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Study of angle-preserving transformations
In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines
Inversive_geometry
Concept in inversive geometry
In inversive geometry, the inversive distance is a way of measuring the "distance" between two circles, regardless of whether the circles cross each other
Inversive_distance
Application of Clifford algebra
by composition of reflections, it is a special case of inversive geometry. Inversive geometry itself can be performed with the larger system known as
Plane-based_geometric_algebra
Geometric system with a finite number of points
their higher-dimensional analogs such as higher finite inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector
Finite_geometry
Mechanical linkage transforming rotary motion into linear
then D is the inverse of B with respect to the circle (O,k) with center O and radius k. Thus, by the properties of inversive geometry, since the figure
Peaucellier–Lipkin_linkage
Study of angle-preserving transformations of a geometric space
flat models are the spaces of inversive geometry. For pseudo-Euclidean of metric signature (p, q), the model flat geometry is defined analogously as the
Conformal_geometry
Overview of and topical guide to geometry
geometry Information geometry Integral geometry Inversive geometry Inversive ring geometry Klein geometry Lie sphere geometry Non-Euclidean geometry Noncommutative
Outline_of_geometry
Concept in geometry including line and circle
and vice-versa. However, generalised circles are fundamental to inversive geometry, in which circles and lines are considered indistinguishable, the
Generalised_circle
Point found separated from another, given a point pair
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following
Projective_harmonic_conjugate
Maths textbook
Geometry of Complex Numbers is an undergraduate textbook on geometry, whose topics include circles, the complex plane, inversive geometry, and non-Euclidean
Geometry_of_Complex_Numbers
theory Inversive geometry the study of invariants preserved by a type of transformation known as inversion Inversive plane geometry inversive geometry that
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
Topics referred to by the same term
hyperbolic planes, elliptic planes two-dimensional spherical geometry. Plane curve Inversive geometry Geometrography This disambiguation page lists mathematics
Plane geometry (disambiguation)
Plane_geometry_(disambiguation)
Geometry without using coordinates
constructed Riemannian geometry, of which elliptic geometry is a particular case. Another example concerns inversive geometry as advanced by Ludwig Immanuel
Synthetic_geometry
In-depth exploration of circles, spheres, and inversive geometry by Julian Coolidge
Circle and the Sphere is a mathematics book on circles, spheres, and inversive geometry. It was written by Julian Coolidge and published by the Clarendon
A Treatise on the Circle and the Sphere
A_Treatise_on_the_Circle_and_the_Sphere
Circles whose tangent lines at the points of intersection are perpendicular
also considered as a kind of generalized circles, for instance in inversive geometry, then an orthogonal pair of lines or line and circle are orthogonal
Orthogonal_circles
English–American mathematician (1860–1937)
Morley, Inversive Geometry". Bull. Amer. Math. Soc. 40 (5): 374–375. doi:10.1090/s0002-9904-1934-05848-8. Henry Forder (1934) Review:Inversive Geometry, The
Frank_Morley
Branch of mathematics concerned with the movement of shapes and sets
inappropriate for lower grades. Thus, inversive geometry, a larger study than grade school transformation geometry, is usually reserved for college students
Transformation_geometry
Inverse functions of sin, cos, tan, etc.
ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. There are several notations for the inverse trigonometric
Inverse trigonometric functions
Inverse_trigonometric_functions
In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, α and β, is a reference circle for which α and β are inverses
Circle_of_antisimilitude
Type of geometric algebra
facilitates exploration of the classical methods of projective geometry and inversive geometry in a concrete, easy-to-manipulate setting. It has also been
Conformal_geometric_algebra
Complex numbers with non-negative imaginary part
metric space is the hyperbolic plane. In terms of the models of hyperbolic geometry, this model is frequently designated the Poincaré half-plane model. Mathematicians
Upper_half-plane
Limiting case which is different from the rest of the class
case of a parabola if the parabola resides on a tangent plane. In inversive geometry, a line is a degenerate case of a circle, with infinite radius. Two
Degeneracy_(mathematics)
Research program on the symmetries of geometry
Firstly, n-dimensional hyperbolic geometry, n-dimensional de Sitter space and (n−1)-dimensional inversive geometry all have isomorphic automorphism groups
Erlangen_program
Set of circles related by tangency
In geometry, a Steiner chain is a set of n circles, all of which are tangent to two given non-intersecting circles (blue and red in Figure 1), where n
Steiner_chain
Fundamental operation on complex numbers
{\displaystyle z} as a variable are illustrated in Frank Morley's book Inversive Geometry (1933), written with his son Frank Vigor Morley. The other planar
Complex_conjugate
In projective geometry, a bijection between projective spaces that preserves collinearity
Projective Geometry / An Introduction, Oxford University Press, ISBN 9780199298860 Morley, Frank; Morley, F.V. (1933), Inversive Geometry, London: G.
Collineation
inversive plane because it is closed under inversion with respect to any generalized circle, and thus a natural setting for planar inversive geometry
Möbius_plane
Topics referred to by the same term
order in a sequence Inverse element Inverse function, a function that undoes the operation of another function. Inversive geometry#Circle inversion, a
Inversion
Isometric automorphisms of a hyperbolic space
of the primitive notions of geometry to merely point and motion. Hyperbolic motions are often taken from inversive geometry: these are mappings composed
Hyperbolic_motion
On reflection in a spherical mirror
This hyperbola can be characterized in many ways; one way involves inversive geometry. The locus of points L {\displaystyle L} at which the two lines to
Alhazen's_problem
Type of quadrilateral
quadrilateral sides to cross. The Apollonius quadrilaterals are important in inversive geometry, because the property of being an Apollonius quadrilateral is preserved
Apollonius_quadrilateral
Curve created by a geometric operation
In inversive geometry, an inverse curve of a given curve C is the result of applying an inverse operation to C. Specifically, with respect to a fixed circle
Inverse_curve
Disproven hypothesis
coordinates, let radius r go to R2/r where R is the Earth's radius; see inversive geometry.) The transformation entails corresponding changes to the forms of
Hollow_Earth
3D coordinate system used in mathematics
{\displaystyle R} -separable for the 3-variable Laplace equation. Multiplicative inverse (for 1-dimensional version) Moon, P. and Spencer, D. E. 6-sphere Coordinates
6-sphere_coordinates
Field of mathematics which studies incidence structures
In mathematics, incidence geometry is the study of incidence structures. A geometric structure such as the Euclidean plane is a complicated object that
Incidence_geometry
short descriptions of redirect targets Inversive distance – Concept in inversive geometry Inversive geometry – Study of angle-preserving transformations
List_of_circle_topics
Number with a real and an imaginary part
described below. Complex conjugation is also employed in inversive geometry, a branch of geometry studying reflections more general than ones about a line
Complex_number
Area of combinatorics
their higher-dimensional analogs such as higher finite inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector
Algebraic_combinatorics
Chain of 6 spheres tangent to 3 given spheres
{1}{r_{5}}}={\frac {1}{r_{3}}}+{\frac {1}{r_{6}}}.} Descartes' theorem Inversive geometry Sangaku Rothman 1998 Soddy 1937. Ogilvy 1990 Coxeter 1952 O'Connor
Soddy's_hexlet
Geometric symmetry operation
(with multiplicity n). The term inversion should not be confused with inversive geometry, where inversion is defined with respect to a circle. In two dimensions
Point_reflection
Book on Japanese temple geometry problems
western mathematics would have been solved using calculus or inversive geometry. Sacred Geometry can be read by historians of mathematics, professional mathematicians
Sacred_Mathematics
American mathematician
where he studied inversive geometry with Frank Morley. In 1926, he wrote a dissertation "Differential Invariants of Inversive Geometry" for his doctoral
Boyd_Crumrine_Patterson
Well studied projective geometries over finite fields
Univ. of North Carolina Press, pp. 426–514 Bruen, A.A. (1978). "Inversive geometry and some translation planes, I". Geometriae Dedicata. 7: 81–98. doi:10
Spread_(projective_geometry)
Circles in two perpendicular families
In geometry, Apollonian circles are two families (pencils) of circles such that every circle in the first family intersects every circle in the second
Apollonian_circles
Describes the general shape and layout of an aircraft wing
here under more than one heading. This is particularly so for variable geometry and combined (closed) wing types. Most of the configurations described
Wing_configuration
Computing joint values of a kinematic chain from a known end position
convenient to exploit the geometry of the system and decompose it using subproblems with known solutions. Other applications of inverse kinematic algorithms
Inverse_kinematics
Line which touches a circle at exactly one point
externally tangent to the other of the two circles. In Möbius or inversive geometry, lines are viewed as circles through a point "at infinity" and for
Tangent_lines_to_circles
Mapping from a Euclidean space to itself
{v\cdot a-c}{a\cdot a}}a.} Additive inverse Coordinate rotations and reflections Householder transformation Inversive geometry Plane of rotation Reflection mapping
Reflection_(mathematics)
Projective construction in ring theory
quaternion-multiplicative-inverse transformation in his 1911 relativity study. In 1947 some elements of inversive quaternion geometry were described by P.G
Projective_line_over_a_ring
German Jewish mathematician (1790–1861)
published a paper about the inversion transformation, which leads to inversive geometry. His reputation as a mathematician was established by 1834 and an
Ludwig_Immanuel_Magnus
Model of hyperbolic geometry
models. Poincaré half-plane model Poincaré disk model Poincaré metric Inversive geometry Beltrami, Eugenio (1868). "Saggio di interpretazione della geometria
Beltrami–Klein_model
Three linked but pairwise separated rings
the impossibility of circular realizations, by Helge Tverberg, uses inversive geometry to transform any three circles so that one of them becomes a line
Borromean_rings
Geometry founded on spheres
infinity (i.e., having infinite radius). This extension is known as inversive geometry with automorphisms known as "Mobius transformations". Second, points
Lie_sphere_geometry
Mathematical formulation of vector pairs used in physics (rigid body dynamics)
a screw transformation. The tradition of inversive geometry borrows some of the ideas of projective geometry and provides a language of transformation
Screw_theory
instrument of Forschungszentrum Jülich at FRM II and EMU at ANSTO. Inverse geometry spectrometers at spallation sources include IRIS and OSIRIS at the
Neutron_backscattering
Soviet mathematician (1921–1988)
include line coordinates in the Euclidean and Lobachevski planes, and inversive geometry. The first three books were originally published in English by Random
Isaak_Yaglom
Rational function of the form (az + b)/(cz + d)
Topic: Klein's Theory of the Icosahedron, p. 66. J.B. Wilker (1981) "Inversive Geometry", MR 0661793 Iwaniec, Tadeusz and Martin, Gaven, The Liouville theorem
Möbius_transformation
Straight figure with zero width and depth
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature. It is a special case of a curve
Line_(geometry)
In geometry, the limiting points of two disjoint circles A and B in the Euclidean plane are points p that may be defined by any of the following equivalent
Limiting_point_(geometry)
Plane curve
the directrices can be constructed by compass and straightedge (see Inversive geometry). Pole-polar relations exist for hyperbolas and parabolas as well
Ellipse
Universality of construction using just a straightedge and a single circle with center
Drafting Geometric algebra Geometric invariant theory Geometrography Inversive geometry Steel square T-Square Eves 1963, p.205 https://www.encyclopedia
Poncelet–Steiner_theorem
Equation for radii of tangent circles
of the original three circles; Coxeter also provided a proof using inversive geometry. Additional proofs involve arguments based on symmetry, calculations
Descartes'_theorem
Model of hyperbolic geometry
hyperbolic plane for its world geometry, and also uses the Poincaré disk model. Poincaré metric Pseudosphere Inversive geometry Uniform tilings in hyperbolic
Poincaré_disk_model
Type of transformations applicable to coordinate space-time
radius R. His work initiated a large body of publications, now called inversive geometry. The most prominently named mathematician became August Ferdinand
Inversion_transformation
Geometrical structure
points, tangent or non-intersecting). This homogeneous geometry is called classical inversive geometry or a Möbius plane. The inhomogeneity of the description
Benz_plane
Geometric inversion of a torus, cylinder or double cone
This property means that Dupin cyclides are natural objects in Lie sphere geometry. Dupin cyclides are often simply known as cyclides, but the latter term
Dupin_cyclide
the momentum and energy transferred by the neutron to the sample. Inverse geometry spectrometers are also possible. In this case, the final position and
Neutron time-of-flight scattering
Neutron_time-of-flight_scattering
Mathematical concept
mathematics, the inverse function of a function f (also called the inverse of f) is a function that undoes the operation of f. The inverse of f exists if
Inverse_function
American mathematician (1899–1980)
published the "stimulating volume," Inversive Geometry. The book develops complex numbers as a tool for geometry and function theory. In 1936 Morley's
Frank_Vigor_Morley
{R^{2}}{|x^{*}|^{2}}}x^{*}\right).} William Thomson, 1st Baron Kelvin Inversive geometry Spherical wave transformation William Thomson, Lord Kelvin (1845)
Kelvin_transform
Generalized sphere of dimension n (mathematics)
spheres – How spheres of various dimensions can wrap around each other Inversive geometry – Study of angle-preserving transformations Möbius transformation –
N-sphere
Ring of circles between two tangent circles
In geometry, the Pappus chain is a ring of circles between two tangent circles investigated by Pappus of Alexandria in the 3rd century AD. Given two circles
Pappus_chain
Field in mathematics
differential geometry have also been examined. The field concerns itself with two kinds of questions: direct problems and inverse problems. Inverse problems
Spectral_geometry
Movement of an object which leaves at least one point unchanged
the motion of the distant stars to the local inertial frame Orientation (geometry) Point reflection Rolling – motion of two objects in contact with each-other
Rotation
Creating a complex 3D surface or object by combining primitive objects
Constructive solid geometry (CSG; formerly called computational binary solid geometry) is a technique used in solid modeling. Constructive solid geometry allows a
Constructive_solid_geometry
Branch of mathematics
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems
Algebraic_geometry
Hyperbola constructed from a given triangle and point
(1953) "An extended inversive geometry", American Mathematical Monthly 60(4):233–7 Isaak Yaglom (1979) A Simple Non-Euclidean Geometry and its Physical Basis
Nine-point_hyperbola
Property shared by codirectional lines
In geometry, direction, also known as spatial direction, vector direction or relative direction, is the common characteristic of all rays which coincide
Direction_(geometry)
Theorem in Euclidean geometry
Mohr–Mascheroni theorem to non-Euclidean geometries. Napoleon's problem Geometrography Inversive geometry Projective geometry Eves 1963, p. 201 Georg Mohr, Euclides
Mohr–Mascheroni_theorem
Geometry problem about finding touching circles
parallel lines can be considered as tangent at a point at infinity in inversive geometry (see below). The solution circle may be either internally or externally
Problem_of_Apollonius
Mathematical functions
inverse hyperbolic functions hyperbolic area functions. Hyperbolic functions occur in the calculation of angles and distances in hyperbolic geometry.
Inverse_hyperbolic_functions
Conjecture in algebraic geometry
publicized by Abhyankar as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus. The
Jacobian_conjecture
Geometric line segment whose endpoints lie on a circular arc
In geometry, a chord (from Latin chorda 'catgut, string') of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord
Chord_(geometry)
roots are all on the unit circle is necessarily self-inversive. The coefficients of self-inversive polynomials satisfy the relations. p k = ω p ¯ n − k
Cohn's_theorem
British sculptor (1934–2016)
embraced abstract art and began exploring the sculptural potential of inversive geometry. From then on, he made art exclusively with the inversion principle
John_M._Pickering
Physical law
free. [citation needed] The inverse-square law, fundamental in Euclidean spaces, also applies to non-Euclidean geometries, including hyperbolic space
Inverse-square_law
Structure in combinatorial mathematics
inversive plane, or Möbius plane, of order n. It is possible to give a geometric description of some inversive planes, indeed, of all known inversive
Block_design
Planar movement within a Euclidean space without rotation
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction
Translation_(geometry)
Construction in algebraic topology
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant”
Inverse_image_functor
Field of algebraic geometry
In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside
Birational_geometry
French mining engineer
École des Mines and contributed to mathematical geometry. The Moutard transformation in inverse geometry is named after him. Moutard was born in Soultz
Théodore_Moutard
Relation between the side lengths and altitude of a right triangle
In geometry, the inverse Pythagorean theorem (also known as the reciprocal Pythagorean theorem or the upside down Pythagorean theorem) is as follows:
Inverse_Pythagorean_theorem
Topics referred to by the same term
(mathematics) or generalised circle, a circle or straight line in inverse geometry Cline (linguistics): Cline of instantiation, a concept in systemic
Cline
Bijection of a set using properties of shapes in space
that its inverse exists. The study of geometry may be approached by the study of these transformations, such as in transformation geometry. Geometric
Geometric_transformation
Concept in projective geometry
In projective geometry, duality or plane duality is a formalization of the striking symmetry of the roles played by points and lines in the definitions
Duality_(projective_geometry)
Department of Mathematics, Darmstadt University of Technology Kahn, Jeff. Inversive planes satisfying the bundle theorem. Journal of Combinatorial Theory
Bundle_theorem
Geometrical property
inversive reflections such as circle reflection on the plane. In Felix Klein's Erlangen program, each possible group of symmetries defines a geometry
Symmetry_(geometry)
1991 book on non-Euclidean geometry
"Dr. Whatif". Its topics include hyperbolic geometry, inversive geometry, and projective geometry, following an arrangement of these topics credited to
Journey_into_Geometries
Type of curve in hyperbolic geometry
parabola in any inversive model of the hyperbolic plane is a harmonic, genus 1 curve. Martin, George E. (1986). The foundations of geometry and the non-euclidean
Hypercycle_(geometry)
Perpendicular line segment from a triangle's side to opposite vertex
In geometry, an altitude of a triangle is a line segment through a given vertex (called apex) and perpendicular to a line containing the side or edge opposite
Altitude_(triangle)
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