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In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces. Let (X, || ||) be a complex Banach space and let B
Complex_geodesic
Iranian mathematician (1977–2017)
that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal. The closures of complex geodesics are
Maryam_Mirzakhani
Straight path on a curved surface or a Riemannian manifold
In geometry, a geodesic (/ˌdʒiː.əˈdɛsɪk, -oʊ-, -ˈdiːsɪk, -zɪk/) is a curve representing in some sense the locally shortest path (arc) between two points
Geodesic
Study of Earth's spatial information
Kath, Khwarezm, using the maximum altitude of the Sun, and solved a complex geodesic equation to accurately compute the Earth's circumference, which was
Geography
Spatial grid based on a geodesic polyhedron
A geodesic grid is a spatial grid based on a geodesic polyhedron or Goldberg polyhedron. The earliest use of the (icosahedral) geodesic grid in geophysical
Geodesic_grid
Mathematics of smooth surfaces
a geodesic of sufficiently short length will always be the curve of shortest length on the surface which connects its two endpoints. Thus, geodesics are
Differential geometry of surfaces
Differential_geometry_of_surfaces
American philosopher, architect and inventor (1895–1983)
known geodesic dome; carbon molecules known as fullerenes were later named by scientists for their structural and mathematical resemblance to geodesic spheres
Buckminster_Fuller
Smooth manifold with an inner product on each tangent space
{\displaystyle \gamma '(0)=v} exists, one obtains a geodesic called a maximal geodesic of which every geodesic satisfying γ ( 0 ) = p {\displaystyle \gamma (0)=p}
Riemannian_manifold
Mathematical concept
in complex projective space, there passes a unique complex line (a CP1). A great circle of this complex line that contains p and q is a geodesic for
Complex_projective_space
Paths of particles in the Schwarzschild solution to Einstein's field equations
In general relativity, Schwarzschild geodesics describe the motion of test particles in the gravitational field of a central fixed mass M , {\textstyle
Schwarzschild_geodesics
a metric space, if and only if all geodesics can be infinitely extended. Complete metric space Completion Complex hyperbolic space Conformal map is a
Glossary of Riemannian and metric geometry
Glossary_of_Riemannian_and_metric_geometry
Theorem in differential geometry
with boundary ∂M. Let K be the Gaussian curvature of M, and let kg be the geodesic curvature of ∂M. Then ∫ M K d A + ∫ ∂ M k g d s = 2 π χ ( M ) , {\displaystyle
Gauss–Bonnet_theorem
18th-century expedition to present-day Ecuador
The Spanish-French Geodesic Mission (French: Expédition géodésique française en Équateur), also called the French Geodesic Mission to Peru, was an 18th-century
French Geodesic Mission to the Equator
French_Geodesic_Mission_to_the_Equator
Algebraic operation on coordinate vectors
. The inner product of two vectors over the field of complex numbers is, in general, a complex number, and is sesquilinear instead of bilinear. An inner
Dot_product
3} . Every totally geodesic submanifold of the complex hyperbolic space of dimension n is one of the following : a copy of a complex hyperbolic space of
Complex_hyperbolic_space
United States historic place
The ASM International Headquarters and Geodesic Dome, at the Materials Park campus in Russell Township, Geauga County, Ohio, United States, are the headquarters
ASM Headquarters and Geodesic Dome
ASM_Headquarters_and_Geodesic_Dome
Environment museum in Montreal, Quebec
the grounds of Parc Jean-Drapeau on Saint Helen's Island. The museum's geodesic dome was designed by Buckminster Fuller. The structure was originally built
Montreal_Biosphere
Three-holed sphere
follows: Given a hyperbolic pair of pants with totally geodesic boundary, there exist three unique geodesic arcs that join the cuffs pairwise and that are perpendicular
Pair_of_pants_(mathematics)
Architectural element similar to the hollow upper half of a sphere; there are many types
and are a type of "circular dome" for that reason. Geodesic domes are the upper portion of geodesic spheres. They are composed of a framework of triangles
Dome
Topological space formed from distances
In a geodesically convex space Y, the Vietoris–Rips complex of any subspace X ⊂ Y for distance δ has the same points and edges as the Čech complex of the
Vietoris–Rips_complex
Array of numbers
{\displaystyle F} . A real matrix and a complex matrix are matrices whose entries are respectively real numbers or complex numbers. More general types of entries
Matrix_(mathematics)
topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected
Busemann_function
Compact astronomical body
inside, points where the curvature of spacetime becomes infinite, and geodesics terminate within a finite proper time. For a non-rotating black hole,
Black_hole
Branch of mathematics
on the Earth's surface. Indeed, the measurements of distance along such geodesic paths by Eratosthenes and others can be considered a rudimentary measure
Differential_geometry
Visitor attraction in Cornwall, United Kingdom
pentagonal ethylene tetrafluoroethylene (ETFE) inflated cells supported by geodesic tubular steel domes. The larger of the two biomes simulates a rainforest
Eden_Project
parallelism Prime geodesic Geodesic flow Exponential map (Lie theory) Exponential map (Riemannian geometry) Injectivity radius Geodesic deviation equation
List of differential geometry topics
List_of_differential_geometry_topics
Branch of mathematics that studies dynamical systems
kind. In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf
Ergodic_theory
Field-equations in general relativity
trajectories of particles and radiation (geodesics) in the resulting geometry are then calculated using the geodesic equation. As well as implying local energy–momentum
Einstein_field_equations
Architechural Model Maker
Cornwall by Grimshaw Architects. This was a challenging model given the complex geodesic design and Armiger describes the model's construction in detail in
Richard_Armiger
American public spaceflight company
satellite payload called Humanity Star, a 1 m-wide (3.3 ft) carbon fiber geodesic sphere made of 65 panels that reflect the Sun's light. Humanity Star re-entered
Rocket_Lab
Movie theater in Hollywood, California
design for theaters that would show its movies. They would be based on the geodesic dome developed by R. Buckminster Fuller, would cost half as much as conventional
Cinerama_Dome
Equations that describe the behavior of a physical system
fictitious force. The relative acceleration of one geodesic to another in curved spacetime is given by the geodesic deviation equation: D 2 ξ α d s 2 = − R α β
Equations_of_motion
Science of measuring the shape, orientation, and gravity of Earth
The general solution is called the geodesic for the surface considered, and the differential equations for the geodesic are solvable numerically. On the
Geodesy
Upper-half plane model of hyperbolic non-Euclidean geometry
} This provides a basic description of the geodesic flow on the unit-length tangent bundle (complex line bundle) on the upper half-plane. Starting
Poincaré_half-plane_model
Four-dimensional number system
}=\|q\|^{x}{\bigl (}\cos(x\varphi )+{\hat {n}}\sin(x\varphi ){\bigr )}.} The geodesic distance dg(p, q) between unit quaternions p and q is defined as: d g (
Quaternion
Concept in mathematics
space X {\displaystyle X} is geodesic, i.e. any two points x , y ∈ X {\displaystyle x,y\in X} are end points of a geodesic segment [ x , y ] {\displaystyle
Hyperbolic_metric_space
parametrised by arclength. If a geodesic arrives at a singularity it is required to stop there. Thus a maximal geodesic is a curve defined on a closed
Translation_surface
Branch of mathematics
Amsterdam: Elsevier. ISBN 978-0-444-88355-1. OCLC 162589397. "geodesic – definition of geodesic in English from the Oxford dictionary". OxfordDictionaries
Geometry
Theme park at Walt Disney World
culture. Epcot is also known for its iconic landmark, Spaceship Earth, a geodesic sphere. The EPCOT name originated as an acronym for Experimental Prototype
Epcot
Skyscraper in George Town, Penang, Malaysia
substation (phase 2A), a department complex (phase 2B), an 11-storey car park with 750 parking spaces (phase 2C), a geodesic dome (phase 2D) and an 11-acre
Komtar
Condition in which spacetime itself breaks down
defined by the scalar invariant curvature becoming infinite or, better, by a geodesic being incomplete. General relativity predicts that any object collapsing
Gravitational_singularity
(pseudo-)Riemannian manifold whose geodesics are reversible
T_{p}M} as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both
Symmetric_space
Length of shortest path between two nodes of a graph
edges in a shortest path (also called a graph geodesic) connecting them. This is also known as the geodesic distance or shortest-path distance. Notice that
Distance_(graph_theory)
Property of a mathematical space
sometimes useful in the study of complex manifolds and algebraic varieties to work over the complex numbers instead. A complex number ( x + i y {\displaystyle
Dimension
Tensor in differential geometry
a curved space locally differs from flat space by tracking how nearby geodesics spread apart or converge. Formally, it is a symmetric rank-two tensor
Ricci_curvature
Parametrizes complex structures on a surface
then f ∗ α {\displaystyle f_{*}\alpha } is homotopic to a unique closed geodesic α x {\displaystyle \alpha _{x}} on M {\displaystyle M} (up to parametrisation)
Teichmüller_space
Algorithm used to solve non-linear least squares problems
step as the velocity v k {\displaystyle {\boldsymbol {v}}_{k}} along a geodesic path in the parameter space, it is possible to improve the method by adding
Levenberg–Marquardt_algorithm
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
mirroring or reversal of the direction of a geodesic. Its differential flips the direction of the tangents to a geodesic. It is a linear operator of norm one;
Killing_vector_field
Mathematical theory of the geometry of space and time
force in Newton's static Euclidean reference frame. Objects move along geodesics—curved paths determined by the local geometry of spacetime—rather than
Curved_spacetime
Mathematical function that preserves angles
sciences (including brain mapping and genetic mapping), in applied math (for geodesics and in geometry), in earth sciences (including geophysics, geography,
Conformal_map
On the structure of complete Riemannian manifolds of non-positive sectional curvature
simply connected then it is a geodesic space in the sense that any two points are connected by a unique minimizing geodesic, and hence contractible. A metric
Cartan–Hadamard_theorem
Cylindrical conformal map projection
circumference of that parallel; i.e., 10,007.5 km. On the other hand, the geodesic between these points is a great circle arc through the pole subtending
Mercator_projection
Polyhedron resembling a soccerball
that are typically patterned with white hexagons and black pentagons. Geodesic dome structures, such as those whose architecture Buckminster Fuller pioneered
Truncated_icosahedron
Diffeomorphism that has a hyperbolic structure on the tangent bundle
flow comes from the realization that g t {\displaystyle g_{t}} is the geodesic flow on P and Q. Lie vector fields being (by definition) left invariant
Anosov_diffeomorphism
Four-dimensional analog of the icosahedron
structure exists in the 600-cell, although it is somewhat more complex. The 10-cell geodesic path in the 120-cell corresponds to the 10-vertex decagon path
600-cell
manifolds, geodesics are paths of shortest distance between two points. The concept of geodesics becomes central in general relativity, since geodesic motion
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
German polymath and scholar (1777–1855)
triangles to geodesic triangles on arbitrary surfaces with continuous curvature; he found that the angles of a "sufficiently small" geodesic triangle deviate
Carl_Friedrich_Gauss
Mathematical space with a notion of distance
becomes a geodesic: a curve which is a distance-preserving function. A geodesic is a shortest possible path between any two of its points. A geodesic metric
Metric_space
American mathematician (born 1958)
1007/S00222-015-0590-Z, S2CID 253742362, Zbl 1364.37103 McMullen, C. T.; et al. (2017), "Geodesic planes in hyperbolic 3-manifolds", Invent. Math., 209 (2): 425–461, Bibcode:2017InMat
Curtis_T._McMullen
Graph operation
's two articles. Regular lattices over the complex plane can be used to create "master polygons". In geodesic dome terminology, this is the "breakdown structure"
Goldberg–Coxeter_construction
Number, approximately 3.14
be defined using properties of the complex exponential, exp z, of a complex variable z. Like the cosine, the complex exponential can be defined in one
Pi
Branch of mathematics
obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related problem is posed by Fermat's principle: light follows the path
Calculus
Rubber products with structural reinforcement
for each radius. In other words, for complex shapes there is not one magic angle but the fibres follow a geodesic path with angles varying with the change
Reinforced_rubber
Zoo and aquarium in Omaha, Nebraska, US
one of the world's largest indoor deserts, as well as the largest glazed geodesic dome. The zoo's mission includes four pillars—conservation, research, recreation
Omaha's Henry Doorly Zoo and Aquarium
Omaha's_Henry_Doorly_Zoo_and_Aquarium
Surface homeomorphic to a sphere
homeomorphic to the 2-sphere, equipped with a Riemannian metric all of whose geodesics are closed and of equal length. While the usual unit-sphere metric on
Zoll_surface
US scientific research station at the South Pole, Antarctica
station was moved in 1975 to the newly constructed Buckminster Fuller geodesic dome 160 feet (50 m) wide by 52 feet (16 m) high, with 46 by 79 feet (14 m
Amundsen–Scott South Pole Station
Amundsen–Scott_South_Pole_Station
Concept in mathematics
the free factor complex (sometimes also called the complex of free factors) is a free group counterpart of the notion of the curve complex of a finite type
Free_factor_complex
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
2016-01-28, retrieved 2011-08-03 Besse, Arthur (1978). Manifolds all of whose Geodesics are Closed. Springer-Verlag. ISBN 978-3-540-08158-6. (§0.26 on page 6)
Hopf_fibration
Entertainment venue in the Las Vegas Valley, United States
it would be the largest spherical building on Earth, if it were not a geodesic dome. The arena cost $2.3 billion, making it the most expensive entertainment
Sphere_(venue)
Polygon associated with a compact Riemann surface
diagonal piecewise geodesic segment in its interior. The final polygon has 4g equivalent vertices, with sides that are piecewise geodesic. The sides are labelled
Fundamental_polygon
Protein shell of a virus
Geometric examples for many values of h, k, and T can be found at List of geodesic polyhedra and Goldberg polyhedra. Many exceptions to this rule exist: For
Capsid
Topics referred to by the same term
X\mapsto \gamma _{X}(1)} , where γ X {\displaystyle \gamma _{X}} is a geodesic with initial velocity X, is sometimes also called the exponential map.
Exponential_map
Conjecture on zeros of the zeta function
an infinite product similar to the Euler product but taken over closed geodesics rather than primes. The Selberg trace formula is the analogue for these
Riemann_hypothesis
Equation for radii of tangent circles
is defined as k j = cot ρ j , {\textstyle k_{j}=\cot \rho _{j},} the geodesic curvature of the circle relative to the sphere, which equals the cotangent
Descartes'_theorem
Length of a line segment
three dimensions, the Euclidean distance should be distinguished from the geodesic distance, the length of a shortest curve that belongs to the surface. In
Euclidean_distance
Part of a line that is bounded by two distinct end points; line with two endpoints
bivector generalizes the directed line segment. Beyond Euclidean geometry, geodesic segments play the role of line segments. A line segment is a one-dimensional
Line_segment
Branch of geometry
a Riemannian n-manifold M {\displaystyle M} , consider its unit-speed geodesic curves (i.e. parameterized by arc length). This produces a transport of
Contact_geometry
Differentiable manifold
curvature. These spaces can be used as comparison spaces in studying geodesics and volume comparison theorems on CR manifolds with zero Webster torsion
CR_manifold
Tool from special relativity
acceleration with gravitation. The geodesic equations in the Rindler chart are easily obtained from the geodesic Lagrangian; they are t ¨ + 2 x x ˙ t
Rindler_coordinates
Function used in computer graphics
path along a line segment in the plane; a great circle is a spherical geodesic. More familiar than the general slerp formula is the case when the end
Spherical linear interpolation
Spherical_linear_interpolation
American mathematician
Conjectures". Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics
Jeremy_Kahn
Particular mapping that projects a sphere onto a plane
inverse stereographic projection from the plane to the sphere defines a geodesic distance between points in the plane equal to the spherical distance between
Stereographic_projection
Functions in mathematics
over to this more general setting, including the mean value theorem (over geodesic balls), the maximum principle, and the Harnack inequality. With the exception
Harmonic_function
Large urban area enclosed within a dome
displaying short descriptions of redirect targets Geodesic dome – Spherical shell structure based on a geodesic polyhedron IBTS Greenhouse – Egyptian desalination
Domed_city
Grouping of neighbouring metropolises
melded into a single mass of urban sprawl. It has been enclosed in several geodesic domes and merged into one megacity. The city has become a separate world
Megalopolis
Simple curve of Euclidean geometry
conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are
Circle
Hypothetical topological feature of spacetime
past for any possible trajectory of a free-falling particle (following a geodesic in the spacetime). In order to satisfy this requirement, it turns out that
Wormhole
Notation in general relativity
direction indicate whether or not the vector is tangent to a geodesic and if so, whether the geodesic has an affine parameter. A null tangent vector T a {\displaystyle
Newman–Penrose_formalism
Concept in geometry
intrinsic metric that arises by measuring geodesic length. The geodesic circles are the parallels in a geodesic coordinate system. More precisely, fix a
Area_of_a_circle
City in Uttar Pradesh, India
cover the main temple. Vrindavan Chandrodaya Mandir is housed in a modern geodesic structure with a traditional gopuram based on Khajuraho style of architecture
Vrindavan
Theorem in general relativity
a ray is geodesic and shear-free, then ε + ε ¯ = κ = σ = 0 {\displaystyle \varepsilon +{\bar {\varepsilon }}=\kappa =\sigma =0} . A complex rotation o
Goldberg–Sachs_theorem
Russian-French mathematician
vanishing theorem asserting that (certain) harmonic mappings must be totally geodesic or holomorphic. Gromov had the insight that the extension of this program
Mikhael Gromov (mathematician)
Mikhael_Gromov_(mathematician)
Area of geometry, about angles and lengths
Projections of the Sphere, Dialling, Astronomy, the Solution of Equations, and Geodesic Operations. Baldwin, Cradock, and Joy. Neugebauer, Otto (1948). "Mathematical
Trigonometry
1972 book by Deleuze and Guattari
money that produces more money? There are socioeconomic "complexes" that are also veritable complexes of the unconscious, and that communicate a voluptuous
Anti-Oedipus
Branch of differential geometry and differential topology
subjects is the analogy between geodesics in Riemannian geometry and pseudoholomorphic curves in symplectic geometry. Geodesics are curves of shortest length
Symplectic_geometry
Generalized manifold
with unique geodesics connecting any two points. Let X be an orbispace endowed with a metric space structure for which the charts are geodesic length spaces
Orbifold
Exposition buildings in Montreal, Quebec
the US pavilion, a geodesic dome designed by Buckminster Fuller. Expo 67 also featured Habitat 67, an urban modular housing complex designed by architect
Expo_67_pavilions
Arc crossing all meridians of longitude at the same angle
great circle is locally "straight" with zero geodesic curvature, whereas a rhumb line has non-zero geodesic curvature. Meridians of longitude and parallels
Rhumb_line
Method of describing higher-order polyhedra
family can be used to produce the Goldberg polyhedra and geodesic polyhedra: see List of geodesic polyhedra and Goldberg polyhedra for formulas. The two
Conway_polyhedron_notation
Domed stadium on the campus of Northern Michigan University in Marquette, Michigan
1 ha) and has a volume of 16,135,907 cubic feet (456,918.0 m3). It is a geodesic dome constructed with 781 Douglas fir beams and 108.5 miles (174.6 km)
Superior_Dome
COMPLEX GEODESIC
COMPLEX GEODESIC
Boy/Male
Tamil
Complete
Girl/Female
Tamil
Complete
Girl/Female
Tamil
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Complete
Sompurna | ஸோமபà¯à®°à¯à®¨à®¾
Boy/Male
Tamil
Complete
Boy/Male
Indian
Complete
Surname or Lastname
English
English : habitational name, probably from Comley in Shropshire or Combley on the Isle of Wight; both are named with Old English cumb ‘valley’ + lēah ‘woodland clearing’.
Girl/Female
Muslim
Complex, Zigzag, Curling
Girl/Female
Arabic, Muslim
Complex; Zigzag; Curling
Surname or Lastname
English
English : habitational name from Coppull in Lancashire, recorded in the 13th century as Cophill, from Old English copp ‘peak’ + hyll ‘hill’.English : nickname from Old French curt peil ‘short hair’.Probably an Americanized spelling of German and Jewish Koppel or German and Dutch Kappel.
Boy/Male
Indian
Complete
Girl/Female
Hindu, Indian
Complex
Girl/Female
Tamil
Shesha Harani | ஷேஷ ஹரணீÂ
Complete
Shesha Harani | ஷேஷ ஹரணீÂ
Surname or Lastname
English (Yorkshire)
English (Yorkshire) : habitational name from any of various places called Copley, for example in County Durham, Staffordshire, and Yorkshire, from the Old English personal name Coppa (apparently a byname for a tall man) or from copp ‘hilltop’ + lēah ‘woodland clearing’.
Boy/Male
Tamil
Poornan | பூரà¯à®¨à®¾à®¨
Complete
Poornan | பூரà¯à®¨à®¾à®¨
Surname or Lastname
English
English : unexplained.Americanized form of German Koppler.
Girl/Female
Tamil
Complete
Girl/Female
Tamil
Complete
Girl/Female
Tamil
Complete
Girl/Female
Bengali, Indian
Good Complex
Boy/Male
Tamil
Complete
COMPLEX GEODESIC
COMPLEX GEODESIC
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Tamil, Telugu
Goddess Laxmi
Boy/Male
Tamil
Of the Aryan race, Ancient, Warrior
Boy/Male
Hindu, Indian
Perfect
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Sanskrit, Sindhi, Tamil, Telugu
Meditative
Boy/Male
Muslim
Bright like morning
Girl/Female
Arabic, Muslim
Teller of Tales of 1001 Nights; Variant of Shahrazad; Modern; City Dweler; Person of the City; City Born; City Dweller
Girl/Female
Tamil
Venus, Flute, Created with immense power
Male
Celtic
, chief priest, or metropolitan (of the Turones).
Male
Arthurian
, chief leader in war.
Girl/Female
Hindu
COMPLEX GEODESIC
COMPLEX GEODESIC
COMPLEX GEODESIC
COMPLEX GEODESIC
COMPLEX GEODESIC
a.
Intricate; entangled; complicated; complex.
n.
One who complies, yields, or obeys; one of an easy, yielding temper.
adv.
In a complex manner; not simply.
a.
Complex, complicated.
n.
Two taken together; a pair or couple; especially two lines of verse that rhyme with each other.
n.
One who couples; that which couples, as a link, ring, or shackle, to connect cars.
n.
A complex; an aggregate of parts; a complication.
a.
Repeatedly compound; made up of complex constituents.
a.
One of the pairs of plates of two metals which compose a voltaic battery; -- called a voltaic couple or galvanic couple.
imp. & p. p.
of Couple
imp. & p. p.
of Compile
a.
See Couple-close.
n.
One who compiles; esp., one who makes books by compilation.
a.
Not complex; uncompounded; simple.
n.
Composed of two or more parts; composite; not simple; as, a complex being; a complex idea.
imp. & p. p.
of Comply
a.
That which joins or links two things together; a bond or tie; a coupler.
pl.
of Couple-close
a.
Finished; ended; concluded; completed; as, the edifice is complete.
v. t.
To bring to a state in which there is no deficiency; to perfect; to consummate; to accomplish; to fulfill; to finish; as, to complete a task, or a poem; to complete a course of education.