various Geometries
		Affine transformations preserve collinearity of points: if three points belong to the same straight line, their images under affine transformations also belong to the same line and, in addition, the middle point remains between the other two points.
		Analytically, affine transformations are represented in the matrix form f(x) = Ax + b, where the determinant det(A) of a square matrix A is not 0.
		Projective Geometry originated in the works of Désargues (1593-1662), B.Pascal, G.Monge (1746-1818) and was further developed in the 19th century by J.V.Poncelet (1788-1867) and C.J.Brianchon (1785-1864).
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	esp@cenet description view
		The projective transformation pulse trains Xs and Ys obtained from the projective transformation operation executed by the arithmetic circuits 106X and 106Y in the above fashion are applied to machine tool servo circuitry (not shown) thereby controlling the movement of the movable element of the machine tool, such as a tool or table.
		Although the projective transformation circuit 22 has been described and illustrated in the form of hardware, it is obvious that the same may be realized by means of a microcomputer or the like.
		The output of the projective transformation circuit 32 is connected to an interpolating circuit 33 which executes pulse distribution operations on the basis of the transformation data obtained from circuit 32.
	v3.espacenet.com /textdes?&DB=EPODOC&IDX=EP71378 (2304 words)

	Transformation of coordinates (Projective; Affine; Metric)
		The metric transformations are a subset of the affine transformations.
		The affine transformations are a subset of the projective transformations.
		The projective properties are a subset of the affine properties.
	www.ping.be /~ping1339/coortf.htm (1901 words)

	cp2tform (Image Processing Toolbox)
		Use this transformation when shapes in the input image are unchanged, but the image is distorted by some combination of translation, rotation, and scaling.
		Linear conformal transformations are a subset of affine transformations.
		In a piecewise linear transformation, linear (affine) transformations are applied separately to each triangular region of the image [1].
	www-rohan.sdsu.edu /doc/matlab/toolbox/images/cp2tform.html (916 words)

	Basics (Site not responding. Last check: 2007-11-06)
		Projective geometry is concerned with incidences, that is, where elements such as lines planes and points either coincide or not.
		This is the Fundamental Theorem of projective geometry.
		Projective transformation in which it is demonstrated that parallelism is not conserved.
	www.anth.org.uk /NCT/basics.htm (1152 words)

	Geometric Transformations
		Note that the matrix form of an affine transformation is a 4-by-4 matrix with the fourth row 0, 0, 0 and 1.
		Projective transformations are the most general "linear" transformations and need to use homogeneous coordinates.
		Therefore, projective transformations are more general than affine transformations because the fourth row does not have to contain 0, 0, 0 and 1.
	www.css.tayloru.edu /~btoll/s99/424/res/mtu/Notes/geometry/geo-tran.htm (1941 words)

	Projective to Affine
		Using projective transformation causes lines that are parallel on the object plane to appear to converge in the image plane, intersecting at a vanishing points.
		In this case the projective transformation from object cordinates to image cordinates also happens to be an affine transformation.
		It is possible to recover a frontal view from the image of tilted object plane using a projective transformation that translates the object's vanishing line to infinity.
	www.cs.technion.ac.il /Labs/Isl/Project/Projects_done/VisionClasses/Vision/Camera_Geometry/node10.html (518 words)

	3.3 Projective Transformations (Site not responding. Last check: 2007-11-06)
		Projective transformations (if not affine) are not defined on all of the plane, but only on the complement of a line (the missing line is ``mapped to infinity'').
		A common example of a projective transformation is given by a perspective transformation (Figure 1).
		Strictly speaking this gives a transformation from one plane to another, but if we identify the two planes by (for example) fixing a cartesian system in each, we get a projective transformation from the plane to itself.
	www.geom.uiuc.edu /docs/reference/CRC-formulas/node16.html (179 words)

	11.3 Projective Transformations (Site not responding. Last check: 2007-11-06)
		A transformation that maps lines to lines (but does not necessarily preserve parallelism) is a projective transformation.
		Any spatial projective transformation can be expressed by an invertible 4×4 matrix in homogeneous coordinates; conversely, any invertible 4×4 matrix defines a projective transformation of the plane.
		Projective transformations (if not affine) are not defined on all of space, but only on the complement of a plane (the missing plane is ``mapped to infinity'').
	www.geom.uiuc.edu /docs/reference/CRC-formulas/node50.html (244 words)

	From projective to affine
		Starting from a projective representation, however, the structure is only determined up to an arbitrary projective transformation.
		In this case upgrading the geometric structure from projective to affine implies that one first has to find the position of the plane at infinity in the particular projective representation under consideration.
		In Figure 2.1 a projective representation of a cube is given.
	www.cs.unc.edu /~marc/tutorial/node28.html (330 words)

	PProjection3dC - Projective transformation in 3D space (Site not responding. Last check: 2007-11-06)
		The projection is represented by a 4x4 rank 3 matrix.
		If this projective transformation is a perspectivity, the singular point is the optical center of the projection.
		The result projects projective objects in the same way as they would be projected by projection 'p' at first and then by this projection.
	www.ee.surrey.ac.uk /Research/VSSP/RavlDoc/share/doc/RAVL/Auto/Basic/Class/RavlN.PProjection3dC.html (190 words)

	m_hcoords
		Recall the definition of a projective transformation: To transform a point, you choose some triple representing the point in homogeneous coordinates, multiply the triple by the matrix to get a new triple, and then take the point represented by the new triple.
		Projective geometry is concerned with theorems that mention only which lines meet at which points, and not with angles and distances.
		One good proof in the two-dimensional case is this: Given the diagram, imagine constructing a three-dimensional diagram whose perpendicular projection in two dimensions is the given diagram.
	www.math.ucla.edu /~baker/149.1.02w/handouts/m_hcoords/node11.html (1604 words)

	[No title]
		The ``stiff'' as well as the projective transformations are both special cases of topological transformations.
		It is based on the concept of projective transformation.
		Theorem: Any projective transformation of a line is fully determined by the fate of the three points on the line.
	www.nbi.dk /~kleppe/random/ggg/geo.html (337 words)

	The action of on (Site not responding. Last check: 2007-11-06)
		From the algebraic viewpoint, there is little difference between the description of the action of the complex projective group on complex projective space and the description of the action of the real projective group on real projective space.
		As a trivial consequence of this exercise we have:
		by stereographic projection from the north pole of the unit sphere.
	www.math.poly.edu /courses/projective_geometry/chapter_three/node2.html (953 words)

	ipedia.com: Fundamental theorem of projective geometry Article (Site not responding. Last check: 2007-11-06)
		In mathematics, the fundamental theorem of projective geometry states that if P n is a projective space and F and F′ are framess of P n, then there exists a unique projective transformation send...
		In case n = 1 this comes down to saying that given two ordered triples of distinct points, there is a projective transformation of the projective line taking the first triple to the second.
		This is a basic result on Möbius transformations, saying that the group they form is "triply" transitive.
	www.ipedia.com /fundamental_theorem_of_projective_geometry.html (165 words)

	Numerical C/C++
		This transformation may be 'linearised' by a change of coordinates but as it stands it is not a linear transformation.
		The transformation y = a x, is a linear transformation where the number a represents the transformation in these coordinates.
		An important application of all of this is for general transformations which may be non-linear such as projective transformation (an example is perspective) - in this case by using homogeneous notation the transformation is represented by a linear transformation and we may thus use the tools of linear algebra for this problem.
	groups.msn.com /NumericalCC/info.msnw (875 words)

	Spatial Transformations (Image Processing Toolbox)
		A transformation that may include translation, rotation, scaling, stretching and shearing.
		A transformation in which straight lines remain straight, but parallel lines converge toward "vanishing points." (The vanishing points may fall inside or outside the image--even at infinity.)
		A special case of an affine transformation where each dimension is shifted and scaled independently.
	www.eecs.umich.edu /dco/faq/matlab-6.5/help/toolbox/images/geom8.html (348 words)

	Appendix G - OpenGL Programming Guide (Addison-Wesley Publishing Company) (Site not responding. Last check: 2007-11-06)
		Vertex transformations (such as rotations, translations, scaling, and shearing) and projections (such as perspective and orthographic) can all be represented by applying an appropriate 4 × 4 matrix to the coordinates representing the vertex.
		Then, the transformation rules for normal vectors are described by the transformation rules for perpendicular planes.
		In other words, normal vectors are transformed by the inverse transpose of the transformation that transforms points.
	rush3d.com /reference/opengl-redbook-1.1/appendixg.html (902 words)

	Homogeneous Transformation Matrices
		Then f is a projective transformation if there exists an n x n matrix T of real numbers such that: (1) for all P in the domain of f, Pf is represented by PT, (2) for all P not in the domain of f, PT = 0.
		A "projection" is just one type of "projective transformation" - the latter including transformations such as translation that are unrelated to projection.
		A projective transformation T may have a hyperplane h as an axis.
	www.silcom.com /~barnowl/HTransf.htm (3989 words)

	PProjection1dC - Projective transformation in 1D space (Site not responding. Last check: 2007-11-06)
		The projection is represented by a 2 x 2 matrix.
		Creates a projective parametrization of the projective plane that transforms the projective point 'i' into the ideal point I(1,0), the projective point 'o' into the origin O(0,1), and the projective point 'u' into the unit point 'u'.
		Scales all elements in order to the determinant of the projection matrix is unit.
	www.ee.surrey.ac.uk /Research/VSSP/RavlDoc/share/doc/RAVL/Auto/Develop/Class/RavlN.PProjection1dC.html (232 words)

	Mathematics of Perspective Drawing
		The isometric projections are that class or parallel projections for which a round sphere projects to a round circle.
		Perspective transformations have the property that parallel lines on the object are mapped to pencils of lines passing through a fixed point in the drawing plane.
		The usual construction is to draw a square around the circle, and then project the perspective view of the square by finding its edges using the vanishing points and measuring points, the center by drawing the diagonals, and then sketching the projected circle by drawing it tangent to the projected square.
	www.math.utah.edu /~treiberg/Perspect/Perspect.htm (5252 words)

	[No title]
		# ############################################################################ # # pptrans(L1, L2) returns a planar projective transform that maps # the four points in L1 to the four points in L2.
		The reference # ellipsoid is specified by a, the equatorial radius in metres, and f, # the flattening.
		# ############################################################################ # # compose(p1, p2,..., pn) returns the projection that is the # composition of the projections p1, p2,..., pn.
	www.cs.arizona.edu /icon/library/src/procs/cartog.icn (583 words)

	Homework 3 (Site not responding. Last check: 2007-11-06)
		This is also known as projective transform, and the resulting value is a projective texture coordinate.
		In general, the act of applying a texture map by using a projective transformation is called projective texturing.
		For our shadow-mapping exercise, the 4x4 projective transform from object space to the light's clip space was provided to us.
	www.eecs.tufts.edu /comp/150GPU/hw3/hw3_2.html (477 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		A defines a forward transformation % such that TFORMFWD(U,T), where U is a 1-by-N vector, returns a 1-by-N % vector X such that X = W(1:N)/W(N+1), where W = [U 1] * A. T has % both forward and inverse transformations.
		X is the row vector to % be transformed, or a matrix with a vector in each row.
		U is the row vector to % be transformed, or a matrix with a vector in each row.
	www.clemson.edu /cle4_share/CWE/COES0915_CLUG/REFERENCE/matlabr14/toolbox/images/images/maketform.m (1831 words)

	maketform (Image Processing Toolbox)
		matrix whose rows are points in the transformation's output space.
		has a defined forward transform function only if all of the component transforms have defined forward transform functions.
		has a defined inverse transform function only if all of the component functions have defined inverse transform functions.
	www-rohan.sdsu.edu /doc/matlab/toolbox/images/maketform.html (348 words)

	klein view page (Site not responding. Last check: 2007-11-06)
		result, this map P is given by a projective transformation.
		projective transformation T. Combining these, we see that the perspectivity is described by ToPoR.
		a projective transformation whose matrix is the product of those for T, P and R. This shows that a perspectivity can be interpreted as a projective transformation.
	www.maths.gla.ac.uk /~wws/cabripages/klein/proofs2.html (237 words)

	Homogeous Coordinates: Methods
		For perspective transformations, an axis of rank r is dual to a center of rank n - r.
		Thus the axes of a projective transformation f are disjoint, and consist of all invariant points that share the same eigenvalue (Pedoe, p.
		Say by geometrical considerations we know an axis and corresponding center of a projective transformation f, and wish to construct a homogeneous matrix representation T of f.
	www.silcom.com /~barnowl/homogcoords.htm (3952 words)

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