
	Where results make sense

Topic: Affine line

	Affine space - Wikipedia, the free encyclopedia
		An affine space is a set with a faithful transitive vector space action, a principal homogeneous space with a vector space action.
		An affine subspace of a vector space V is a subset closed under affine combinations of vectors in the space.
		An affine transformation between two vector spaces is a combination of a linear transformation and a translation.
	en.wikipedia.org /wiki/Affine_space (1714 words)

	Proper map - Wikipedia, the free encyclopedia
		, is the affine line minus the origin and thus not closed.
		For example, the projective line is proper over a field (or even over Z) since one can always scale homogeneous co-ordinates by their least common denominator.
		A deep property of proper morphisms is the existence of a Stein factorization, namely the existence of an intermediate scheme such that a morphism can be expressed as one with connected fibres, followed by a finite morphism.
	en.wikipedia.org /wiki/Proper_map (626 words)

	Various Geometries
		Affine Geometry is not concerned with the notions of circle, angle and distance.
		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.
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	PlanetMath: affine transformation (Site not responding. Last check: 2007-09-09)
		An affine property is a geometry property that is preserved by an affine transformation.
		One can more generally define an affine transformation to be an order-preserving bijection between two affine geometries.
		This is version 23 of affine transformation, born on 2004-10-24, modified 2006-06-16.
	planetmath.org /encyclopedia/AffineTransformation.html (387 words)

	Math 371 Affine Geometry Notes (Site not responding. Last check: 2007-09-09)
		There is a one to one correspondence between the points this line and the points on a semicircle tangent at its midpoint to the point P0 on the line.
		T assigns the point Px on the line to the unique point Qx on the semicircle that lies on the segment PxO where O is the center of the semicircle.
		T assigns the point Px on the line to the unique point Qx on the circle that lies on the segment PxQ* where Q* is the point on the diameter of the circle opposite P0.
	www.humboldt.edu /~mef2/Courses/AffineGeometryNotes.html (1692 words)

	Summary
		N-dimensional affine space is invariant under the n(n+1)parameter affine group of translations and linear deformations (rotations, non-isotropic scalings and skewings).
		Affine space is obtained from projective space by fixing an arbitrary hyperplane to serve as the hyperplane of points `at infinity': requiring that this be fixed puts n constraints on the allowable projective transformations, reducing them to the affine subgroup.
		Affine space has notions of `at infinity', sidedness/betweenness, and parallelism (lines meeting at infinity), but no notion of rigidity, angle or absolute length.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/MOHR_TRIGGS/node45.html (326 words)

	COMP 290-001: Lecture Notes: Design Example for an Affine-Geometry Kernel
		An affine transformation is a linear mapping from an affine space to an affine space that preserves affine combinations.
		This implies that the affine transformation of the difference of two points (a vector) is the same as the difference of two affine transformed points.
		Assuming the reference frame for the new space is given in the reference frame of the points, the affine transformation is easily specified as follows; the base point of the reference frame maps to the origin and each vector in the basis of the reference frame maps to the unit vectors of the standard basis.
	photon.poly.edu /~hbr/cs903-F00/lib_design/notes/gkernel.html (1098 words)

	Affine Transformations
		The transformations that move lines into lines, while preserving their intersection properties, are special and interesting, because they will move all polylines into polylines and all polygons into polygons.
		Every affine transformation can be expressed as a transformation that fixes some special point (the "origin") followed by a simple translation of the entire plane.
		The fill-in text lines in the top half of the dialog specify parameters for the buttons: scale factors, translation components, and rotation angle (in degrees counterclockwise).
	www.quantdec.com /GIS/affine.htm (3073 words)

	Math 371 Geometry Notes on Line (Site not responding. Last check: 2007-09-09)
		The line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is congruent to one half of the third side.
		If a line, l, (or circle, O'A') has at least one point inside a given circle OA and one point outside the same given circle then there is of a point on the line (circle) that is also on the given circle.
		Homogenous coordinates for an affine line or plane (in 1 and 2 dimensions).
	www.humboldt.edu /~mef2/Courses/m371notes04.html (8713 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.
		Parallel lines on the object translate to lines in the image that converge to a vanishing point, all of which lie on the vanishing line associated with the object plane.
		For a pinhole camera image, the location and orientation of the vanishing line of a an object plane determins the orientation of the plane with respect to the camera's line of sight.
	www.cs.technion.ac.il /Labs/Isl/Project/Projects_done/VisionClasses/Vision/Camera_Geometry/node10.html (518 words)

	[No title] (Site not responding. Last check: 2007-09-09)
		In particular, it is concerned with projective and affine coordinates of points and lines.
		The line whose first two coordinates are zero is the line at infinity.
		The affine coordinates (1/3,2/3) are expressed as follows for the point A with projective coordinates (1,2,3).
	www.win.tue.nl /~amc/oz/om/cds/plangeo4.html (183 words)

	M6221 Lecture Notes 1e
		The lines of A are the lines of P not in
		The rank 2 geometry of points and lines of A is called an affine space of dimension d.
		The line labeled a of the projective space becomes the affine line whose points are 0001 and 1001.
	www-math.cudenver.edu /~wcherowi/courses/m6221/pglc1e.html (432 words)

	The Point-Set and Line-Set of a Plane
		An affine or projective plane created by Magma consists of three objects: the point-set V, the line-set L and the plane P itself.
		Given a line l of the plane P, return a representive point of P which is incident with l.
		Given a line l of the plane P, return a random point of P which is incident with l.
	www.math.ufl.edu /help/magma/text512.html (680 words)

	The Connection between Projective and Affine Planes (Site not responding. Last check: 2007-09-09)
		The projective completion of the affine plane P, together with the point set and line set of the projective plane, plus the embedding map from P to the projective plane.
		Given any point or line of the projective plane, provided that it is not on the adjoined line at infinity, the preimage in the affine plane can be found.
		is the line at infinity for this embedding.
	www.umich.edu /~gpcc/scs/magma/text1227.htm (345 words)

	Joe Stegman's WebBlog
		There are lots of resources on affine transformations, so I’m not going to go into great details but will provide the very basics you need to know to make use of a MatrixTransform in WPF/E (or WPF).
		An affine transformation is a linear transformation - meaning if the input is a line, the output will be a line.
		Affine transformations are used to rotate, skew, scale and translate a set of points and are described using a matrix based mathematical formula.
	blogs.msdn.com /jstegman (529 words)

	Involutions
		Deduce that on the projective line, all involutions with one fixed point have two fixed points, and thence that they map points to their harmonic conjugates with respect to these two fixed points.
		Given the invariance of the cross ratio under complex collineations, show that all involutions of the projective line map points to their harmonic conjugates with respect to the two fixed points of the involution.
		Also, any line through any two points of the figure meets the line through the two opposite points on the axis: hence the axis is easy to find given a few correspondences.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/MOHR_TRIGGS/node36.html (362 words)

	[No title]
		Lines that were parallel in the ordinary plane will intersect at one of the points at infinity.
		In other words, draw a line N though 1' and x, draw a line M' parallel to M through the point y, and draw a line N' parallel to N through the point where L' and M' intersect.
		A "collineation" is a map from a projective plane to itself that preserves all lines.
	math.ucr.edu /home/baez/twf_ascii/week145 (2920 words)

	Matlab tutorial and Linear Algebra Review
		•Line: sum of a point and a vector
		• Line Segment: For 0 £ a1, a2 £ 1, P lies between P1 and P2
		•Line: set of points equidistant from the origin in the direction of a unit vector.
	www.cs.umd.edu /~djacobs/CMSC427/Geometry_files/slide0050.htm (46 words)

	The Point-Set and Line-Set of a Plane
		An affine or projective plane in Magma consists of three objects: the plane P itself, the point--set V of P, and the line--set L of P. Although called the point--set and line--set, V and L are not actual Magma sets.
		Given the line set L of a classical plane P defined over a finite field K, and elements a, b, c of K, create the line (i.e.
		Given the line--set L of a plane P and a line l of a (possibly) different plane (generally a subplane of P), return the line of P corresponding to l.
	www.umich.edu /~gpcc/scs/magma/text1219.htm (930 words)

	VIVA (Viewpoint Invariant Visual Acquisition) (Site not responding. Last check: 2007-09-09)
		For curved-line fragments subjected to affine transformations, the law is equivalent to requiring affine relative invariance.
		In collaboration with Leuven and Lund, eight geometric attributes of curved contours having one symmetry axis and parallel chords were analysed for invariance of the Weber fraction under symmetry-preserving affine transformations.
		For each of the curved-line attributes, increment threshold Dc was determined as a function of c under combinations of enlargements or reductions and simple elongations or compressions of the curved lines.
	www.esat.kuleuven.ac.be /~konijn/news3.3.html (400 words)

	[No title]
		is a curve of degree 2, and lines have degree 1.
		In other words, we add one point for each direction of a line through the origin of the affine plane.
		In the affine case this was easy: we simply took lists of polynomials (that clearly gave maps from one affine space to another), restricted them to an algebraic subset of the domain, and looked at conditions that ensured that the image landed in another algebraic set.
	odin.mdacc.tmc.edu /~krc/agathos/proj.html (999 words)

	Standard Properties
		Dual basis of the affine hull of the polyhedron.
		Dimension of the affine hull of the polyhedron = dimension of the polyhedron.
		Coordinates in affine 3-space of the vertices which correspond to a 3-dimensional (Schlegel-) projection of a 4-polytope.
	cgm.cs.mcgill.ca /labdocs/polymake-1.5.1/apps/polytope/std_sections.html (1874 words)

	Glossary
		An affine or projective algebraic set is called a variety if it is irreducible in its Zariski topology i.e.
		The functor X → A(X) is an antiequivalence of the full subcategory of affine varieties and the category of affine domains over k (finitely generated k-algebras which are integral domains).
		The function field of an affine variety is the quotient field of its coordinate ring.
	www.math.purdue.edu /~dvb/algeom2.html (786 words)

	Operations on Points and Lines
		Given a line l of the plane P, return a representative point of P which is incident with l.
		Recall that in a classical plane (where a, b, c in K) represents the line given by the equation ax + by + cz = 0 in a projective plane or ax + by + c = 0 in an affine plane.
		Given a line l = from a classical plane P (projective or affine), return the sequence [a, b, c] of coordinates of l.
	magma.maths.usyd.edu.au /magma/htmlhelp/text1453.htm (1138 words)

	ee_linear_geom
		Explain how any line written functionally can be easily rewritten in (a) the usual relational form; (b) in parametric form.
		The relational and parametric representations of a line can each be interpreted in terms of affine functions between spaces.
		Break the plane into regions (some of which could be single points or pieces of lines) based on outcomes that actually occur.
	www.math.ucla.edu /~baker/149.1.03s/handouts/ee_linear_geom/node11.html (972 words)

	IM v6 Examples -- Drawing
		Note that if the line is too long to fit the ellipse at the angle given, the size of the ellipse will be enlarged to fit the line, with the ellipse centered on the line.
		The line connecting the control point to the final point on the path of that path segment (control line) basically defines the direction of the curve though that point on the path.
		However these command line versions are operators and are applied immediately to images already existing in memory rather that to a drawn surface only which vector objects have yet to be drawn.
	www.cit.gu.edu.au /~anthony/graphics/imagick6/draw (7888 words)

	On-Line Geometric Modeling Notes
		These notes discuss affine combinations of points, barycentric coordinates of points and vectors, convex combinations, convex sets, and the convex hull of a set of points.
		Points in an affine space are utilized to position ourselves within the space.
		The operations on the vectors of an affine space are numerous - addition, scalar multiplication, dot products, cross products - but the operations on the points are limited.
	www.css.tayloru.edu /~btoll/s99/424/res/ucdavis/CAGDNotes/Affine-Barycentric-and-Convex/Affine-Barycentric-and-Convex.html (673 words)

	The Point-Set and Line-Set of a Plane
		An affine or projective plane in Magma consists of three objects: the plane P itself, the point-set V of P, and the line-set L of P. Although called the point-set and line-set, V and L are not actual Magma sets.
		For efficency and clarity, the points and lines of a plane are given special types in Magma.
		Given the line-set L of a plane P and a line l of a (possibly) different plane (generally a subplane of P), return the line of P corresponding to l.
	www.math.wisc.edu /help/magma/text657.html (938 words)

	Copyright (c) 1996, 1997, 1998 Thomas E. Burge. All rights reserved.
		The Affine Toolkit is published to provide some useful tools to folks using the RenderMan Standard.
		Included in the Affine Toolkit is a set of libraries for reading and writing RIB files.
		The Affine Toolkit is based on only published materials regarding the RenderMan Standard and the general subject of computer graphics.
	www.affine.org (1735 words)

	Sketch
		has split the line into two pieces and ordered the three resulting objects so that the correct portion of the line is hidden.
		Again, 60 segments of the helix are produced by connecting 61 instances of the swept line.
		The options of the swept line, if any, are applied to the faces produced by sweeping the line, but not the end polygons.
	www.math.utah.edu /pub/ctan/tex-archive/graphics/sketch/Doc/manual.html (6915 words)

	Equaffine
		A line in the plane can be described under one of 4 following forms: an explicit function, an implicit equation, a pair of parametric equations, or two points on the line.
		A situation that generalises over affine subspaces of higher dimension (such as a line or a plane in the space).
		With the variation of dimension, it can be used either in very elementary levels (line in the plane), or right up to situations requiring complicated computations of linear algebra.
	wims.unice.fr /wims/en_H5~geometry~equaffine.en.html (245 words)

Try your search on: Qwika (all wikis)