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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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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)
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
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
Type of area of skin
fossa, just proximally to the elbow. C6 – On the dorsal surface of the proximal phalanx of the thumb. C7 – On the dorsal surface of the proximal phalanx
Dermatome_(anatomy)
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
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
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
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
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)
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
Knot which lies on the surface of a torus in 3-dimensional space
kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way
Torus_knot
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
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
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
Terrain measurement with light beams
represents the maximum inclination of a plane to the horizontal, dip direction is the direction of the intersection line between horizontal and the inclined
Geological structure measurement by LiDAR
Geological_structure_measurement_by_LiDAR
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
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
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
On reflection in a spherical mirror
involving the intersection of conic sections. According to Roberto Marcolongo, Leonardo da Vinci invented a mechanical device to solve the problem. Later mathematicians
Alhazen's_problem
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
Branch of algebraic geometry
there are 27 {\displaystyle 27} lines on a cubic surface. Enumerative geometry Chow ring Intersection theory Grassmannian Giambelli's formula Pieri's formula
Schubert_calculus
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
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
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
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
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
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
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
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
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
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
Vibrational energy transfer in Earth or other planetary body
three dimensions. Particle motion of surface waves is larger than that of body waves, so surface waves tend to cause more damage. Body waves travel through
Seismic_wave
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
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
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
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
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
(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
American 3D graphics software company
One of the major tasks was curve/curve intersection. It became evident that if the general intersection problem could be solved for the Bézier/Bézier case
Solid_Modeling_Solutions
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)
Geographic coordinate specifying north-south position
a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at the south pole to 90° at the north
Latitude
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
Toroidal polyhedron with 7 faces
symmetry as the polyhedron. Unsolved problem in mathematics Is there a non-convex polyhedron without self-intersections with more than seven faces, all of
Szilassi_polyhedron
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
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
modeled as geometric ray with no thickness to perform efficient geometric queries such as a ray-triangle intersection. From a physics of light transport point
Cone_tracing
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
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
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
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