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Topic: Collinear points

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	Show that three points are collinear (Site not responding. Last check: 2007-10-08)
		Determine whether 3 Points are Collinear: Are these points collinear?
		If the points are collinear, the sum of the distances between two pairs of points equals the distance between the third pair of points.
		Hence, we have AB + AC = BC, and the points are collinear.
	www.jtaylor1142001.net /calcjat/Solutions/3Dim/Collinear.htm (65 words)

	Xah: Introduction to Real Projective Plane
		The points on a line are said to form a range, especially when we regard them as the possible positions of a variable point X (which runs along the line).
		Theorem: The correspondence between the points of a range and their harmonic conjugates with respect to two fixed points M and N is an opposite correspondence with invariant points M and N. Xah's Note: Given H(MN,XX'), there exist a quadrangle P,Q,R,S such and such by the definition of Harmonic sets.
		All normal point on B is map to P of A except where A and B intersect is map to a pencil of P. Point at infinity of B is map to P too.
	xahlee.org /projective_geometry/projective_geometry.html (6440 words)

	Conics (Site not responding. Last check: 2007-10-08)
		Suppose CDEF are distinct points and distinct from A and B on a nondegenerate conic determined by pencils centered at A and B.
		That is, transformations of points to points and lines to lines in the projective plane that preserve the relation of incidence [1].
		A conic is the locus of self-conjugate points of a hyperbolic polarity [3].
	www-math.cudenver.edu /~dlholder/proj_geom/conic/node1.html (2419 words)

	Sylvester–Gallai theorem - Wikipedia, the free encyclopedia
		This theorem was posed as a problem by James Joseph Sylvester in 1893, and solved by Tibor Gallai in 1944.
		Suppose we have a finite number of points which are not collinear (in particular, we must have at least three points).
		Since the points are not collinear, there is at least one point P which is not on l.
	en.wikipedia.org /wiki/Sylvester-Gallai_theorem (356 words)

	Machine vision methods for identifying collinear sets of points from an image - Patent 6075881
		In the case of an image with one or more perpendicular sets of collinear points, the estimated angle is the angle of the predominant line defining one axis of the sets, e.g., the predominant angle of the "horizontal" lines or of the "vertical" lines.
		In one embodiment of the invention, the estimated angle of the collinear points is assumed to be equal to the candidate angle of the single projection with the highest density peak.
		In this phase, the method identifies the sets of collinear points in the data points aligned with the estimated angle (i.e., the angle of the sets of predominant collinear points, as determined in the angle-finding phase), as well as lines at 90.degree.
	www.freepatentsonline.com /6075881.html (6240 words)

	Pappus's hexagon theorem - Wikipedia, the free encyclopedia
		Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A, B, C, and another set of collinear points a, b, c, then the intersection points x, y, z of line pairs Ab and aB, Ac and aC, Bc and bC are collinear.
		The dual of this theorem states that given one set of concurrent lines A, B, C, and another set of concurrent lines a, b, c, then the lines x, y, z defined by pairs of points resulting from pairs of intersections A∩b and a∩B, A∩c and a∩C, B∩c and b∩C are concurrent.
		A generalization of this theorem is Pascal's theorem, which was discovered by Blaise Pascal at the age of 16.
	en.wikipedia.org /wiki/Pappus's_hexagon_theorem (387 words)

	InterMath / Dictionary / Description
		Collinear: Points are collinear if they lie on the same straight line.
		In addition, points that are collinear must satisfy the same linear equation.
		So without even looking at a graph, if we are given the linear equation y = 4x - 2, we know that the points (2,6) and (0,-2) are collinear because 6 = 42 - 2 and -2 = 40 - 2.
	www.intermath-uga.gatech.edu /dictnary/descript.asp?termID=74 (65 words)

	1-2 - Points, Lines, & Planes (Prentice Hall) (Site not responding. Last check: 2007-10-08)
		Two fundamental objects in geometry are points and lines.
		Thus we often determine if three points are collinear.
		All points plotted in the Cartesian Plane are coplanar.
	www.e-zgeometry.com /classph/sec1/1.2/1.2.htm (768 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The problem to be solved is to determine the 3-D coordinates of feature points on a curved surface from as few as a single digital image generated by as few as a single 2-D sensor such as a camera.
		Note that given an image point (u, v), it is possible to determine the line that passes through the camera's focal point and the object point (x, y, z).
		An example of control points are the corner points of object 14 (A, B, C and D).
	www.wipo.int /cgi-pct/guest/getbykey5?KEY=01/22365.010329&ELEMENT_SET=DECL (2852 words)

	Projective Geometry: Axioms and Definitions (Site not responding. Last check: 2007-10-08)
		The set of all lines through a point P is called a pencil of lines with centre P; the set of all points on a line p is called a pencil of points with axis p.
		A one-to-one mapping between 2 pencils of points with axes p and p' is called a perspectivity if the lines joining corresponding points on p and p' are concurrent at some point O. is called the centre of the perspectivity.
		The point C is called the harmonic conjugate of D with respect to A and B (and D is called the harmonic conjugate of C etc).
	s13a.math.aca.mmu.ac.uk /geometry/M23Geom/ProjGeometry/ProjGeomAxsDefs.html (559 words)

	Lecture Notes 20 - Math 4220
		Its coordinates are of the form (a,b,1) with neither a nor b being 0 or 1, and a not equal to b.
		If only 4 points are collinear, then the repeated line is no longer a possibility, and the second line of the pair can be any line through the 5th point.
		Points C = (a,b,1) and C' = (c,d,1) are such that none of a,b,c,or d is 0 or 1, a is not equal to b or c, and d is not equal to c or b, by the assumption that no three of the points are collinear.
	www-math.cudenver.edu /~wcherowi/courses/m4220/hg2lec20.html (1109 words)

	The cross ratio
		For a set of collinear points, we can always select a coordinate such that at least three of the points have nonzero entries for that coordinate.
		As a final comment on the cross ratio, it is worth noting that it does not require that the original points be collinear.
		For example, given five points in a star configuration, as shown in figure 6, we can connect the dots as shown in (a) to yield lines containing four collinear points, the points of intersection, whose cross ratio can be used.
	homepages.inf.ed.ac.uk /rbf/CVonline/LOCAL_COPIES/BIRCHFIELD/node10.html (481 words)

	The Construction: Observation Three (Site not responding. Last check: 2007-10-08)
		Observation 3 The triple of points on the line l constructed from the triples of pointsx and yby the construction described in Observation Two is exactly the same asthe triple of points constructed using the triples of points x and z.
		However, the green points are fixed to remain on the same horizonal line and the blue points must lie on a line that does not change the intersection point of the two lines.
		Notice that whereever the blue points are moved,the red points that lie on the same line as the green points are not affected.
	www.math.umd.edu /~wphooper/pappus/obs3 (214 words)

	MSN Encarta - Search Results - collinear
		Collinear, lying on or passing through a single straight line.
		Determine the coordinates of any three points that lie on the graph of this equation.
		Plot these points on the coordinate plane, and graph the line that represents this equation.
	encarta.msn.com /collinear.html (130 words)

	Families of Conics
		The configuration of Pappus is obtained by taking two distinct lines, any three distinct points on one line and any three distinct points on the other (subject to the condition that none of the six points lies at the point of intersection of the two lines).
		Label the three points on the first line A, C, and E, and the three points on the second line by B, D, and F. We use the notation X=AB.DE to mean that X is the point of intersection of the lines AB and DE.
		Each of the these 5 points is either one of the original six points that defined the configuration, or is a cross point that comes directly from the original six points.
	mathforum.org /dynamic/submissions/familyofconics (755 words)

	Informal Geometry Review
		Collinear points are points that belong to the same line.If three distinct points are collinear, then one of the points is between the other two.
		A ray is the union of a point and a half-line determined by the point (i.e.
		Non-standard angles are straight angles (generated by collinear points with a specified point designated as the vertex) and zero degree angles (an angle generated by coincident rays with a common end point).
	www.csupomona.edu /~vmsmith/GeoRev.html (1391 words)

	2 The axiomatic approach (Site not responding. Last check: 2007-10-08)
		If two points of a line are on a plane then every point of the line is on the plane (we then say that the line is on the plane).
		Then there is a point O on the line and a pair of points P and N such that [NOP] and the two classes consists of all points between N (respectively P) and O.
		Now we will outline how the points, lines and planes of such a geometry can be realised as the points, lines and planes of a convex region in coordinate 3-space preserving all the relations of incidence and between-ness; that is to say we have an embedding of our geometry into that of coordinate 3-space.
	www.imsc.res.in /~kapil/papers/krp/node2.html (1709 words)

	AoPS Math Forum :: View topic - Very very difficult combinatorics problem
		Prove that for each n there are n collinear points on the frog's path.
		collinear points on the frog's path, and we are done in that case.
		In this point of view, for the two-dimensional case you are asking for, I think you probably have to ask for a set with at least
	www.artofproblemsolving.com /Forum/post-17354.html (865 words)

	GEOMETRY CHAPTER 1 (Site not responding. Last check: 2007-10-08)
		Collinear points are points that lie on the same line.
		Noncollinear points are points that do not lie on the same line
		In this Collinear Points graph we say point B is b_tween point A and point C. If the three points are not collinear we cannot say a point is b-tween two other points.
	www.ndsu.nodak.edu /trio/ub/UB2002/students/vivianne_dort/viv_math.htm (123 words)

	fuselier
		A parabola is the locus of all points P such that the distance from P to F (called the focus) is equal to the distance from P to l(also known as the directrix).
		Also, if the points are collinear, the constant A is zero and the function ceases to be quadratic.
		Given n points in the xy-plane with distinct x-coordinates, there is exactly one polynomial function of degree n-1 or less that passes through all n points.
	www.selu.edu /Academics/Depts/Math/students/fuselier/fuselier.htm (1097 words)

	ez1-1 Coordinate Geometry (Glencoe Geometry) (Site not responding. Last check: 2007-10-08)
		A pair of numbers (x coordinate, y coordinate) indicating the position of a point in the Cartesian Plane, for example P(6,3), a positive 6 x-value and a positive 3 y-value.
		All points in Quadrant II have a negative x-value and a positive y-value, K(-5,7) or Y(-77,54).
		All points in Quadrant IV have a positive x-value and a negative y-value, D(2,-3) or T(62,-4).
	www.e-zgeometry.com /class/class1/1.1/1.1.htm (430 words)

	Geomtest1_1 (Site not responding. Last check: 2007-10-08)
		Name a point that is collinear to the points N and Z. Name a point that lies on the line c.
		Name all the points that are collinear to and Q. Name three non-collinear points.
		Point H is between G and I. Draw a sketch to represent the line segments. Use the segment postulate to find the length of each segment and the total length of the line.
	home.att.net /~l.b.gee/Geomtest1_1.htm (201 words)

	The Construction: Observation Two
		In observation one we found that we could apply Pappus' theorem six times to each pair of triples of collinear points in the Projective Plane.
		The results are six new lines in the plane and we found that ofthese six lines, there are two sets of three of them which are coincident.From what we know of duality, two pairs of triples of coincident linesis dual to two pairs of triples of collinear points.
		We can apply thedual to observation one and it is obvious that these six points can bebroken up into two pairs of three collinear point.
	www.math.umd.edu /~wphooper/pappus/obs2 (268 words)

	COS 226 Pattern Recognition Assignment (Site not responding. Last check: 2007-10-08)
		The algorithm works because points that make the same angle with p are collinear and sorting brings such points together.
		Assume the points are given as pairs of integers (x, y) between 0 and 32,768.
		Note that the points are scaled down by a factor of 64.0 so that they fit inside the 512-by-512 turtle graphics window.
	www.cs.princeton.edu /courses/archive/spring03/cs226/assignments/lines.html (515 words)

	Surgical instrument actuator with non-collinear hydraulic pistons - Patent 4586502
		2a, the needle 50 is passed into the approximated and folded tissue 40 from a point on one side of the wound and on through the tissue until the tip of the needle 50 exits the tissue on the opposite side of the wound.
		As long as the flexible pusher member 74 extends beyond the front of the magazine 100, the next fastener 30 in the magazine 100 is prevented from being fed from the magazine to the channel 72 in the housing piece 62.
		At this point the first piston 321 is at the second position illustrated in FIG.
	www.freepatentsonline.com /4586502.html (6957 words)

	EPGY Theorem Proving Environment
		Axiom P2 For any point P and line l there is at most one line through P parallel to l.
		P, Q, R are collinear to indicate that the points P, Q, and R lie on the same line.
		Theorem 3.22 For any three points A, B, and C, two points are on the same side of the line l.
	www-epgy.stanford.edu /tpe?exercises/M015.html (1494 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Among all triples (p,q,r) of non-collinear points in L, pick one which minimizes the distance between r and the line pq, and let p' be a point witnessing (*) for the line pq.
		Both this problem and the previous one (N red, N blue points) are projectively invariant assertions but the short proofs use metric concepts.
		The result follows from a purely combinatorial theorem of Haim Hanani stating that if the N points are not collinear then they determine at least N distinct "lines".
	www.math.niu.edu /~rusin/papers/known-math/94/geometry.lines (938 words)

	Collinear points (Site not responding. Last check: 2007-10-08)
		Which of the 4 points are collinear when you construct the following concurrent lines or rays of a triangle?
		P(2), the point where the altitudes (or extensions) intersect (inside or outside of the triangle).
		P(4), the point where the perpendicular bisectors (or extensions) of the three sides of a triangle intersect.
	mathcentral.uregina.ca /qq/database/QQ.09.02/gary1.html (216 words)

	Geometry Definitions - Unit 1 (Site not responding. Last check: 2007-10-08)
		disk – The union of the interior points of a circle and the points on the circle is a
		of a circle is the distance from the center to a point on the circle.
		Ray AB consists of points A and B and all the points X such that AXB and ABX.
	arapaho.nsuok.edu /~wilkinso/geo_def.html (404 words)

	Miscellaneous Questions - Numericana
		Lattice points are points of the plane with integer coordinates.
		For any sequence S of points (X(n),Y(n)) where two consecutive points are at most D units apart, we may consider the sequence C obtained by removing from the sequence (floor(X(n)/D),floor(Y(n)/D)) any consecutive elements which happen to be equal.
		A lattice polygon is a polygon whose vertices are lattice points.
	home.att.net /~numericana/answer/misc.htm (2172 words)

	Re: Maximum number of collinear points
		Ofcourse, there is the natural O(n^2 logn) way > where we goto each point, make it the origin, sort the points in order > of increasing angle, and find the largest continuous equal subsequence > (equal meaning angles equal).
		Assuming that the point locations are exact (e.g., integer coordinates) and you are looking for exact collinear points, you could compute the exact slope and intercept for each pair, then express those values in canonical form (cf), such as reduced p/q.
		Each time a cf match is found, increment the number of collinear points and compare the number of collinear points in the cf list with the maximum found so far and update, if needed.
	www.talkaboutprogramming.com /group/comp.programming/messages/202319.html (241 words)

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