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Finding the smallest circle that contains all given points
The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, least bounding circle problem, smallest enclosing
Smallest-circle_problem
LP-type problems include many important optimization problems that are not themselves linear programs, such as the problem of finding the smallest circle containing
LP-type_problem
Two-dimensional packing problem
Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle. If
Circle_packing_in_a_circle
Index of articles associated with the same name
quadrilateral, a special case of a cyclic polygon. Smallest-circle problem, the related problem of finding the circle with minimal radius containing an arbitrary
Circumscribed_circle
Two-dimensional packing problem
Circle packing in a square is a packing problem in recreational mathematics where the aim is to pack n unit circles into the smallest possible square
Circle_packing_in_a_square
Israeli mathematician and computer scientist
various computational geometric optimization problems, in particular to solve the smallest-circle problem in linear time. His former doctoral students
Nimrod_Megiddo
Theorem relating the diameter of a point set to the minimum radius of an enclosing ball
studied this inequality in 1901. Algorithms also exist to solve the smallest-circle problem explicitly. Consider a compact set K ⊂ R n {\displaystyle K\subset
Jung's_theorem
Combinatorial optimization problem
Euclidean facility location problem, Euclidean 1-center problem in the plane, etc.). It is also known as the smallest circle problem. Its generalization to
1-center_problem
Field of geometry closely arranging circles on a plane
numbers of circles. Specific problems of this type that have been studied include: Circle packing in a circle Circle packing in a square Circle packing in
Circle_packing
Topics referred to by the same term
Minimum bounding circle may refer to: Bounding sphere Smallest circle problem This disambiguation page lists articles associated with the title Minimum
Minimum_bounding_circle
Problems which attempt to find the most efficient way to pack objects into containers
of 2-dimensional packing problems have been studied. People are given n unit circles, and have to pack them in the smallest possible container. Several
Packing_problems
Mathematical counting-out question
the Josephus problem, a number of people are standing in a circle waiting to be executed. Counting begins at a specified point in the circle and proceeds
Josephus_problem
Sphere that contains a set of objects
ISBN 978-3-540-20064-2 miniball open-source project Smallest Enclosing Circle Problem – describes several algorithms for enclosing a point set,
Bounding_sphere
Two-dimensional packing problem
packing problem where the objective is to determine how many congruent squares can be packed into some larger shape, often a square or circle. Square
Square_packing
Mathematical problem
Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer
Squaring_the_square
theorem – Relates to a chain of six circles together with a triangle Smallest circle problem – Finding the smallest circle that contains all given pointsPages
List_of_circle_topics
Arrangement of points on a sphere
The objective of the Thomson problem is to determine the minimum electrostatic potential energy configuration of N electrons constrained to the surface
Thomson_problem
Tree connecting given points by short paths
this tree can be found in linear time using algorithms for the smallest-circle problem and its generalizations. Ho, Jan-Ming; Lee, D. T.; Chang, Chia-Hsiang;
Minimum-diameter spanning tree
Minimum-diameter_spanning_tree
Open question in philosophy of how abstract minds interact with physical bodies
The mind–body problem is a philosophical problem concerning the relationship between thought and consciousness in the human mind and body. It addresses
Mind–body_problem
Two-dimensional packing problem
Unsolved problem in mathematics What is the smallest possible equilateral triangle which an amount n of unit circles can be packed into? More unsolved
Circle packing in an equilateral triangle
Circle_packing_in_an_equilateral_triangle
On point sets with no small-area triangles
Unsolved problem in mathematics What is the asymptotic growth rate of the area of the smallest triangle determined by three out of n {\displaystyle n}
Heilbronn_triangle_problem
midpoint x ^ {\displaystyle {\hat {x}}} . Bounding sphere Smallest-circle problem Circumscribed circle (covers circumcenter) Centre (geometry) Centroid Boyd
Chebyshev_center
Unsolved problem in mathematics What is the smallest real number r ( n ) {\displaystyle r(n)} such that n {\displaystyle n} disks of radius r ( n ) {\displaystyle
Disk_covering_problem
On reflection in a spherical mirror
touches the circle, or for an ellipse that is tangent to the circle and has the given points as its foci. Although special cases of this problem were studied
Alhazen's_problem
Geometry problem about finding touching circles
Euclidean plane geometry, Apollonius's problem is to construct circles that are tangent to three given circles in a plane (Figure 1). Apollonius of Perga
Problem_of_Apollonius
programs as well as certain nonlinear programs such as the smallest circle problem. The problem of finding the sink in a unique sink orientation of a hypercube
Unique_sink_orientation
Moser's worm problem – what is the smallest area of a shape that can cover every unit-length curve in the plane? The moving sofa problem – what is the
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Distance estimation problems in computational geometry
a largest circle centered within their convex hull and enclosing none of them Smallest enclosing rectangle: unlike the bounding box problem mentioned
Proximity_problems
Two-dimensional packing problem
Circle packing in a right isosceles triangle is a packing problem where the objective is to pack n unit circles into the smallest possible isosceles right
Circle packing in an isosceles right triangle
Circle_packing_in_an_isosceles_right_triangle
Computer scientist
geometric problems such as the development of space-efficient range searching data structures. He devised linear time randomized algorithms for the smallest circle
Emo_Welzl
Number, approximately 3.14
transcendence of π implies that it is impossible to solve the ancient problem of squaring the circle with a compass and straightedge. The decimal digits of π appear
Pi
Minimum dimension for a vehicle to make a turn
without skidding. The Oxford English Dictionary describes turning circle as "the smallest circle within which a ship, motor vehicle, etc., can be turned round
Turning_radius
Summatory function of the divisor-counting function
Dirichlet in 1849. The Dirichlet divisor problem, precisely stated, is to improve this error bound by finding the smallest value of θ {\displaystyle \theta }
Divisor_summatory_function
Unsolved geometry problem
Lebesgue's universal covering problem is an unsolved problem in geometry that asks for the convex shape of smallest area that can cover every planar set
Lebesgue's universal covering problem
Lebesgue's_universal_covering_problem
Category of mathematical proof
Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e
Proof_of_impossibility
Geometric inequality applicable to any closed curve
L^{2},} and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area
Isoperimetric_inequality
Problem in geometry
In geometry, the napkin-ring problem involves finding the volume of what remains after a circular hole is drilled through a sphere. Specifically, the
Napkin_ring_problem
Natural number
natural number following 5 and preceding 7. It is a composite number and the smallest perfect number. A six-sided polygon is a hexagon, one of the three regular
6
Equation for radii of tangent circles
of pairwise tangent spheres or hyperspheres. Geometrical problems involving tangent circles have been pondered for millennia. In ancient Greece of the
Descartes'_theorem
Three tangent circles in a triangle
the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within
Malfatti_circles
Problem in discrete geometry
In discrete geometry, the Erdős distinct distances problem states that every set of points in the plane has a nearly linear number of distinct distances
Erdős distinct distances problem
Erdős_distinct_distances_problem
A kinetic smallest enclosing disk data structure is a kinetic data structure that maintains the smallest enclosing disk of a set of moving points. In 2
Kinetic smallest enclosing disk
Kinetic_smallest_enclosing_disk
Mathematical puzzle game
The Tower of Hanoi (also called the problem of Benares Temple, Tower of Brahma or Lucas's Tower, and sometimes pluralized as Towers, or simply the pyramid
Tower_of_Hanoi
Theorem in plane geometry
cannot account for cases where the smallest circle is located between the other two, nor any case where one circle is fully contained by another. It can
Monge's_theorem
Method of drawing geometric objects
single circle and its center. Ancient Greek mathematicians first conceived straightedge-and-compass constructions, and a number of ancient problems in plane
Straightedge and compass construction
Straightedge_and_compass_construction
Geometry problem on grid points
no-three-in-line problem and then scaling down the integer grid to fit within a unit square produces solutions to the Heilbronn triangle problem where the smallest triangle
No-three-in-line_problem
variants of this problem. In many areas of computer graphics, the bounding box (often abbreviated to bbox) is understood to be the smallest box delimited
List of combinatorial computational geometry topics
List_of_combinatorial_computational_geometry_topics
Bound on eigenvalues
In mathematics, the Gershgorin circle theorem (also called Gershgorin Disk Theorem) may be used to bound the spectrum of a square matrix. It was first
Gershgorin_circle_theorem
Area of discrete mathematics
finding the problem of a graph's group automorphism, bend minimization, angular resolution, and slope number. Tools for graph drawings are the circle packing
Graph_theory
Unique knot with a crossing number of four
with a crossing number of four. This makes it the knot with the third-smallest possible crossing number, after the unknot and the trefoil knot. The figure-eight
Figure-eight knot (mathematics)
Figure-eight_knot_(mathematics)
Set of points equidistant from a center
are great circles. Many other surfaces share this property. Of all the solids having a given volume, the sphere is the one with the smallest surface area;
Sphere
On surrounding polygons by layers of copies
general problem. For example, a square may be surrounded by infinitely many layers of congruent squares in the square tiling, while a circle cannot be
Heesch's_problem
Polynomial whose coefficients are all 1 or −1
Littlewood, who studied their values on the unit circle and posed several influential extremal problems about them in the 1960s. The natural size of a Littlewood
Littlewood_polynomial
Branch of computer science
two with the smallest distance from each other. Farthest pair of points Largest empty circle: Given a set of points, find a largest circle with its center
Computational_geometry
Curved triangle with constant width
inscribed circle and the smallest circumscribed circle are concentric, and their radii sum to the constant width of the curve. Unsolved problem in mathematics
Reuleaux_triangle
Hashing technique
algorithms, and Daniel Lewin as their inventor, with solving the slashdotting problem which plagued the World Wide Web in the 1990s. The term "consistent hashing"
Consistent_hashing
Graph with sign-labeled edges
frustration index (early called the line index of balance) of Σ is the smallest number of edges whose deletion, or equivalently whose sign reversal (a
Signed_graph
Characterizes spherical triangles with fixed base and area
of Lexell's theorem: the Lexell circles through the points antipodal to the base vertices representing the smallest and largest triangle areas are those
Lexell's_theorem
inside a circle of radius 20. 1259 is a prime number and a highly cototient number. 1260 = 22 × 32 × 5 × 7. It is a pronic number, the smallest vampire
1000_(number)
Wooden tablets inscribed with geometrical theorems in Edo Japan
gaku (Japanese: 算額, lit. 'calculation tablet') are Japanese geometrical problems or theorems on wooden tablets which were placed as offerings at Shinto
Sangaku
Shape with three equal sides
triangle is equilateral. A packing problem asks the objective of n {\displaystyle n} circles packing into the smallest possible equilateral triangle. Optimal
Equilateral_triangle
Distance along a curve
differential geometry, or for curves that might not necessarily be smooth as a smallest upper bound of lengths of polygonal chains. The curves for which this limit
Arc_length
Extends the Jordan curve theorem to characterize the inner and outer regions
without changing it on the unit circle. This diffeomorphism then provides the smooth solution to the Schoenflies problem. The Jordan-Schoenflies theorem
Schoenflies_problem
On tangency patterns of circles
ε {\displaystyle \varepsilon } times the radius of the smallest circle. The concept of circle packings is used in load‑balanced position-based routing
Circle_packing_theorem
Two congruent circles within an arbelos
diameter of each twin circle is d = s ( 1 − s ) . {\displaystyle d=s(1-s).\,} The smallest circle that encloses both twin circles has the same area as
Twin_circles
Planar maps require at most four colors
four-color conjecture were false, there would be at least one map with the smallest possible number of regions that requires five colors. The proof showed
Four_color_theorem
Counterintuitive observation
tiny fractions of a millimeter and below, there is no obvious size of the smallest feature that should be taken into consideration when measuring, and hence
Coastline_paradox
Shape with four equal sides and angles
to proofs of the Pythagorean theorem. Square packing problems seek the smallest square or circle into which a given number of unit squares can fit. A
Square
Intersection of triangle altitudes
the inscribed triangle with the smallest perimeter is the orthic triangle. This is the solution to Fagnano's problem, posed in 1775. The sides of the
Orthocenter
Mathematical problem
(2018), Coloring Problems for Arrangements of Circles (and Pseudocircles) Grime, James (February 27, 2019), "A Colorful Unsolved Problem", Numberphile,
Hadwiger–Nelson_problem
Integer side lengths of a right triangle
divisible by 2 or 3. For the smallest case v = 5, hence k = 25, this yields the well-known cannonball-stacking problem of Lucas, 0 2 + 1 2 + 2 2 + ⋯
Pythagorean_triple
Minor planet found within the inner Solar System
Strickland, A. (28 October 2019). "It's an asteroid! No, it's the new smallest dwarf planet in our solar system". CNN. Retrieved 28 October 2019. "About
Asteroid
Fractal composed of tangent circles
, d ) {\displaystyle (a,b,c,d)} are a root quadruple (the smallest in some integral circle packing) if a < 0 ≤ b ≤ c ≤ d {\displaystyle a<0\leq b\leq
Apollonian_gasket
Computational complexity class
n^{\Omega (\log n)}} under the exponential time hypothesis. Finding the smallest dominating set in a tournament. This is a subset of the vertices of the
Quasi-polynomial_time
On distances between points on a circle
lengths. The two problems are equivalent under a linear correspondence between the unit interval and the circumference of the circle, and a correspondence
Three-gap_theorem
human foot. Mathematics: 6 is the smallest perfect number. Mathematics: 𝜏 ≈ 6.283185307179586476, the ratio of a circle's circumference to its radius. Biology:
Orders_of_magnitude_(numbers)
Game Theory variant
to the condition that the value of all circles centered at the origin is 0. 3. Find the disc D1 with the smallest radius, r1. There are two cases. 4. If
Hill–Beck land division problem
Hill–Beck_land_division_problem
Length of a line segment
distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known
Euclidean_distance
Proposed lower bound on the Mahler measure for polynomials with integer coefficients
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts
Lehmer's_conjecture
In geometry, set whose intersection with every line is a single line segment
diameter D, its inradius r (the biggest circle contained in the convex body) and its circumradius R (the smallest circle containing the convex body). In fact
Convex_set
Method of crop irrigation
nozzle sizes are smallest at the inner spans and increase with distance from the pivot point. Aerial views show fields of circles created by tracings
Center-pivot_irrigation
Cubic graph with 10 vertices and 15 edges
counterexample for many problems in graph theory. The Petersen graph is named after Julius Petersen, who in 1898 constructed it to be the smallest bridgeless cubic
Petersen_graph
Computational problem in graph theory
maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate. The maximum flow problem can be seen
Maximum_flow_problem
Triangle center minimizing sum of distances to each vertex
the point is the smallest possible or, equivalently, the geometric median of the three vertices. It is so named because this problem was first raised
Fermat_point
Problem in geometric probability
Sylvester's four point problem in geometric probability asks for the probability that four randomly chosen points in the Euclidean plane form a convex
Sylvester's four point problem
Sylvester's_four_point_problem
numbers have qualities that could arguably make them notable. Even the smallest "uninteresting" number is paradoxically interesting for that very property
List_of_numbers
Plane curve
whispering gallery). It also serves to formulate Alhazen's problem of reflection on a circle tangent to the ellipse. Additionally, because of the focus-to-focus
Ellipse
Smallest positive number divisible by two integers
lowest common multiple, or smallest common multiple (SCM) of two integers a and b, usually denoted by lcm(a, b), is the smallest positive integer that is
Least_common_multiple
Skyscraper in Manhattan, New York
Center (also known as One Columbus Circle and formerly Time Warner Center) is a mixed-use building on Columbus Circle in Manhattan, New York City, United
Deutsche_Bank_Center
On finding a repeating loop in a sequence
computer science, cycle detection or cycle finding is the algorithmic problem of finding a cycle in a sequence of iterated function values. For any function
Cycle_detection
Largest distance between two points
the space. This generalizes the diameter of a circle, the largest distance between two points on the circle. This usage of diameter also occurs in medical
Diameter_of_a_set
Austrian-Canadian mathematician
the age of 48. In 1966, Moser posed the question "What is the region of smallest area which will accommodate every planar arc of length one?" Rephrased
Leo_Moser
Interpretation of sensory information
quantitative laws in psychology are Weber's law, which states that the smallest noticeable difference in stimulus intensity is proportional to the intensity
Perception
Optimization problem
known as the smallest enclosing circle problem. For one facility in three dimensional space, it is known as the smallest enclosing sphere problem or 1-center
Optimal_facility_location
Triangulation method
other points in the set are outside of it. This maximizes the size of the smallest angle in any of the triangles, and tends to avoid sliver triangles. The
Delaunay_triangulation
finding antiderivatives) Closest pair problem: find the pair of points (from a set of points) with the smallest distance between them Collision detection
List_of_algorithms
one city to the other. This problem is a generalization of the above, and the solution is not as obvious. Given two circles which will serve as top and
Infinite-dimensional optimization
Infinite-dimensional_optimization
Process of producing small rectangular items of fixed dimensions
target rectangles using the smallest possible number of sheets. It is a variant of the two-dimensional bin-packing problem. k-staged guillotine cutting
Guillotine_cutting
Shape with three sides
base of length a {\displaystyle a} is equal to a {\displaystyle a} . The smallest possible ratio of the side of one inscribed square to the side of another
Triangle
Circles tangent to all three sides of a triangle
In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches (is tangent to) the
Incircle_and_excircles
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