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Topic: Inversion (geometry)

	GM-SYS 3D Modeling - 3D Inversion (Site not responding. Last check: 2007-11-05)
		Inversion is essential in 3D modeling, in order to rapidly sort through the vast number of possible models.
		Inversion is performed on one block at a time, in order to reduce the number of variables and thus produce more useful results.
		Inversion may be performed on internal property variance as well, either laterality or by depth.
	www.geosoft.com /pinfo/partners/gmsys-3d-inversion.asp (265 words)

	Essay Number 1- Circle Inversion (Site not responding. Last check: 2007-11-05)
		A line that is tangent to the circle of inversion inverts to a circle (that goes through the center of the circle of inversion) that is also tangent to the circle of inversion at the same point of tangency as the original line.
		A line that intersects the circle of inversion in two points, but does not go through the center of the circle of inversion, inverts to a circle (that goes through the center of the circle of inversion) that intersects the circle of inversion at the same two points as the original line.
		A line that is the perpendicular bisector of the radius of the circle of inversion inverts to a circle that is congruent to the circle of inversion.
	jwilson.coe.uga.edu:16080 /EMT668/EMAT6680.F99/Challen/inversion (1700 words)

	Xah: Special Plane Curves: Inversion
		The pedal and inversion of cissoid of Diocles.
		The inversion of a curve is the inverse of all points on the curve.
		The inversion of {x,y} with respect to a circle centered at {a,b} and radius r is {a + (r^2(-a + x))/((a - x)^2 + (b - y)^2), b + (r^2(-b + y))/((a - x)^2 + (b - y)^2)}.
	xahlee.org /SpecialPlaneCurves_dir/Inversion_dir/inversion.html (1662 words)

	Inversion -- from Wolfram MathWorld
		inversion circle itself, circles orthogonal to it, and lines through the inversion center are invariant under inversion.
		Coolidge, J. "Inversion." §1.2 in A Treatise on the Geometry of the Circle and Sphere.
		Johnson, R. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.
	mathworld.wolfram.com /Inversion.html (617 words)

	Inversion: Reflection in a Circle (Site not responding. Last check: 2007-11-05)
		Lines through the center of inversion (with the exception of the center itself) also have that property.
		The center of inversion is often [Coxeter] left over as a point with no inverse image, but sometimes [Courant] is said to be mapped to the point at infinity.
		Caution must be exercised as this is not the same point at infinity where, for example, all parallel lines meet.
	www.cut-the-knot.org /Curriculum/Geometry/SymmetryInCircle.shtml (550 words)

	Tangencies: Inversion (Site not responding. Last check: 2007-11-05)
		An inversion is performed with respect to a particular center point and choice of scale; then each point in the plane is transformed to another point, forming the same angle to the center but with distance inversely proportional to its original distance.
		The key properties of inversion are that it transforms circles to circles, and preserves the angles of crossings between circles; in particular inversion preserves the tangencies of a collection of circles, such as the three tangent circles shown in the animation.
		When interpreting geometric figures using inversion, lines should be viewed as infinite-radius circles that go through a special "point at infinity" which is the image under inversion of the inversion circle's center.
	www.ics.uci.edu /~eppstein/junkyard/tangencies/inversion.html (248 words)

	Count On (Site not responding. Last check: 2007-11-05)
		A very rich geometry of the circle is the transformation known as inversion.
		Inversion is a transformation where a point P is mapped with respect to a circle such that OP.OP = r
		Inversion was discovered in the nineteenth century and enables many theorems to be proved in a concise and elegant way, as well as offering simple constructions for seemingly impossible problems.
	www.mathsyear2000.org /explorer/circles/inversion (369 words)

	Inversion (geometry) - Wikipedia, the free encyclopedia
		In geometry, an inversion is a transformation that maps all circles into circles, where by a circle one may also mean a line (a circle with infinite radius).
		As said, in inversive geometry there is no distinction made between a straight line and a circle (or hyperplane and hypersphere): a line is just nothing more and nothing less than a circle in its particular embedding in a Euclidean geometry (with a point added at infinity) and one can always be transformed into another.
		This is therefore true in general of orthogonal spheres, and in particular inversion in one of the spheres orthogonal to the unit sphere maps the unit sphere to itself.
	en.wikipedia.org /wiki/Inversion_(geometry) (1233 words)

	Inversion - Interpreting the Deformation path - Why Does it Matter?, A. D. Gibbs, #40034
		Inversion is dominantly a 3D process, whether it is driven by the extension - compression cycle, transtension or halokinetic movement.
		Inversion is defined as occurring when the tectonic regime moves from extension to compression or from subsidence to uplift (Figure 1.1).
		Inversion caused by changes in slip direction on faults or flow within the model due to salt is a 3D effect.
	www.searchanddiscovery.com /documents/gibbs02/index.htm (1602 words)

	Inversion with Negative Power (Site not responding. Last check: 2007-11-05)
		The applet is supposed to remind of the symmetry in circle - inversion.
		OA×OA' = OR which appears exactly like the inversion identity, but is not quite the same.
		The difference is in that formerly the points A and A' lay on the same side from O, while now O separates the two.
	www.cut-the-knot.org /Curriculum/Geometry/Inversion2.shtml (284 words)

	Inversion in a point - Wikipedia, the free encyclopedia
		In Euclidean geometry, the inversion of a point X in respect to a point P is a point X* such that P is the midpoint of the line segment with endpoints X and X*.
		Inversion with respect to the origin corresponds to additive inversion of the position vector, and also to scalar multiplication by −1.
		"Inversion" without indicating "in a point", "in a line" or "in a plane", means this inversion, also called parity transformation.
	en.wikipedia.org /wiki/Inversion_in_a_point (318 words)

	triangle Index
		Geometry tutorials (pdf's) When I was learning triangle geometry about 10 years ago, I wrote articles for myself and for friends.
		As geometry goes this is pretty advanced, using barycentric coordinates and general knowledge of triangle geometry, especially projective triangle geometry.
		Abstract Algebra and Triangle Geometry We are in the third great period of discovery in triangle geometry.
	www.paideiaschool.org /Teacherpages/Steve_Sigur/geometryIndex.htm (2167 words)

	Shi's Abstract
		To implement the regularized inversion on a 3-D resistivity model, one faces a computational challenge owing to the nonlinear nature of the resistivity problem and the large number of model parameters and data which can possibly exist in a moderate 3-D model.
		While in the IP inversion, the imaginary component of the complex resistivity is much smaller than the real part, the objective function is constructed in a complex form, and the minimization is solved directly in the complex domain using a bi-conjugate gradient method.
		This new survey geometry is investigated with sensitivity analysis and model correlation estimation, and it appears to be more effective than traditional pseudo-section acquisition geometry in cases where structure has an extended lateral variation.
	eaps.mit.edu /erl/research/theses/abstracts/Shi.html (1106 words)

	Inversion (Site not responding. Last check: 2007-11-05)
		Because straight lines and circles are mixed up with each other by inversion, when we do inversive geometry we may tend to think of lines as just a special kind of circle.
		Because inversion preserves angles, in particular it preserves the relation of orthogonality; and H' is orthogonal to the sides of A'B'C'.
		Hyperbolic geometry, in particular, is the subject of Anderson.
	www.monmouth.com /~chenrich/Inversion.html (1357 words)

	Math 371 Geometry Notes on Line
		Euclid's division algorithm is stated in geometry that of ON is a segement and OD is a segment that is contained as a subsegment of ON, then ON is congruent to q*OD with possibly a remaining segment RN which is congruent to a subsegment of OD.
		The impact of this on geometry was that one could not presume that all of geometry could be handled by using simple ratios of whole numbers for measurements.
		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.
	www.humboldt.edu /~mef2/Courses/m371notes04.html (8713 words)

	Rift Basin Architecture & Evolution (Site not responding. Last check: 2007-11-05)
		Basin inversion occurs in a variety of tectonic environments (e.g., Buchanan and Buchanan, 1995), including several passive margins related to the breakup of Pangea (e.g., Doré and Lundin, 1996; Vagnes et al., 1998; Withjack et al., 1995, 1998; Hill et al., 1995; Withjack and Eisenstadt, 1999).
		Although the inversion is obvious in this model, erosion of material down to the level of the red line would remove the most obvious evidence of inversion in the half graben.
		Thus, the end of rifting, the initation of inversion, and probably the initiation of seafloor spreading are diachronous along the central Atlantic margin (i.e., during earliest Jurassic time in the southeastern United States and Early to Middle Jurassic time in the northeastern United States and Maritime Canada) (Withjack et al., 1998).
	www.ldeo.columbia.edu /~polsen/nbcp/breakupintro.html (3417 words)

	How to Use History to Clarify Common COnfusions in Geometry
		One of the first open questions about hyperbolic geometry was whether it is the geometry of any surface in Euclidean space, in the same sense that spherical geometry is the geometry of the sphere in Euclidean 3-space.
		This aspect of hyperbolic geometry belongs in the Navigation/Star Gazing strand of geometry, in the sense that differential geometry of surfaces (and higher-dimensional manifolds) uses calculus to study the geometric properties that are intrinsic – properties of the surface that a bug crawling on the surface could detect.
		Inversions in a circle are a necessary component of the analytic study of hyperbolic geometry.
	www.math.cornell.edu /~dtaimina/MAA/MAA.htm (6700 words)

	Chain Reaction: Inversion and the Pappus Chain Theorem
		The inverse of the center point of the circle of inversion is a point at infinity.
		The inverse of a circle whose circumference passes through the center of the circle of inversion is a line.
		Your goal is to gain a sufficient understanding of the principles of circle inversion and their application to the arbelos so that you can demonstrate to yourself and others that the statements in the preceding two paragraphs are indeed true.
	www.sciencebuddies.org /mentoring/project_ideas/Math_p011.shtml (1408 words)

	Finite Fault Inversion of the Loma Prieta Earthquake Incorporating Complex Structure
		The spatial distribution of aftershocks suggest that the rupture geometry was complex and may have occurred on multiple planes with distinctly different orientations.
		We perform inversions using Green's functions calculated from 3 distinct one-dimensional velocity models to evaluate whether the choice in velocity structure significantly impacts the slip distribution.
		Additional fault geometries will be tested, including a change in dip along strike and a delay in rupture to the northwest patch.
	seismo.berkeley.edu /annual_report/ar00_01/node25.html (926 words)

	Inversion
		The circle is called the circle of inversion, and point O is the center of inversion.
		The inverse of a line (not through the center of inversion) is a circle through the center of inversion.
		In this sketch, the circle on the left is being inverted with respect to the red circle, with center O and radius r.
	whistleralley.com /inversion/inversion.htm (1675 words)

	The Triangle Figure (Site not responding. Last check: 2007-11-05)
		Hyperbolic geometry is the result of replacing the parallel axiom of Euclidean Geometry with the alternative of there being, through a given point, at least two lines parallel to a given line.
		Another use for circle inversions is in the fact that a circle passing through both points of a circle inversion pair will be orthogonal to the circle of inversion.
		So circle inversion may be used to aid in the construction of a type-2 h-line through a given point that is perpendicular to a given type-2 h-line.
	www.math.clemson.edu /~rsimms/triangle/hyperbolic.html (367 words)

	Numerical models of the inversion of half-graben basins
		We investigate the dynamic evolution of fold and thrust structures which form by compression and inversion of a sequence of half-graben basins.
		The choice for a half-graben geometry is motivated by seismic studies and reconstructions of preinversion geometry of inverted regions, which show that rifting often leads to a series of half-grabens.
		With continuing shortening, further inversion is more difficult owing to relative strengthening of the half-graben region.
	www.agu.org /pubs/crossref/2003/2002TC001417.shtml (338 words)

	inversion - OneLook Dictionary Search
		Inversion, inversión : AllWords.com Multi-Lingual Dictionary [home, info]
		Inversion : National Weather Service Glossary [home, info]
		Phrases that include inversion: sexual inversion, pericentric inversion, paracentric inversion, thermal inversion, visceral inversion, more...
	www.onelook.com /?w=inversion (486 words)

	perplexus.info :: Geometry : Inversion Distance
		A circle (of radius a), a line, and a point are mapped by inversion into two concentric circles and the center of those concentric circles.
		Let O be the center of a circle of radius k.
		An inversion with respect to circle O is a mapping f:R
	perplexus.info /show.php?pid=4451 (125 words)

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