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Topic: Distance geometry

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	Distance geometry - Wikipedia, the free encyclopedia
		Distance geometry is the characterization and study of sets of points based only on given values of the distances between member pairs.
		Therefore distance geometry has immediate relevance where distance values are determined or considered, such as in surveying, cartography and physics.
		Therefore the distance from A to B is no bigger than the length of the straight-line path from A to C plus the length of the straight-line path from C to B.
	en.wikipedia.org /wiki/Distance_geometry (376 words)

	space-time
		Now, Euclidean geometry, in the form in which it has been handed down to us from Euclid, uses the fundamental concepts "straight line" and "plane" which do not appear to correspond, or at any rate, not so directly, with experiences concerning the position of rigid bodies.
		This leads to the concepts: producing a distance by an amount equal to itself; dividing a distance into equal parts; expressing a distance in terms of a number by means of a measuring-rod (definition of the space-interval between two points).
		This signifies analytically: the relations of Euclidean geometry are covariant with respect to linear orthogonal transformations of the co-ordinates.
	www.britannica.com /nobel/classic/C00010.html (3807 words)

	Geometry and Topology - Numericana (Site not responding. Last check: 2007-10-25)
		Geometry from the Land of the Incas by Antonio Gutierez.
		The centroid G is between the orthocenter H and the circumcenter I. The distance HG is twice the distance GI.
		/50, the parameter is 25 and the focal distance is 12.5.
	home.att.net /~numericana/answer/geometry.htm (7724 words)

	Math Forum - Ask Dr. Math
		I used the distance formula theorem when I was given two sets of coordinates to find the radius of a circle, which I got right (that was easy).
		You can think about the left side as the square of the distance from (x,y) to (3,-4), which according to the equation is 3^2.
		So the distance from (x,y) to (3,-4) is 3, and (x,y) is on a circle with center (3,-4) and radius 3.
	mathforum.org /library/drmath/view/61371.html (601 words)

	Taking geometry seriously
		Distance is then plotted on the later position line and finally the line of the course bearing can be drawn using the known course bearing and the calculated distance.
		This is a problem in three dimensional geometry with the angle formed at the centre of the earth to be found.
		The distance remains in nautical miles, the heights of the objects become feet, and 2.08 is replaced by 1.15.
	education.qld.gov.au /curriculum/area/maths/compass/html/geoseriously/tatak.html (1713 words)

	struc_c (Site not responding. Last check: 2007-10-25)
		The actual distances obtained from the NOE data are used to define lower and upper limits to each element of the distance matrix.
		Before embedding the distance matrix, it is advisable to first smooth the distance matrix and then select distances between the upper and lower bounds to generate the distance matrix.
		Distance geometry is often (always) combined with simulated annealing to improve the quality of the structure.
	stingray.bio.cmu.edu /~web/nmr/struc_c (837 words)

	Access Excel Geo Functions (Site not responding. Last check: 2007-10-25)
		In aerial, shipboard and land-based surveys, the distance is often measured between the observer and the observation (e.g., animal).
		These distances are measured as a vertical angle from the horizon with reticle binoculars in shipboard or shore-based surveys or from horizontal with an inclinometer in aerial surveys.
		From shipboard and shore-based surveys, distance is measured with binoculars that contain reticles which measure an angle from the horizon.
	nmml.afsc.noaa.gov /Accessibility/AccessExcelGeoFunctions.html (2954 words)

	A Unified Algebraic Framework for Classical Geometry
		With roots in ancient times, the great flowering of classical geometry was in the 19th century, when Euclidean, non-Euclidean and projective geometries were given precise mathematical formulations and the rich properties of geometric objects were explored.
		Though fundamental ideas of classical geometry are permanently imbedded and broadly applied in mathematics and physics, the subject itself has practically disappeared from the modern mathematics curriculum.
		Because the three geometries are obtained by interpreting null vectors of the same Minkowski space differently, natural correspondences exist among geometric entities and constraints of these geometries.
	modelingnts.la.asu.edu /html/UAFCG.html (2056 words)

	Geometry Distance of Triangles using a Protractor (Site not responding. Last check: 2007-10-25)
		Use the protractor to determine the measure of angle BAC.
		We now attempt to determine X, the distance (AC) from the point A to the Point C. Make a model (as described in Part 1) keeping the angles found but reduce the size of line segment ab.
		To verify, measure the distance from point A to point C. Compare with the calculated answer.
	www.iit.edu /~smile/ma9707.html (314 words)

	Inquire Geometry-Geometry Distance (3D)
		Use this command to dynamically inquire the distance between one geometry entity and another.
		The Variable command option can be used to assign the queried distance to a variable and log the variable to the active part's history.
		This contributes to the "family-of-parts" functionality by allowing the part to be driven during a history regen by variables derived from queried distances.
	www.vx.com /help/0090.htm (220 words)

	MARDIGRAS-Matrix Analysis of Relaxation for DIscerning the Geometry of an Aqueous Structure (Site not responding. Last check: 2007-10-25)
		In all cases the distance is calculated to a pseudoatom which is the geometric mean position of the three methyl protons.
		It assumes that the geometry of the residue itself as given in the model PDB file is essentially correct regardless of the final conformation of the molecule.
		The distance constraints are calculated by the worst case analysis based on motional averaging values, noise and relative fitting of experimental and converged intensity values.
	picasso.ucsf.edu /mardigras/OLD_mardi/mardi.html (4150 words)

	UDN - Two - LevelOptimization
		Distance Fog is the last of the tools a level designer has for optimizing his or her level, and it should be used last as well.
		How Distance Fog works to occlude objects from the renderer is that the further away things are from the player’s perspective, the more saturated with the fog color they become.
		Once the geometry is far enough away to be at 100% saturated in the fog color, the renderer ceases to draw it with no noticeable popping for the player.
	udn.epicgames.com /Two/LevelOptimization (1448 words)

	A formal theory for geometry (Site not responding. Last check: 2007-10-25)
		Euclid's geometry was still regarded as a model of logical rigor, a shining example of what a well-organized scientific discipline ideally ought to look like.
		An outcome of all this foundational activity was a thorough reworking of geometry, this time as a collection of formal theories within the predicate calculus.
		This axiom is typical of two-dimensional (i.e., plane) geometry and does not apply to geometries of dimension greater than two.
	www.math.psu.edu /simpson/papers/philmath/node15.html (452 words)

	Taxicab geometry (Site not responding. Last check: 2007-10-25)
		-distance as the distance between two points measured along axes right angles.
		Manhattan distance is also known as city block distance.
		It is so named because it the distance a car would drive in city laid out in square blocks like Manhattan (discounting the facts that in Manhattan are one-way and oblique streets and that streets only exist at the edges of - there is no 3.14th Avenue).
	www.freeglossary.com /Block_distance (169 words)

	Ray Tracing Point Sampled Geometry (Site not responding. Last check: 2007-10-25)
		Intersections with the point--sampled geometry are detected by tracing a ray through the scene until the local density of points is above a predefined threshold.
		We then use all the points within a fixed distance of the ray to interpolate the position, normal and any other attributes of the intersection.
		The considered distance from the ray must be larger than the largest ``hole'' among the points.
	graphics.stanford.edu /~henrik/papers/psp (187 words)

	Analytic geometry Article, Analyticgeometry Information (Site not responding. Last check: 2007-10-25)
		Analytic geometry, also called coordinate geometry and earlier referred to asCartesian geometry, is the study of geometry using the principlesof algebra.
		Usually the Cartesian coordinate system is applied to manipulate equations for planes, lines, curves, and circles, often in two and sometimes in three dimensions of measurement.Some consider that the introduction of analytic geometry was the beginning of modern mathematics.
		Analytic geometry, for algebraic geometers,is also the name for the theory of (real or) complex manifolds andthe more general analytic spaces defined locally by the vanishing of analytic functions of severalcomplex variables (or sometimes real ones).
	www.anoca.org /two/algebraic/analytic_geometry.html (308 words)

	Geometry -- Taxicab Treasure Hunt (Site not responding. Last check: 2007-10-25)
		Taxicab geometry is a special kind of geometry that works on city streets.
		In taxicab geometry the shortest distance between two points is not a straight line, but rather the number of blocks a taxi has to travel along the streets.
		So, in the street plan of Arborville (shown below), the distance from the blue star to the red star is 4 blocks.
	www.learner.org /teacherslab/math/geometry/shape/taxicab (123 words)

	Cynthia Lanius' Lesson: School-Bus Geometry Introduction (Site not responding. Last check: 2007-10-25)
		I'm sure you've heard that the shortest distance between 2 points is a straight line.
		Calculating distance is tied to a famous theorem.
		Since these distances are irrelevant on city streets, we will apply a different measurement, a measurement that doesn't allow you to leave the streets.
	math.rice.edu /~lanius/Geom/schbus1.html (172 words)

	Re: Homegrown geometry - FWIW
		My observation, from a few years ago, is that an OLD hardtail (e.g., from the early 90s) whose rigid fork is replaced with a suspension fork will often have the "same" resting geometry as a contemporary hardtail.
		If you subtract 100mm from the distance between your fork's dropouts & the fork's shoulder, you may find the distance is close-to being the same as the distance between the dropouts & fork's shoulder of a rigid mtb fork.
		By inference, you may deduce that putting almost any rigid mtb fork on your Homegrown will mimic/approximate the geometry of a "classic" rigid mountain bike -- I think that used to be about 71º for both the seatpost & headtube angles.
	www.bikeschool.com /technical-board/messages/10874.htm (520 words)

	Taxicab geometry (Site not responding. Last check: 2007-10-25)
		Taxicab geometry, considered by Hermann Minkowski in the 19th century, is aform of geometry in which the usual metric of Euclidean geometry is replaced bya new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates.
		-distance, asthe distance between two points measured along axes at right angles.
		It is so named because it is the distance a carwould drive in a city laid out in square blocks, like Manhattan (discounting thefacts that in Manhattan there are one-way and oblique streets and that real streets only exist at the edges of blocks - there isno 3.14th Avenue).
	www.therfcc.org /taxicab-geometry-25487.html (174 words)

	Inquire Point-Geometry Distance (3D)
		Use this command to dynamically inquire the shortest distance between a point and geometry.
		The shortest distance to the untrimmed curve or face is calculated.
		The shortest distance to the trimmed curve or face only is calculated.
	www.vx.com /help/0088.htm (208 words)

	Excel Geometry Functions (Site not responding. Last check: 2007-10-25)
		To obtain the distance along the ground (D) as in Lerczak and Hobbs (1998) use the approximation: D~sqrt(D02 - h2), where h is the height of the observer’s eye measured in the same units as D0.
		For most observation platforms D and D0 are very close when used with reticle binoculars because the closest distance within the field of view is much larger than the observation height.
		For the Reticle-Distance functions, distance units are nautical miles (1 nautical mile = 1852 meters) and platform heights units are meters.
	nmml.afsc.noaa.gov /Software/ExcelGeoFunctions/excelgeofunc.htm (1487 words)

	DGSOL (Site not responding. Last check: 2007-10-25)
		Distance geometry problems are interesting mathematical problems with important applications in computational biology, the interpretation of NMR data, and the determination of protein structure.
		From a complexity viewpoint, obtaining an approximate solution to a distance geometry is NP-hard.
		DGSOL solves distance geometry problems with a global continuation algorithm, with Gaussian smoothing of a merit function that only depends on the sparse distance data.
	www.mcs.anl.gov /~more/dgsol (340 words)

	NonEuclid: Postulates and Proofs
		Most pairs of points (A and B) in Spherical Geometry, lie on one and only one great circle; however if A and B happen to be antipodal (on opposite ends of any single axis), then there are an infinite number of different great circles that pass through them.
		In Euclidean Geometry, for any triangle ABC, there exists a unique parallel to BC that passes through point A. Additionally, it is a theorem in Euclidean Geometry that when two parallel lines are cut by a transversal, then the opposite interior angles are congruent; therefore, ∠NAB ≅ ∠ABC and ∠MAC ≅ ∠ACB.
		In Hyperbolic Geometry, however, there are an infinite number of lines that are parallel to BC and pass through point A, yet there does not exist any line such that both: ∠NAB ≅ ∠ABC and ∠MAC ≅ ∠ACB.
	www.cs.unm.edu /~joel/NonEuclid/proof.html (1392 words)

	CIT Training Course #407 (Site not responding. Last check: 2007-10-25)
		One of the important applications of distance geometry is the realization of molecular conformations consistent with experimental data (such as distance constraints obtained from NMR data, along with known bond lengths and bond angles and other sterochemical information).
		Distance Geometry in the setting of converting a set of constraints on the locations of atoms in a molecule into an ensemble of possible molecular conformations
		Viewing an initial step in a distance geometry algorithm as the linear projection of a high dimensional dataset into 3 dimensions that preserves as much of the distance variation as possible
	training.cit.nih.gov /coursepicfull.asp?cnumber=407&term=05G (513 words)

	Jorge J. Mor'e and Zhijun Wu (ResearchIndex) (Site not responding. Last check: 2007-10-25)
		Abstract: We study the performance of the dgsol code for the solution of distance geometry problems with lower and upper bounds on distance constraints.
		The dgsol code uses only a sparse set of distance constraints, while other algorithms tend to work with a dense set of constraints either by imposing additional bounds or by deducing bounds from the given bounds.
		Our computational results show that protein structures can be determined by solving a distance geometry problem with dgsol and that the...
	citeseer.ist.psu.edu /268704.html (441 words)

	Taxicab Geometry (Site not responding. Last check: 2007-10-25)
		Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the (absolute) differences of their coordinates.
		It is so named because it is the distance a car would drive in a city laid out in square blocks, like Manhattan (discounting the facts that in Manhattan there are one-way and oblique streets and that real streets only exist at the edges of blocks - there is no 3.14th Avenue).
		Any route from a corner to another one that is 3 blocks East and 6 blocks North, will cover at least 9 blocks.
	www.wikiverse.org /taxicab-geometry (183 words)

	51: Geometry
		Solid geometry is placed here (actually in 51M05) because it mirrors elementary plane geometry, but spherical geometry is primarily on the page for general convex geometry.
		Cabri-geometry is used for teaching secondary school geometry, but, equally important, is its use for university level instruction and as a tool by mathematicians in their research work.
		A useful collection of Geometry Formulas and Facts is taken from the CRC Standard Mathematical Tables and Formulas, and available at the The Geometry Center.
	www.math.niu.edu /~rusin/known-math/index/51-XX.html (828 words)

	Geometry
		This definition implies that non-simple geometries which are arguments to spatial analysis methods must be subjected to a line-dissolve process to ensure that the results are simple.
		The buffer of a Geometry is the Minkowski sum or difference of the Geometry with a disc of radius
		The buffer of a Geometry is the Minkowski sum of the Geometry with a disc of radius
	www.vividsolutions.com /Jts/javadoc/com/vividsolutions/jts/geom/Geometry.html (2132 words)

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