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Basic workflow in computer-aided geometric design
The surface-to-surface intersection (SSI) problem is a basic workflow in computer-aided geometric design: Given two intersecting surfaces in R3, compute
Surface-to-surface intersection problem
Surface-to-surface_intersection_problem
Type of smooth complex surface of kodaira dimension 0
surface means a curve isomorphic to P 1 {\displaystyle \mathbf {P} ^{1}} with self-intersection −2.) The moduli space of quasi-polarized K3 surfaces of
K3_surface
Species of mathematical spline
Bézier patch meshes are difficult to render directly. One problem with Bézier patches is that calculating their intersections with lines is difficult, making
Bézier_surface
Orientable surface whose boundary is a knot or link
Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be
Seifert_surface
Mathematics of smooth surfaces
geodesics are fundamental to the optimization problem of determining the shortest path between two given points on a regular surface. One can also define parallel
Differential geometry of surfaces
Differential_geometry_of_surfaces
Set of points equidistant from a center
A sphere (from Ancient Greek σφαῖρα (sphaîra) 'ball') is a surface analogous to the circle, a curve. In solid geometry, a sphere is the set of points that
Sphere
Submanifold of Lorentzian manifold
{\mathcal {S}}} . The Cauchy surface is defined rigorously in terms of intersections with inextensible curves in order to deal with this case of circular
Cauchy_surface
Algebraic surface defined by a cubic polynomial
^{7}} , with its intersection form (coming from the intersection theory of curves on a surface). For a smooth complex cubic surface, the Picard lattice
Cubic_surface
Periodic minimal surface
Consider the following minimal surface problem on a square in the Euclidean plane: for a natural number n, find a minimal surface Σn as the graph of some function
Scherk_surface
definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can be used to model billiards, and
Translation_surface
Molecular dynamics simulation technique
of systems that include tunneling, conical intersections and electronic excitation. In practice, surface hopping is computationally feasible only for
Surface_hopping
then to add these local indices up to get a global intersection index. The theory starts to diverge from that of algebraic surfaces when we try to ensure
Arithmetic_surface
Periodic minimal surface
surfaces. The surfaces were generated using symmetry arguments: given a solution to Plateau's problem for a polygon, reflections of the surface across the
Schwarz_minimal_surface
Location of a discrete degeneracy between two electronic states
a conical intersection of two or more potential energy surfaces is the set of molecular geometry points where the potential energy surfaces are degenerate
Conical_intersection
Shape formed from points common to other shapes
(such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either
Intersection_(geometry)
Methodological basis for 3D CAD/CAM solid modeling and image rendering
mathematical problem is finding line-surface intersection points. So, surfaces as planes, quadrics, tori, and probably even parametric surface patches may
Ray_casting
Infrastructure for draining excess rain and ground water from impervious surfaces
highway drain, surface water drain/sewer (United Kingdom), or stormwater drain (Australia and New Zealand) is infrastructure designed to drain excess rain
Storm_drain
Road covered with durable surface material
A road surface (British English) or pavement (North American English) is the durable surface material laid down on an area intended to sustain vehicular
Road_surface
Concept in differential geometry
minimal surfaces without self-intersections, NASA Technical Note TN D-5541 (1970)"Infinite periodic minimal surfaces without self-intersections by Alan
Triply periodic minimal surface
Triply_periodic_minimal_surface
Curve that is common to two geometric objects
general, an intersection curve consists of the common points of two transversally intersecting surfaces, meaning that at any common point the surface normals
Intersection_curve
Intersection of cylinders
Proteus Steinmetz, who solved the problem of determining the volume of the intersection. However, the same problem had been solved earlier, by Archimedes
Steinmetz_solid
Computer graphics rendering algorithm
algorithm in the field of computer graphics to render implicit surfaces. It clarifies a minor ambiguity problem of the marching cubes algorithm with some
Marching_tetrahedra
Geometric shape
Maths Is Fun Paper model cone Lateral surface area of an oblique cone Cut a Cone An interactive demonstration of the intersection of a cone with a plane
Cone
Geometry calculation
have no intersection at all. Or there may be one or two points of intersection. Or a line may lie along the surface of a cylinder, parallel to its axis
Line-cylinder_intersection
Problem of finding obscured edges in a wire-frame 3D model
problem whether the same optimal O(n2) upper bound existed for hidden-surface removal. This problem was solved by McKenna in 1987. The intersection-sensitive
Hidden-line_removal
Branch of algebraic geometry
In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties
Intersection_theory
Concept in algebraic geometry
manifolds or Riemann surfaces. They were named Zariski–Riemann spaces after Oscar Zariski and Bernhard Riemann by Nagata (1962) who used them to show that algebraic
Zariski–Riemann_space
Three-dimensional solid
much greater than its height. A cylindric section is the intersection of a cylinder's surface with a plane. They are, in general, curves and are special
Cylinder
quasi-projective scheme classifying the surfaces of general type with those Chern numbers. It remains a very difficult problem to describe these schemes explicitly
Surface_of_general_type
Concept in mathematics
the topology of hyperbolic surfaces, and especially in the study of the intersections of closed curves on these surfaces. The earliest contributors were
Mapping class group of a surface
Mapping_class_group_of_a_surface
Locus of the zeros of a polynomial of degree two
three-dimensional quadric surface, one obtains a quadratic equation for t {\displaystyle t} . If there are two real roots, this gives the two intersection points of the
Quadric
Machining techniques for complex surfaces
is done by intersection of surfaces with parallel plane in Cartesian space. Adaptive iso-planar tool path method: In this process the surface are partitioned
Freeform_surface_machining
Form of decision making in machine learning
The multi-surface method (MSM) is a form of decision making using the concept of piecewise-linear separability of datasets to categorize data. Two datasets
Multi-surface_method
Rendering method
ray tracing was used to make shaded pictures of solids. At the ray-surface intersection point found, they computed the surface normal and, knowing the
Ray_tracing_(graphics)
Non-orientable surface with one edge
minimal surface with the topology of the Möbius strip, can be constructed using solutions to the Björling problem, which defines a minimal surface uniquely
Möbius_strip
Flat-sided three-dimensional shape
may refer either to a solid figure or to its boundary surface. The terms solid polyhedron and polyhedral surface are commonly used to distinguish the two
Polyhedron
Mathematical classification of surfaces
quasi-projective scheme classifying the surfaces of general type with those Chern numbers. However it is a very difficult problem to describe these schemes explicitly
Enriques–Kodaira classification
Enriques–Kodaira_classification
Creating a complex 3D surface or object by combining primitive objects
Bloomenthal, Jules; Bajaj, Chandrajit (1997), "5.2.5 Intersection with CSG Trees", Introduction to Implicit Surfaces, Morgan Kaufmann, pp. 178–180, ISBN 9781558602335
Constructive_solid_geometry
Shape formed by intersecting four balls
tetrahedron is the intersection of four balls of radius s centered at the vertices of a regular tetrahedron with side length s. The spherical surface of the ball
Reuleaux_tetrahedron
Angle between a liquid–vapor interface and a solid surface
tangent on the solid–liquid interface at their intersection. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given
Contact_angle
Generalized notion of counting curve intersections
a surface along a line, the intersection number along the line should be at least two. These questions are discussed systematically in intersection theory
Intersection_number
Area of discrete mathematics
this problem by Peter Tait, Percy John Heawood, Frank P. Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with
Graph_theory
Section of a sphere
equal to the radius of the sphere, the spherical cap is called a hemisphere. The volume of the spherical cap and the area of the curved surface may be
Spherical_cap
Description of how spaces intersect in mathematics
tangent plane to the surface are distinct. In a three-dimensional space, two curves can be transverse only when they have empty intersection, since their
Transversality
Natural satellite orbiting Earth
moon, as well as the largest and most massive in relation to its parent planet. Its surface gravity is about one-sixth of Earth's, about half that of
Moon
a common intersection in the Veronese surface, corresponding to the set of double lines in the plane, all of which have double intersection points with
Steiner's_conic_problem
Largest independent set of paired elements
vector spaces. The problem was formulated by Lawler (1976) as a common generalization of graph matching and matroid intersection. It is also known as
Matroid_parity_problem
Optical phenomenon
other words, it is where the intersection of the surface with a plane factors into a perfect square. For each Kummer surface, there exists a two-dimensional
Conical_refraction
Common elements of two or more sets
determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between
Intersection
Branch of algebraic geometry concerned with counting solutions
geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.[citation needed] While mathematicians mostly
Enumerative_geometry
Generalization of the concept of parallel lines
similarity to the geometric properties of parallel curves. The problem generalizes fairly obviously to higher dimensions e.g. to offset surfaces, and slightly
Parallel_curve
Science of measuring the shape, orientation, and gravity of Earth
geoid is irregular and too complicated to serve as the computational surface for solving geometrical problems like point positioning. The geometrical
Geodesy
Branch of computer science
detection algorithms: check for the collision or intersection of two given solids Cone algorithm: identify surface points Convex hull algorithms: determining
Computational_geometry
Type of area of skin
At the apex of the axilla. T3 – Intersection of the midclavicular line and the third intercostal space T4 – Intersection of the midclavicular line and the
Dermatome_(anatomy)
Embedding a graph in a topological space, often Euclidean
genus problem is fixed-parameter tractable, i.e., polynomial time algorithms are known to check whether a graph can be embedded into a surface of a given
Graph_embedding
In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open
Planar_Riemann_surface
Geometric condition for workpieces and tools
cross-section (the plane curve resulting from the intersection of the surface of interest and a plane spanned by the surface normal): least squares reference line
Flatness_(manufacturing)
Effort to birationally classify algebraic varieties
of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry. The basic idea of the theory is to simplify
Minimal_model_program
Geometry of the surface of a sphere
two-dimensional surface of a sphere or the n-dimensional surface of higher dimensional spheres. Long studied for its practical applications to astronomy, navigation
Spherical_geometry
Computer graphics method
CosineWeightedSample(intersection.surfaceNormal) throughput ← throughput × π × intersection.BRDF(−ray.direction, intersection.surfaceNormal, newDirection)
Path_tracing
Study of geometric properties of sets through measure theory
there exists a surface of least area among all surfaces whose boundary equals the given curve. Such surfaces mimic soap films. The problem had remained
Geometric_measure_theory
Existence of geodesic circles on surfaces
the polyhedron by a smooth surface and applying the theorem of the three geodesics to this surface. It is an open problem whether any of these quasigeodesics
Theorem of the three geodesics
Theorem_of_the_three_geodesics
Producing images of 3D scenes
branching" rule was broken by tracing additional rays from each surface intersection point to randomly chosen points on each light source). Kajiya suggested
Rendering_(computer_graphics)
Study of geometry using a coordinate system
dimensions, a single equation usually gives a surface, and a curve must be specified as the intersection of two surfaces (see below), or as a system of parametric
Analytic_geometry
Thin coating of glazed ice on a surface
on a surface, for example on streets or on lakes. The ice itself is not black, but visually transparent, allowing the often black road below to be seen
Black_ice
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
applying first assertion to γ and U in place of γ and X. The intersection number of two closed curves γ1, γ2 in a Riemann surface X can be defined analytically
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Relation between genus, degree, and dimension of function spaces over surfaces
{C} } to the surface X {\displaystyle X} . For the purposes of the Riemann–Roch theorem, the surface X {\displaystyle X} is always assumed to be compact
Riemann–Roch_theorem
packing problem Circle packing theorem – On tangency patterns of circles Introduction to Circle Packing – a book by Kenneth Stephenson Circular surface Clifford's
List_of_circle_topics
Study of complex manifolds and several complex variables
methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of
Complex_geometry
Input and output device
used to determine if a finger is present or not at an intersection. The capacitance change at every intersection on the grid can be measured to accurately
Touchscreen
Relationship between two lines that meet at a right angle
intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be
Perpendicular
(geometry) Art gallery problem (The museum problem) Visibility graph Watchman route problem Computer graphics applications: Hidden surface determination Hidden
List of combinatorial computational geometry topics
List_of_combinatorial_computational_geometry_topics
Roads built with water-pervious materials
Permeable paving surfaces are made of either a porous material that enables stormwater to flow through it or nonporous blocks spaced so that water can
Permeable_paving
Class of algorithms which use a moving line to solve geometrical problems
algorithmic paradigm that uses a conceptual sweep line or sweep surface to solve various problems in Euclidean space. It is one of the critical techniques in
Sweep_line_algorithm
Iranian mathematician (1977–2017)
the three geodesics for spherical surfaces. Mirzakhani solved this counting problem by relating it to the problem of computing volumes in moduli space—a
Maryam_Mirzakhani
Term in mathematics
this Riemann surface X is trivial. In particular, every line bundle is trivial. This is related to the solution of the second Cousin problem. The standard
Stein_manifold
Branch of mathematics
almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental
Geometry
3D computer graphics procedure
elimination of polygon cycles in the depth sorting required in hidden surface removal. It was proposed in 1972 by brothers Martin Newell and Dick Newell
Newell's_algorithm
Expressway or freeway with only one lane in each direction
[citation needed] A super-2 expressway is a high-speed surface road with at-grade intersections, depending on the common usage of the term expressway in
Two-lane_expressway
Mathematical treatise by Archimedes
and HE is parallel to the axis of symmetry of the parabola. Call the intersection of HE and the parabola F and the intersection of HE and the lever G
The Method of Mechanical Theorems
The_Method_of_Mechanical_Theorems
Lens with a thickness that is negligible
optical axis between the two surfaces of the lens) that is negligible compared to the radii of curvature of the lens surfaces. Lenses whose thickness is
Thin_lens
Area of depressed road surface
A pothole is a pot-shaped depression in a road surface, usually asphalt pavement, where traffic has removed broken pieces of the pavement. It is usually
Pothole
Topmost part of the soil layer
expensive compared to other alternatives. Topsoil is also used for proper surface grading near residential buildings. In order to protect against flooding
Topsoil
Determining where a point is in relation to a coplanar polygon
that we count one intersection for the algorithm to work correctly. A similar problem arises with horizontal segments that happen to fall on the ray. The
Point_in_polygon
On tangency patterns of circles
extended to arbitrary Riemannian surfaces including the sphere, the hyperbolic plane, and to surfaces of bounded genus. More generally, intersection graphs
Circle_packing_theorem
Subdivision of a planar object into triangles
entire surface, the intersection on any pair of subsets is either empty, an edge or a vertex and if the intersection the intersection T α ∩ T β {\displaystyle
Triangulation_(geometry)
Mathematical term for squaring a plane figure
the circle) is a famous old problem that has been shown, in the 19th century, to be impossible with the methods available to the Ancient Greeks. Integral
Quadrature_(mathematics)
Feature in programming languages
described how to use double dispatching in Smalltalk, calling it multiple polymorphism. The general problem addressed is how to dispatch a message to different
Double_dispatch
Technique for determining surface structures
bulk impurities diffuse to the surface and therefore give rise to a re-contamination after each cleaning cycle. The problem is that impurities that adsorb
Low-energy electron diffraction
Low-energy_electron_diffraction
Function of the coefficients of a polynomial that gives information on its roots
defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing a n {\displaystyle a_{n}} by 1 in the first
Discriminant
(LEIS), sometimes referred to simply as ion scattering spectroscopy (ISS), is a surface-sensitive analytical technique used to characterize the chemical
Low-energy_ion_scattering
Smallest convex set containing a given set
arbitrarily small surface area, smaller than the surface area of the convex hull. However, in higher dimensions, variants of the obstacle problem of finding
Convex_hull
Road building method by John Loudon McAdam
keep the running surface level with the countryside, this road was put in a trench, which created drainage problems. These problems were addressed by
Macadam
Cone- or frustum-shaped gears for shafts whose axes intersect
pitch surfaces of meshed external bevel gears are coaxial with the gear shafts; the apexes of the two surfaces are at the point of intersection of the
Bevel_gear
Method for specifying point positions
based on the intersection of curves. Orthogonal coordinates: coordinate surfaces meet at right angles Skew coordinates: coordinate surfaces are not orthogonal
Coordinate_system
modifies the area of the surface nor the length of the curves drawn on it, which are the magnitudes relevant to the problem. The surface can be removed from
Filling_area_conjecture
Computer graphics technique
discussion of techniques used to combat the fill time problem. Heidmann proposed that if the front surfaces and back surfaces of the shadows were rendered
Shadow_volume
Region where Earth's magnetic field is weakest relative to an idealised dipole
radiation belt comes closest to Earth's surface, dipping down to an altitude of 200 kilometres (120 mi). This leads to an increased flux of energetic
South_Atlantic_Anomaly
American mathematician
Minkowski Theory, his notion of intersection body and his contribution to the solution of the Busemann–Petty problem, for proving the long-conjectured
Erwin_Lutwak
sixteenth problem Hyperelliptic curve cryptography Inflection point Inscribed square problem intercept, y-intercept, x-intercept Intersection number Intrinsic
List_of_curves_topics
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
Boy/Male
German
Surname relating to falconry.
Boy/Male
English
Surname relating to falconry.
Boy/Male
American, British, English
Falconry; Surname Relating to Falconry
Female
Egyptian
, another form of Ratta or Ritho.
Boy/Male
German
Surname relating to falconry.
Boy/Male
Irish
Surname.
Surname or Lastname
English (Cumbria and Durham)
English (Cumbria and Durham) : variant spelling of Furness.
Boy/Male
German
Surname relating to falconry.
Surname or Lastname
Probably an Americanized spelling of the Swiss German surname Bunz (see Bunce).English
Probably an Americanized spelling of the Swiss German surname Bunz (see Bunce).English : possibly a variant of Bunt.
Boy/Male
Irish
Surname.
Boy/Male
Indian
Part of Sun
Boy/Male
Australian, Danish, French, German, Hebrew
Surname Relating to Falconry; Falconer
Boy/Male
British, English
Surname Related to Paul; Small
Boy/Male
Irish American Welsh Scandinavian Scottish English
Surname.
Boy/Male
Indian, Sanskrit
Surface of the Earth
Surname or Lastname
English
English : unexplained; possibly related to another unexplained English surname, Brookshaw.
Boy/Male
Australian, German, Italian, Latin
Surname Relating to Falconry
Female
Egyptian
, the wife of the priest Anhur-mes.
Boy/Male
Scottish American English
Surname.
Boy/Male
Irish
Surname.
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
Girl/Female
Arabic, Muslim
One who is Like a Delicate Flower
Boy/Male
Arthurian Legend
Eldest son of Lot.
Surname or Lastname
English
English : variant of Wakeley.
Boy/Male
Arabic, Swahili
Stranger; Visitor; Traveller; Poor; Alien
Girl/Female
Greek
Named for Dionysus god of wine.
Girl/Female
Egyptian
Born of a troubled father.
Female
English
Pet form of English Beatrix, BEE means "voyager (through life)."
Male
English
Variant spelling of English Isidore, ISADOR means "gift of Isis."
Biblical
shadow; the tingling of the ear
Girl/Female
Indian
Part of Goodly
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
SURFACE TO-SURFACE-INTERSECTION-PROBLEM
v. t.
To satisfy; to content; to be equal to the wants or demands of.
v. t.
To work over the surface or soil of, as ground, in hunting for gold.
a.
Having the surface smooth and polished; -- said of leaves, the surfaces of shells, etc.
v. t.
To give a surface to; especially, to cause to have a smooth or plain surface; to make smooth or plain.
n.
A form of machine for dressing the surface of wood, metal, stone, etc.
n.
The exterior part of anything that has length and breadth; one of the limits that bound a solid, esp. the upper face; superficies; the outside; as, the surface of the earth; the surface of a diamond; the surface of the body.
n.
Hence, outward or external appearance.
imp. & p. p.
of Surface
n.
The act of intercepting; as, interception of a letter; interception of the enemy.
n.
A magnitude that has length and breadth without thickness; superficies; as, a plane surface; a spherical surface.
n.
The act, state, or place of intersecting.
v. t.
To harass; to fatigue.
n.
The point or line in which one line or surface cuts another.
a.
Pertaining to, or formed by, intersections.
n.
That part of the side which is terminated by the flank prolonged, and the angle of the nearest bastion.
n.
To throw out, or exhale, as from a furnace; also, to put into a furnace.
n.
An inclosed place in which heat is produced by the combustion of fuel, as for reducing ores or melting metals, for warming a house, for baking pottery, etc.; as, an iron furnace; a hot-air furnace; a glass furnace; a boiler furnace, etc.
v. t.
To name or call by an appellation added to the original name; to give a surname to.