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Topic: Pencil projective geometry

	Geometry - LoveToKnow 1911 (Site not responding. Last check: 2007-11-04)
		Pythagoras, seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the, known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization, fractional, it is true, of the amorphous collection of material at hand.
		Pythagorean geometry was essentially a geometry of areas and solids; its goal was the regular solids the tetrahedron, cube, octahedron, dodecahedron and icosahedronwhich symbolized the five elements of Greek cosmology.
		Pencils in InvolutionThe theory of involution may at once be extended from the row to the flat and the axial pencilviz, we say that there is an involution in a flat or in an axial pencil if any line cuts the pencil in an involution of points.
	www.1911encyclopedia.org /G/GE/GEOMETRY.htm (20221 words)

	Geometry - Search View - MSN Encarta
		The Bolyai-Lobachevsky geometry, often called hyperbolic non-Euclidean geometry, describes the geometry of a plane consisting only of the points on the inside of a circle in which all possible straight lines are chords of the circle.
		Another important 17th-century development was projective geometry, the investigation of the properties of geometrical figures that do not vary when the figures are projected from one plane to another.
		Descriptive geometry is the science of making accurate, two-dimensional drawings, or representations, of three-dimensional geometrical forms and of graphically solving problems relating to the size and position in space of such forms.
	encarta.msn.com /text_761569706__1/Geometry.html (5322 words)

	PROHIBITION (Lat. proh... - Online Information article about PROHIBITION (Lat. proh...
		The projections of the points of intersection of two curves are the points of intersection of the projections of the given curves.
		The projection of a tangent to a curve is a tangent to the projection of the curve.
		It follows In two projective planes there are in general two and only two pencils in either such that angles in one are equal to their corresponding angles in the other.
	encyclopedia.jrank.org /PRE_PYR/PROHIBITION_Lat_prohibere_to_pr.html (5566 words)

	What Is Geometry?
		The power of geometry, in the sense of accuracy and utility of these deductions, is impressive, and has been a powerful motivation for the study of logic in geometry.
		Although the word geometry derives from the Greek geo (earth) and metron (measure) [Words], which points to its practical roots, Plato already knew to differentiate between the art of mensuration which is used in building and philosophical geometry [Philebus (57)].
		However, depending on intuition may be misleading, as, for example, in projective geometry, according to the Duality Principle, all occurrences of the two terms in the axioms and theorems are interchangeable.
	www.cut-the-knot.org /WhatIs/WhatIsGeometry.shtml (1348 words)

	PlanetMath: pencil
		A pencil is a set of geometric objects, usually either congruent or similar to each other, that share a common incidence property.
		A pencil of lines usually means a set of straight lines that are incident with one point.
		This is version 2 of pencil, born on 2005-06-22, modified 2005-06-22.
	planetmath.org /encyclopedia/FlatPencil.html (181 words)

	Projective Geometry -- from Wolfram MathWorld
		In older literature, projective geometry is sometimes called "higher geometry," "geometry of position," or "descriptive geometry" (Cremona 1960, pp.
		The most amazing result arising in projective geometry is the duality principle, which states that a duality exists between theorems such as Pascal's theorem and Brianchon's theorem which allows one to be instantly transformed into the other.
		More generally, all the propositions in projective geometry occur in dual pairs, which have the property that, starting from either proposition of a pair, the other can be immediately inferred by interchanging the parts played by the words "point" and "line."
	mathworld.wolfram.com /ProjectiveGeometry.html (376 words)

	Various Geometries
		Another approach to defining and classifying various geometries was introduced, in 1872, by Felix Klein (1849-1925) in the inaugural address he gave upon appointment to the Faculty and Senate of the University of Erlanger.
		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).
		By analogy with the 3-dimensional case, a (planar) pencil of lines is the set of lines in the same plane that pass through the same point.
	www.cut-the-knot.org /triangle/pythpar/Geometries.shtml (2183 words)

	Fundamental Theorem
		A', B', C' are elements of a pencil with axis p'. Further, assume the points are distinct and the axes p and p' are distinct.
		Axiom 6 implies that a projectivity on a pencil that leaves three elements of the pencil invariant is the identity mapping.
		A projectivity between two distinct pencils of points with a common element that corresponds to itself is a perspectivity.
	www.mnstate.edu /peil/geometry/C4ProjectiveGeometry/11fundthm3.htm (675 words)

	Projection - LoveToKnow 1911 (Site not responding. Last check: 2007-11-04)
		Projection in this sense, when treated by co-ordinate geometry, leads in its algebraical aspect to the theory of linear substitution and hence to the theory of invariants and co-variants (see Algebraic Forms).
		The projection of a line (straight line) is a line; for all points in a line are projected by rays which lie in the plane determined by S and the line, and this plane cuts the plane 7r' in a line which is the projection of the given line.
		These curves appear thus as sections of a circular cone, for in case that the two planes of projection are separated the rays projecting the circle form such a cone.
	www.1911ency.org /P/PR/PROJECTION.htm (9350 words)

	Projective Geometry (3) (Site not responding. Last check: 2007-11-04)
		Projection center O, segment ST and segment S'T' correspond to the camera, ground and Panel-screen that we saw in chapter 29.
		And the fl pencil of lines a, b, c and d, and the red pencil of lines a' b' c' and d' are drawn.
		The defference between axioms in projective geometry and axioms in Euclidean geometry is as follows.
	www1.kcn.ne.jp /~iittoo/us27b_infi.htm (3268 words)

	Xah: Introduction to Real Projective Plane
		The dual of a range is a pencil, consisting of the lines through one point: the possible positions of a variable line x (which rotates about the point).
		In affine (or Euclidean) geometry, the line p (through O) parallel to o would be exceptional, for it would have no corresponding point on o; but when we have extended the affine plane to the projective plane, the corresponding point P is just the point at infinity on o.
		In affine geometry the point X makes an infinite jump; but in projective geometry its motion, through the single point at infinity, is continuous.
	xahlee.org /projective_geometry/projective_geometry.html (6397 words)

	Math 371 Geometry Notes on line (Site not responding. Last check: 2007-11-04)
		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.
		For a projectivity (the composition of a finite number of perspectivities) the result is that a projectivity is completely determined by the correspondence of three points.
		This is not an isometry in this geometry because it is orientation reversing.
	www.humboldt.edu /~mef2/Courses/m371notes.html (9950 words)

	Projective Geometry
		The cross ratio of a given pencil of lines is equal to the cross ratio of any pencil of points related to the pencil of lines by a perspectivity.
		The cross ratio of a given pencil of points is equal to the cross ratio of any pencil of lines related to the pencil of points by a perspectivity.
		We know that the pencil of lines and the pencil of points in the figure have the same cross ratio.
	www.bsu.edu /web/mdlade/Project (823 words)

	Amazon.com: Projective Geometry: Books: H.S.M. Coxeter (Site not responding. Last check: 2007-11-04)
		Projective geometry is simpler: its constructions require only a ruler.
		In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity.
		The plane geometry of the first six books of Euclid's Elements may be described as the geometry of lines and circles: its tools are the straight-edge (or unmarked ruler) and the compasses.
	www.amazon.com /Projective-Geometry-H-S-M-Coxeter/dp/0387965327 (745 words)

	Introduction to Projective Geometry. Sixth page (Site not responding. Last check: 2007-11-04)
		Since the nondegenerate conics are all equivalent to the circle under projective (and even perspective) transformations, we may think that the projective geometry of conics is rather dull.
		Projective duality not only interchanges points and lines, it interchanges the set of points lying on a curve for the set of lines tangent to the curve.
		The converses of the previous propositions are the master theorems in the projective study of conics.
	www.math.poly.edu /~alvarez/teaching/projective-geometry/Inaugural-Lecture/page_6.html (487 words)

	Exercises 8
		Show that any three distinct points on a projective line can be mapped to any three distinct points on a second line by positioning the lines suitably in a plane and projecting from a point not on either line.
		A pencil of lines is a set of lines through a common point.
		If a line l meets the pencil OA, OB, OC, OD as shown, show that the cross-ratio of the four points of intersection does not depend on which line l we take.
	www-groups.dcs.st-and.ac.uk /~john/geometry/Tutorials/T8.html (319 words)

	Fields of Mathematics (Site not responding. Last check: 2007-11-04)
		Geometry is a skill of the eyes and the hands as well as of the mind.
		Whereas at the outset geometry is reported to have concerned herself with the measurement of muddy land, she now handles celestial as well as terrestrial problems: she has extended her domain to the furthest bounds of space.
		Geometry, which is the only science that it hath pleased God hitherto to bestow on mankind.
	www.chemistrycoach.com /fields_of_mathematics.htm (11624 words)

	Citations: Projective Geometry - Veiblen, Young (ResearchIndex) (Site not responding. Last check: 2007-11-04)
		We use cross ratios as the invariants of the perspective projection in the representation scheme for 2 D planar objects.
		....3.2 Projective Invariant For the projective case, geometric properties of the shape of a planar object should be invariant under a change in the point of view.
		From the classical projective geometry we have that the cross ratio of sines between five points on a plane is a projective invariant
	citeseer.ist.psu.edu /context/160982/0 (915 words)

	Projective Geometry (Site not responding. Last check: 2007-11-04)
		In Euclidean geometry can be described with lines and circles, its tolls being the straight-edge and the compass.
		In Euclidean geometry there is also a measure of distance and compare figures by measuring them.
		Before looking at the concept of duality in projective geometry, let's look at a few theorems that result from these axioms.
	cgm.cs.mcgill.ca /~athens/cs507/Projects/2002/ChrisElliott/PrGeo.html (492 words)

	Research/Chain Geometry
		The aim of the project is to investigate various aspects of chain geometry, in particular the (generalized) chain geometries arising from the projective line P(R) over a ring R containing a field F which is not necessarily commutative.
		A projective model is given by a hyperbolic linear congruence of lines in a projective 3-space over F, i.e., the set of all lines that meet two skew axes.
		The projective line over this ring is represented by lines of a special linear complex, i.e., the set of all lines in a 3-dimensional projective space that meet a fixed axis; only this axis represents no point of the projective line over the ternions.
	www.geometrie.tuwien.ac.at /havlicek/proj303.html (1607 words)

	Projective Geometry
		The aim of this text is to introduce the basic concepts of projective geometry in their relations to some important computer vision problems.
		However, once the fundamental notions are fully understood, it will be appreciated as a particularly convenient setting for the treatment of some of the important fundamental vision problems, such as the calibration, stereo motion estimation problems.
		, the projective space, the duality is between points (thought of as the pencils of planes) and planes (thought of as the range of points) where a plane is defined by a four-tuple
	www.lems.brown.edu /~leymarie/Hakan/projective.html (1061 words)

	Projective Geometry
		We have a set of points known as a plane with special subsets known as lines, sets which are unions of points and are denoted as lower-case italic letters.
		A plane is said to be a projective plane if it satisfies axioms 1 through 4 in section 3.
		Two pencils are related to each other by a perspectivity with axis O if the points formed by corresponding lines on the pencils are collinear on O.
	halogen.note.amherst.edu /~wing/project/content.php?page=2 (618 words)

	Infinity--Projective Geometry #5
		In fact, it satisfies the axioms of geometry which we desire, so long as we're careful to interpret "point" to mean "pair of opposing points on the sphere," and "line" as "great circle on the sphere:"
		There is widespread suspicion (among physicists and astronomers) that elliptical geometry is closer than Euclidean geometry to the geometry of our universe.
		Find a triangle in Projective geometry whose angle measures are 90, 90, and 90 degrees, as measured on the sphere.
	www.math.lsa.umich.edu /mmss/coursesONLINE/infinity/Geometry/Lesson5.shtml (735 words)

	Problem 27: Desargues' Theorem in the Projective Plane
		Notice also that a projective plane will be Desarguesian if and only if it satisfies the converse of Desargue's Theorem.
		Let x, y, z and x', y', z' be two sets of three distinct collinear points on two distinct lines such that no one of these points is on the intersection of the two lines.
		For every point c and line l of a Desarguesian Projective plane (that is different from the Fano plane), there is a (c,l)-collination that is different from the identity.
	home.wlu.edu /~mcraea/Finite_Geometry/ProjectiveGeometry/Prob27DTinProjPlane/problem27.html (724 words)

	Pencil (mathematics) - Wikipedia, the free encyclopedia
		A pencil is a family of geometric objects, such as lines, that have a common property, such as passage through a given line in a given plane.
		In more technical language, a pencil is the special case of a linear system of divisors in which the parameter space is a projective line.
		Typical pencils of curves in the projective plane, for example, are written as
	en.wikipedia.org /wiki/Pencil_(mathematics) (106 words)

	pascal (Site not responding. Last check: 2007-11-04)
		Let 1, 2, 3, 4, 5, 6 denote six points of C such that points 1,..., 5 are ordinary points of the projective plane.
		Construct five ordinary points of the projective plane such that no three of them are collinear.
		Comment: Let P denote the pencil of lines through point 5 and as earlier let C be the desired conic.
	www.isu.edu /~fishrobe/geometry/sketchpad/pascal.htm (510 words)

	Mathematics (Site not responding. Last check: 2007-11-04)
		The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space (also applying to both more and fewer dimensions), later also generalized to non-Euclidean geometries which play a central role in general relativity.
		The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness, and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations.
		In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system.
	games.abcworld.net /Mathematics.html (2840 words)

	Publications on Geometry
		A Treatise on the Analytic Geometry of Three Dimensions - v.I
		Pencil : Let L and M denote two planes, then the linear combination a L + b M denotes a plane passing through the line of intersection of L and M.
		This pencil of planes is called "büschel" in German.
	www.lems.brown.edu /vision/people/leymarie/Refs/Maths/Geometry.html (939 words)

	Infinity--Projective Geometry #4
		This suggests that to understand the geometry of the eye, we should study the geometry of the sphere.
		We have seen before that the geometry of the eye is the geometry of a plane (with infinite points).
		Of course, both descriptions of the geometry of vision are correct.
	www.math.lsa.umich.edu /mmss/coursesONLINE/infinity/Geometry/Lesson4.shtml (703 words)

	Conics on the real projective plane
		The first important result about conics is that, up to projective transformations, they are all the same.
		Use exercise 4.2 to show that this correspondence preserves cross-ratios and is, therefore, a projective transformation between the pencils.
		This condition, discovered by the mathematician and philosopher Blaise Pascal, is one of the earliest and prettiest results on projective geometry.
	www.math.poly.edu /courses/projective_geometry/chapter_five/node4.html (651 words)

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