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In computational geometry, a Steiner point is a point that is not part of the input to a geometric optimization problem but is added during the solution
Steiner point (computational geometry)
Steiner_point_(computational_geometry)
Swiss mathematician (1796–1863)
Parallel axes rule Steiner–Lehmus theorem Steiner inellipse Steinerian Steiner point (computational geometry) Steiner point (triangle) "Steiner (print-only)"
Jakob_Steiner
Topics referred to by the same term
A Steiner point (named after Jakob Steiner) may refer to: Steiner point (computational geometry), a point added in solving a geometric optimization problem
Steiner_point
Geometry without using coordinates
Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic
Synthetic_geometry
list of books in computational geometry. There are two major, largely nonoverlapping categories: Combinatorial computational geometry, which deals with
List of books in computational geometry
List_of_books_in_computational_geometry
On short connecting nets with added points
In combinatorial mathematics, the Steiner tree problem, or minimum Steiner tree problem, named after Jakob Steiner, is an umbrella term for a class of
Steiner_tree_problem
Overview of and topical guide to geometry
Absolute geometry Affine geometry Algebraic geometry Analytic geometry Birational geometry Complex geometry Computational geometry Conformal geometry Constructive
Outline_of_geometry
Shape with three sides
Discrete and Computational Geometry: Proceedings of the 1996 AMS-IMS-SIAM Joint Summer Research Conference, Discrete and Computational Geometry—Ten Years
Triangle
Type of geometry
projective conic, and in acknowledgement of the work of Jakob Steiner, it is referred to as a Steiner conic. Suppose a projectivity is formed by two perspectivities
Projective_geometry
Method in geometry for representing a polygon by a topological skeleton
Computational Geometry (CCCG'14).. Erickson, Jeff. "Straight Skeleton of a Simple Polygon". 2D Straight Skeleton in CGAL, the Computational Geometry Algorithms
Straight_skeleton
Subdivision of a planar object into triangles
Kreveld, Marc van; Overmars, Mark H.; Schwarzkopf, Otfried (2000). Computational geometry: algorithms and applications (2 ed.). Berlin Heidelberg: Springer
Triangulation_(geometry)
Point on a line segment which is equidistant from both endpoints
In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and
Midpoint
(plane geometry) Pivot theorem (circles) Pompeiu's theorem (Euclidean geometry) Poncelet's closure theorem (conics) Poncelet–Steiner theorem (geometry) Ptolemy's
List_of_theorems
Triangulation method
In computational geometry, a Delaunay triangulation or Delone triangulation of a set of points in the plane subdivides their convex hull into triangles
Delaunay_triangulation
In computational geometry, the multiple line segment intersection problem supplies a list of line segments in the Euclidean plane and asks whether any
Multiple line segment intersection
Multiple_line_segment_intersection
statistics and computational geometry, simplicial depth is a measure of central tendency determined by the simplices that contain a given point. For the Euclidean
Simplicial_depth
Analysis of datasets using techniques from topology
Cohen-Steiner, David; Edelsbrunner, Herbert; Harer, John (2006-12-12). "Stability of Persistence Diagrams". Discrete & Computational Geometry. 37 (1):
Topological_data_analysis
Computational geometry problem
pair of points problem or closest pair problem is a problem of computational geometry: given n {\displaystyle n} points in metric space, find a pair of
Closest pair of points problem
Closest_pair_of_points_problem
Tree connecting given points by short paths
In metric geometry and computational geometry, a minimum-diameter spanning tree of a finite set of points in a metric space is a spanning tree in which
Minimum-diameter spanning tree
Minimum-diameter_spanning_tree
Field of mathematics which studies incidence structures
pg(s, t, α). If α = 1 these partial geometries are generalized quadrangles. If α = s + 1 these are called Steiner systems. For n > 2, a generalized n-gon
Incidence_geometry
Points usable to draw any planar graph
Squarcella, Claudio (2018), "Small Universal Point Sets for k-Outerplanar Graphs", Discrete & Computational Geometry, 60 (2): 430–470, doi:10.1007/s00454-018-0009-x
Universal_point_set
Algorithm for computing convex hulls in a set of points
textbook example of what and how may fail due to floating-point computations in computational geometry. Later D. Jiang and N. F. Stewart elaborated on this
Graham_scan
Theorem in projective geometry
a time, through 20 Steiner points. There are 20 Cayley lines which consist of a Steiner point and three Kirkman points. The Steiner points also lie, four
Pascal's_theorem
Set of primitive shapes whose union equals a polygon
for a given polygon. This is an important class of problems in computational geometry. There are many different polygon covering problems, depending on
Polygon_covering
Collection of subsets covering all t-element subsets
of a Steiner system and are dual to the concept of a packing design. They have connections to coding theory, Turán theory, and finite geometry. Let V
Covering_design
Type of random mathematical object
fields, including spatial point processes, stochastic geometry, spatial statistics and continuum percolation theory. The point process depends on a single
Poisson_point_process
Set of basic shapes which assemble into a polygon
perimeters). Polygon partitioning is an important class of problems in computational geometry. There are many different polygon partition problems, depending
Polygon_partition
Johnson–Lindenstrauss lemma (Euclidean geometry) Margulis lemma Lebesgue's number lemma (dimension theory) Gauss's lemma (Riemannian geometry) Craig interpolation lemma
List_of_lemmas
Shortest network connecting points
Handbook of Computational Geometry, Elsevier, pp. 425–461, MR 1746681 Georgakopoulos, George; Papadimitriou, Christos H. (1987), "The 1-Steiner tree problem"
Euclidean minimum spanning tree
Euclidean_minimum_spanning_tree
Quadrilateral symmetric across a diagonal
"Quadrilateral meshing by circle packing", International Journal of Computational Geometry and Applications, 10 (4): 347–360, arXiv:cs.CG/9908016, doi:10
Kite_(geometry)
Method for computing topological features of a space at different spatial resolutions
Persistent Homology". Discrete & Computational Geometry. 33 (2): 249–274. doi:10.1007/s00454-004-1146-y. ISSN 0179-5376. Cohen-Steiner, David; Edelsbrunner, Herbert;
Persistent_homology
Three tangent circles in a triangle
Reprinted in Steiner, Jacob (1881), Weierstrass, K. (ed.), Gesammelte Werke, Berlin: Druck und Verlag von G. Reimer, pp. 17–76 and separately as Steiner, Jacob
Malfatti_circles
Polygon intersected up to twice by lines orthogonal to a given line
monotone polygons Preparata, Franco P.; Shamos, Michael Ian (1985), Computational Geometry – An Introduction, Springer-Verlag, ISBN 0-387-96131-3, 1st edition;
Monotone_polygon
Class of algorithms in computational geometry
In computational geometry, numerous algorithms are proposed for computing the convex hull of a finite set of points, with various computational complexities
Convex_hull_algorithms
Branch of geometry
In mathematics, contact geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying
Contact_geometry
Radoslav; Pach, János (2011). "A computational approach to Conway's thrackle conjecture". Computational Geometry. 44 (6–7): 345–355. arXiv:1002.3904
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Algorithm for computing convex hulls in a set of points
In computational geometry, the gift wrapping algorithm is an algorithm for computing the convex hull of a given set of points. In the two-dimensional case
Gift_wrapping_algorithm
Shape that blocks all lines of sight
Minghui (2014), "The opaque square", Proc. 30th Annual Symposium on Computational Geometry (SoCG'14), New York: Association for Computing Machinery, pp. 529–538
Opaque_set
Tiling of the hyperbolic plane
Wolfgang; Phillips, Jeff M. (eds.). 40th International Symposium on Computational Geometry, SoCG 2024, June 11-14, 2024, Athens, Greece. LIPIcs. Vol. 293.
Binary_tiling
Point set triangulation minimizing total length
In computational geometry and computer science, the minimum-weight triangulation problem is the problem of finding a triangulation of minimal total edge
Minimum-weight_triangulation
Subdivision of the plane by lines
2017-08-08, retrieved 2024-10-16 Tóth, G. (2001), "Point sets with many k-sets", Discrete & Computational Geometry, 26 (2): 187–194, doi:10.1007/s004540010022
Arrangement_of_lines
Point minimizing sum of distances to given points
In geometry, the geometric median of a discrete point set in a Euclidean space is the point minimizing the sum of distances to the sample points. This
Geometric_median
Method of drawing geometric objects
In geometry, straightedge-and-compass construction – also known as ruler-and-compass construction, Euclidean construction, or classical construction –
Straightedge and compass construction
Straightedge_and_compass_construction
conic sections Jakob Steiner (1796–1863) – champion of synthetic geometry methodology, projective geometry, Euclidean geometry Karl Wilhelm Feuerbach
List_of_geometers
Study of complex manifolds and several complex variables
geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry
Complex_geometry
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
Carathéodory's theorem is a theorem in convex geometry. It states that if a point x {\displaystyle x} lies in the convex hull C o n v ( P ) {\displaystyle
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Graph-theoretic description of polyhedra
Hsien-Chih; Erickson, Jeff (2017), "Untangling planar curves", Discrete & Computational Geometry, 58 (4): 889–920, arXiv:1702.00146, doi:10.1007/s00454-017-9907-6
Steinitz's_theorem
Shape in hyperbolic geometry
Discrete & Computational Geometry, 64 (1): 63–108, arXiv:1707.06848, doi:10.1007/s00454-019-00132-8, MR 4110530, S2CID 203035718 Steiner, Jakob (1832)
Ideal_polyhedron
Triangle with at least two sides congruent
"Reptilings and space-filling curves for acute triangles", Discrete & Computational Geometry, 60 (1): 170–199, arXiv:1603.01382, doi:10.1007/s00454-017-9953-0
Isosceles_triangle
German mathematician (1826–1866)
point can be reduced to a number (scalar), with the surfaces of constant positive or negative curvature being models of the non-Euclidean geometries.
Bernhard_Riemann
Five sporadic simple groups
equivalence a unique S(5,8,24) Steiner system W24 (the Witt design). The group M24 is the automorphism group of this Steiner system; that is, the set of
Mathieu_group
Line constructed from a triangle
"Circumcenter of Mass and Generalized Euler Line", Discrete and Computational Geometry, 51 (4): 815–836, arXiv:1301.0496, doi:10.1007/s00454-014-9597-2
Euler_line
Points with no line through exactly two points
13-element Steiner triple systems. 157353, the parameters of a three-dimensional projective space over a two-element field and of 79 other Steiner triple
Sylvester–Gallai configuration
Sylvester–Gallai_configuration
Samples". Discrete & Computational Geometry. 39 (1–3): 419–441. doi:10.1007/s00454-008-9053-2. ISSN 0179-5376. S2CID 1788129. Cohen-Steiner, David; Edelsbrunner
Offset_filtration
Geometric concept of a 2D space with "points at infinity" adjoined
plane of order N is a Steiner S(2, N + 1, N2 + N + 1) system (see Steiner system). Conversely, one can prove that all Steiner systems of this form (λ
Projective_plane
Problem in computational complexity theory
Mark H. (1995), "On a class of O(n2) problems in computational geometry", Computational Geometry: Theory and Applications, 5 (3): 165–185, doi:10
3SUM
Shape with three equal sides
Yushi (eds.). Discrete and Computational Geometry and Graphs. Japanese Conference on Discrete and Computational Geometry and Graphs. Kyoto. doi:10
Equilateral_triangle
1960 article by Eugene Wigner
notion of the supreme wise being. Cosmology Foundations of mathematics Mark Steiner Mathematical universe hypothesis Philosophy of science Quasi-empiricism
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
Variant of the Steiner tree problem in geometry and combinatorics
The rectilinear Steiner tree problem, minimum rectilinear Steiner tree problem (MRST), or rectilinear Steiner minimum tree problem (RSMT) is a variant
Rectilinear_Steiner_tree
Sporadic simple group
T. (1984), "The Steiner system S(5, 6, 12), the Mathieu group M₁₂ and the "kitten"", in Atkinson, Michael D. (ed.), Computational group theory. Proceedings
Mathieu_group_M12
Flat-sided three-dimensional shape
as well as appearing in biological creatures, nature, and modern computational geometry. The Original Sin in the theory of polyhedra goes back to Euclid
Polyhedron
Branch of mathematics
Newton and Leibniz became the starting point for much of later analysis. In 17th century Europe, new methods for geometry and mathematical physics laid the
Mathematical_analysis
Scheucher, Manfred; Schröder, Felix; Steiner, Raphael (2020), "Topological drawings meet classical theorems from convex geometry", Proceedings of the 28th International
Kirchberger's_theorem
notion of a plane does not carry over.) It is now recognized that Euclidean geometry can be studied as a mathematical abstraction, but that the universe is
List_of_conjectures
Invariant in projective geometry
Cross-Ratio Geometry with Historical Notes, Cambridge University Press. Olver, Peter J. (2001), "Joint Invariant Signatures", Foundations of Computational Mathematics
Cross-ratio
Convex polygon with pairs of equal, parallel sides
distance problem for centrally symmetric convex polygons", Discrete & Computational Geometry, 28 (4): 467–473, doi:10.1007/s00454-002-2882-5, MR 1949894
Zonogon
Circle that passes through the vertices of a triangle
coordinates are trilinear coordinates: Steiner point: the non-vertex point of intersection of the circumcircle with the Steiner ellipse. b c b 2 − c 2 : c a c
Circumcircle
Partition of a polygon into triangles of equal area
equal areas", Discrete & Computational Geometry, 4 (1): 375–381, doi:10.1007/BF02187738, Zbl 0675.52005 Kasimatis, Elaine A.; Stein, Sherman K. (1 December
Equidissection
Inverse function to a tower of powers
{\displaystyle O(\log ^{*}n)} . In computational complexity theory, Santhanam shows that the computational resources DTIME — computation time for a deterministic
Iterated_logarithm
Theorem in geometry
Erickson, Jeff (ed.). Proceedings of the 23rd ACM Symposium on Computational Geometry, Gyeongju, South Korea, June 6–8, 2007. pp. 302–305. doi:10.1145/1247069
Brunn–Minkowski_theorem
Geometry problem about finding touching circles
In Euclidean plane geometry, the problem of Apollonius (also called Apollonius's problem or the Apollonian problem) is to construct circles that are tangent
Problem_of_Apollonius
whole net with multiple nodes is better represented by the rectilinear Steiner tree, the RMST provides a reasonable approximation and wire length estimate
Rectilinear minimum spanning tree
Rectilinear_minimum_spanning_tree
Measure of difference between two points
p ( i ) {\displaystyle F(p)=-\sum _{i}\log p(i)} A key tool in computational geometry is the idea of projective duality, which maps points to hyperplanes
Bregman_divergence
Area of discrete mathematics
continuous curved edges in Euclidean space. As part of discrete geometry and computational geometry, geometric graph theory studies planar graphs, relationship
Graph_theory
Symposium on Geometry Processing (SGP) is an annual symposium hosted by the European Association For Computer Graphics (Eurographics). The goal of the
Symposium on Geometry Processing
Symposium_on_Geometry_Processing
Esther Arkin, Israeli-American researcher in operations research and computational geometry Sandra Arlinghaus, founder of the Institute of Mathematical Geography
List_of_women_in_mathematics
American mathematician and Nobel Laureate (1928–2015)
made fundamental contributions to game theory, real algebraic geometry, differential geometry, and partial differential equations. Nash and fellow game theorists
John_Forbes_Nash_Jr.
Relationship between fields of study
the limitation of a single Euclidean geometry. A version of non-Euclidean geometry, called Riemannian geometry, enabled Albert Einstein to develop general
Relationship between mathematics and physics
Relationship_between_mathematics_and_physics
On tangency patterns of circles
Vogtenhuber, Birgit (2012), "Pointed drawings of planar graphs", Computational Geometry, 45 (9): 482–494, doi:10.1016/j.comgeo.2010.08.001, MR 2926292 Alam
Circle_packing_theorem
Sporadic simple group
T. (1984), "The Steiner system S(5, 6, 12), the Mathieu group M₁₂ and the "kitten"", in Atkinson, Michael D. (ed.), Computational group theory. Proceedings
Mathieu_group_M11
and some devoted to the detail elaboration of algorithm either from geometry point of view (Li Huang) or from Tian yuan shu (Zhang Dunren). In 1963, Chinese
Jigu_Suanjing
Number, approximately 3.14
Jonathan M. (2016). "15.2 Computational records". Pi: The Next Generation, A Sourcebook on the Recent History of Pi and Its Computation. Springer International
Pi
Computational geometry concept
In computational geometry, a bitonic tour of a set of point sites in the Euclidean plane is a closed polygonal chain that has each site as one of its vertices
Bitonic_tour
NP-hard problem in combinatorial optimization
In the theory of computational complexity, the travelling salesman problem (TSP) asks the following question: "Given a list of cities and the distances
Travelling_salesman_problem
Overview of and topical guide to algorithms
NP-completeness and computational complexity theory Juris Hartmanis — computational complexity theory Richard E. Stearns — computational complexity theory
Outline_of_algorithms
Combinatorics problem proposed by Thomas Penyngton Kirkman
triple systems later became known as Steiner triple systems. In 1859, Michel Reiss answered the questions raised by Steiner, using both methodology and notation
Kirkman's_schoolgirl_problem
Mathematical space with a notion of distance
implications for various computational problems: Network design: Improves approximation algorithms for problems like the Group Steiner tree problem (a generalization
Metric_space
Assembly of systems connected to manage forces and movement
VII:213–216, 1876 Jordan, D.; Steiner, M. (1999). "Configuration Spaces of Mechanical Linkages". Discrete & Computational Geometry. 22 (2): 297–315. doi:10
Linkage_(mechanical)
Series of books published by Springer-Verlag
(2015). Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra (4th ed.). doi:10.1007/978-3-319-16721-3
Undergraduate Texts in Mathematics
Undergraduate_Texts_in_Mathematics
Characterizes spherical triangles with fixed base and area
polygones sphériques, d'après M. Steiner" [Lexell's theorem, and transformation of spherical polygons, after Mr. Steiner], Nouvelles Annales de Mathématiques
Lexell's_theorem
German mathematician (1862–1943)
variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations
David_Hilbert
Branch of mathematics studying functions of a complex variable
many branches of mathematics, including functional analysis, algebraic geometry, number theory, analytic combinatorics, and applied mathematics, as well
Complex_analysis
Directed graph with no directed cycles
and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling)
Directed_acyclic_graph
Tree data structure to hold intervals
Preparata and Michael Ian Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985 CGAL : Computational Geometry Algorithms Library in C++ contains
Interval_tree
Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for taxation, commerce, trade, and in astronomy, to record time and formulate
History_of_mathematics
Mathematical condition
physics, particularly in the context of electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is always
Poincaré_lemma
In physics and geometry: conjectured relation between pairs of Calabi–Yau manifolds
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers
Mirror symmetry (string theory)
Mirror_symmetry_(string_theory)
1920s books on mathematical history by Felix Klein
Moebius, Plücker and Steiner), is followed by a chapter on "The Development of Algebraic Geometry after Moebius, Plücker and Steiner", with sections on
Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert
Vorlesungen_über_die_Entwicklung_der_Mathematik_im_19._Jahrhundert
Multi-dimensional generalization of triangle
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex
Simplex
Optimization problem
is as large as possible". OFL is studied in operations research, computational geometry (a branch of computer science), and location theory (a branch of
Optimal_facility_location
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STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
STEINER POINT-COMPUTATIONAL-GEOMETRY
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