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Topic: Projective line

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	PlanetMath: projective geometry
		The study of projective planes extends beyond the consideration of the poset of vector spaces however, and is the beginning of many interesting combinatorial problems.
		Indeed, already for 1-dimensional geometries, so called projective lines, i.e.: a set of points, it is clear that not all such geometries can be captured as the subspaces of a vector space.
		This is version 21 of projective geometry, born on 2006-04-18, modified 2006-06-25.
	planetmath.org /encyclopedia/ProjectiveGeometry3.html (880 words)

	Springer Online Reference Works
		Parallel lines are completed by one and the same improper point, non-parallel lines by distinct improper points, parallel planes by one improper line, and non-parallel planes by distinct lines.
		Projective space is regarded as a set of elements of three kinds: points, lines and planes, between which an incidence relation, basic for projective geometry, satisfying appropriate axioms, is established.
		One says that a point and a line (a line and a plane, a point and a plane) are incident if the point lies on the line (the line passes through the plane, etc.).
	eom.springer.de /p/p075240.htm (826 words)

	Transformation of coordinates (Projective; Affine; Metric)
		Projective properties are for instance collinearity of points; concurrency of lines.
		The projective properties are a subset of the affine properties.
		If the lines defined by the three pairs of corresponding vertices of two triangles are concurrent, then the intersection points of the three pairs of corresponding sides of the triangles are collinear.
	www.ping.be /~ping1339/coortf.htm (1901 words)

	Question Corner -- Understanding Projective Geometry
		The lines of projective space are lines l in Euclidean space together with the extra object f(l) attached.
		Projective geometry can be thought of as the collection of all lines through the origin in three-dimensional space.
		What all this means is that, in projective space, the "line" corresponding to l is actually a family of lines through the origin consisting of: (1) the lines that pass through l, and (2) the limiting horizontal line.
	www.math.toronto.edu /mathnet/questionCorner/projective.html (2444 words)

	Homogenous and inhomogenous coordinates
		Points on the Euclidean line may be described by a single number, after we choose a point to call 0 and a point to call 1.
		The equation of a line in homogeneous coordinates is ax+by+cz = 0.
		The projections on the x-axis are 1/2, 1, 2/3 and 0, whose cross ratio is
	www.math.fau.edu /Richman/Geometry/coords.htm (1180 words)

	Math 371 Geometry Notes on line
		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.
		The axioms for projective geometry in a plane uses two basic objects: points and lines, and a relation between those: a point is on a line, or a line passes through a point.
		Consider lines connecting corresponding points in a pencil of points on a line related by a projectivity (not a perspectivity) and noticed that the envelope of these lines seemed to be a conic, a line conic.
	www.humboldt.edu /~mef2/Courses/m371notes.html (9950 words)

	Coordinate Projective Geometry
		That gives us a way to think of the projective plane without using coordinates: Pick a fixed point O in Euclidean space, consider the lines O to be the points of a projective plane, and the planes through O to be the lines of that projective plane.
		A projective point, which is a Euclidean line L, is on a projective line, which is a Euclidean plane P, exactly when the line L is in the plane P.
		Each line in E is in exactly one Euclidean plane through (0,0,0), so can be thought of as the projective line corresponding to that plane.
	www.math.fau.edu /richman/Geometry/projcoor.htm (995 words)

	Finite Geometries?
		A model of a line segment, such as a stick of thin spaghetti, can be subdivided into smaller and smaller pieces, suggesting that there are infinitely many points that make up the line segment or spaghetti stick.
		In an abstract framework where points and lines are treated as undefined terms and rules reflect the desired properties of these points and lines, then in a geometry where there are parallel classes of lines, that is, sets of lines where no pair of lines within a class meet, then one can proceed as follows.
		If one is given a projective plane, then one can reverse this construction by taking any line m and removing this line and modifying each of the lines of the plane by removing the one point on any line where that line meets m.
	www.ams.org /featurecolumn/archive/finitegeometries.html (5303 words)

	On-Line Computer Graphics Notes
		The primary use of clipping in computer graphics is to remove objects, lines or line segments that are outside the viewing volume.
		This test is used to construct a clipping algorithm for line segments, and the line-segment clipping is used to develop the polygon-clipping algorithm.
		But we have all the machinery now: We have shown how to clip line segments when our planes and points are in three-dimensional space and have shown how to clip against the image-space cube; and we have also seen that we can (at least in a few cases) clip in projective space.
	graphics.cs.ucdavis.edu /education/GraphicsNotes/Clipping/Clipping.html (2723 words)

	Homogeneous coordinates
		To calculate with points and figures in the projective plane we introduce homogeneous coordinates which allow us to see the relationship between points at infinity and ordinary points.
		Then every line through P is identified with a point of the line y = 1 except the x-axis itself.
		Of course, if one projects so that the line at infinity is projected into an ordinary line, it is possible to project a parabola into an ellipse, etc.
	www-groups.dcs.st-and.ac.uk /~john/MT4521/Lectures/L17.html (498 words)

	Octonionic Projective Geometry
		Projective geometry is a venerable subject that has its origins in the study of perspective by Renaissance painters.
		We have already met one example of a projective plane in Section 2.1: the smallest one of all, the Fano plane.
		The dimension of a projective space is defined to be one less than the minimal cardinality of a set that spans the whole space.
	math.ucr.edu /home/baez/octonions/node8.html (1306 words)

	Projective stratum
		The group of projective transformations or collineations is the most general group of linear transformations.
		Relations of incidence, collinearity and tangency are projectively invariant.
		A similar cross-ratio invariant can be derived for four lines intersecting in a point or four planes intersecting in a common line.
	www.cs.unc.edu /~marc/tutorial/node26.html (199 words)

	On the Witt ring of a relative projective line, by Marek Szyjewski (Site not responding. Last check: 2007-09-07)
		A ring homomorphism e^0 : W(X) --> E^+(X) from the Witt ring of a scheme X into suitable subfactor of K_0(X) and E^+(X) itself are studied for general projective bundle and split affine quadric.
		As an application non-extended Witt class on a projective line over a coordinate ring of affine quadric of dimension congruent to 6 mod 8 is constructed.
		This shows that Arason theorem that Witt ring af a projective space over a field equals to Witt ring of this field can not be generalized to projective spaces over regular rings.
	www.math.uiuc.edu /K-theory/0149/index.html (116 words)

	Projective geometry - Wikipedia, the free encyclopedia
		This is, a projectivity is any conceivable invertible linear transform of homogeneous co-ordinates.
		The non-Euclidean geometries discovered shortly thereafter were eventually demonstrated to have models, such as the Klein model of hyperbolic space, relating to projective geometry.
		Varying a single parameter, lambda, metamorphoses the interaction of what are known in projective geometry as growth measures into surprisingly accurate representations of many organic forms not otherwise easily describable mathematically; negative values of the same parameter produce inversions representing vortices of both water and of air.
	en.wikipedia.org /wiki/Projective_geometry (2674 words)

	Charles Olson, Projective Verse 1950
		I want to do two things: first, try to show what projective or OPEN verse is, what it involves, in its act of composition, how, in distinction from the non-projective, it is accomplished; and II, suggest a few ideas about what that stance does, both to the poet and to his reader.
		Each of these lines is a progressing of both meaning and the breathing forward, and then a backing up, without a progress or any kind of movement outside the unit of time local to the idea.
		The dimension of his line itself changes, not to speak of the change in his conceiving, of the matter he will turn to, of the scale in which he imagines that matter’s use.
	www.angelfire.com /poetry/jarnot/olson.html (3119 words)

	Real projective line - Wikipedia, the free encyclopedia
		The geometric interpretation is this: a vertical line may have, so to speak, infinite gradient.
		The real projective line extends the field of real numbers in the same way that the Riemann sphere extends the field of complex numbers, by adding a single point called conventionally ∞.
		The terminology projective line is appropriate, because the points are in 1-1 correspondence with one-dimensional linear subspaces of R
	en.wikipedia.org /wiki/Real_projective_line (937 words)

	Introduction to Informal Projective Geometry
		A projective plane is the geometric object made up of the collection of P-points and P-lines.
		A P-point lies on a P line or a P -line passes through the the P-point if and only if Aa+Bb+Cc= 0 where [A,B,C] are homogeneous coordinates for the P-line and are homogeneous coordinates for the P-point.
		All of the discussion works as long as the symbols A,B,C, a,b, and c represent elements of a field, that is, a set with two operations that work like the real or rational numbers in terms of addition and multiplication.
	www.humboldt.edu /~mef2/Courses/m371_Informal_Proj_Geom.html (473 words)

	Projective spaces
		These ideas were developed into the notion of projective geometry which was studied in the 17th century by (among other) Girard Desargues (1591 to 1661).
		A line from P parallel to the line m does not meet m and so no point on m corresponds to the point x on l.
		Thus a projective line is really rather like circle and when we add a line at infinity to the plane this is more like a circle at infinity.
	www-groups.dcs.st-and.ac.uk /~john/MT4521/Lectures/L16.html (460 words)

	Mathematics of Perspective Drawing
		Because light reflecting off the object travels in straight lines, the object point is seen on the drawing plane at the point where the line from the eyepoint to the object point intersects the drawing plane.
		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)

	Springer Online Reference Works
		The study of line bundles on projective varieties is a classical problem in algebraic geometry (cf.
		Certain special facts are known for algebraic vector bundles on algebraic surfaces and projective spaces [5].
		Many new ideas in the theory of algebraic vector bundles on algebraic curves and projective spaces were inspired by theoretical physics (twistor theory, Yang–Mills theory and string theory).
	eom.springer.de /v/v096390.htm (1313 words)

	Isomorphisms
		Returns a 3 x 3 matrix M defining a parametrization of the conic C via as a projective change of variables from the 2-uple embedding of a projective line in the projective plane, i.e.
		Given a projective line P, a curve C of genus zero, and a point or degree 1 place p on C, returns an isomorphism X -> C parametrizing C. If no rational point or place is given, and the curve C is not over Q or has no point, then an error results.
		The parametrization function takes a projective line as first argument, which will be used as the domain of the parametrization map.
	www.umich.edu /~gpcc/scs/magma/text1050.htm (1125 words)

	Polynomials, symmetry, and dynamics: An undertaking in aesthetics
		This pair of lines corresponds to a pair of antipodal vertices of the dodecahedron.
		The attracting line is the intersection of R, a 10-plane with two zero coordinates and the 10-line at infinity-the light gray basin.
		The green horizontal line corresponds to the intersection of the reflection plane R and the 36-line passing through the pair of green 72-points from the basin plot.
	www.mi.sanu.ac.yu /vismath/crass/index.html (5557 words)

	The Livingston Group: About Our Training
		For example, I now am better able to ad-lib a projective line of questioning to explore an unexpected attitude that pops up in a focus group.
		In Sharpen the Focus, I learned that projective techniques can be applied in many more settings and situations than I had been using them for.
		Their creativity in presenting the projective techniques as well as the development of The Seven Dwarfs personality types will definitely give me the confidence and edge I was looking for when conducting focus groups.
	www.tlgonline.com /tes/1.shtml (1648 words)

	Matt Baker - Abstracts (Site not responding. Last check: 2007-09-07)
		After describing in detail the topological structure of the Berkovich projective line, we introduce the Hsia kernel, the fundamental kernel for potential theory, and define a Laplacian operator on the Berkovich projective line.
		We then develop a theory of capacities, harmonic, and subharmonic functions, all of which are strikingly similar to their classical complex counterparts.
		Finally, we give some applications to non-archimedean dynamics, including the construction of a canonical probability measure on the Berkovich projective line attached to a rational function of degree at least 2.
	www.math.gatech.edu /~mbaker/BERKLINE (316 words)

	Notes on Projective Geometry by B. Csikós (Site not responding. Last check: 2007-09-07)
		The projective space associated to a linear space.
		The incidence axioms of an n-dimensional projective space.
		Collineations induced by automorphisms of F. The Fundamental Theorem of projective geometry.
	www.cs.elte.hu /geometry/csikos/proj/proj.html (104 words)

	Amazon.com: Projective Ornament: Books: Claude Bragdon (Site not responding. Last check: 2007-09-07)
		Complete with charming line drawings of historical architecture and new, geometrically playful forms, this is a book artists and beauty-seekers today will continue to find provocative.
		It introduces the reader to the simple concepts of projective geometry and set theory as it has been applied to such devices as magic squares and Celtic knotwork.
		Beyond that, it is illustrated with fantastic line drawings in Art Nouveau style showing how these aesthetic sources can be applied not just to architecture, but costume and ornament design.
	www.amazon.com /Projective-Ornament-Claude-Bragdon/dp/048627117X (1087 words)

	Projective Story Telling Cards
		You have just discovered what on the surface may appear to be just another in a long line of projective
		story telling techniques, but be absolutely assured that these Projective
		The limtations for their use lies only in the hands of the user.
	www.projectivestorytelling.com /index.html (104 words)

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