An isosurface is a three-dimensional analog of an
isoline. It is a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
that represents points of a constant value (e.g.
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
,
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
,
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
) within a
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
of space; in other words, it is a
level set
In mathematics, a level set of a real-valued function of real variables is a set where the function takes on a given constant value , that is:
: L_c(f) = \left\~,
When the number of independent variables is two, a level set is calle ...
of a continuous
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
whose
domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
* Do ...
is
3-space.
The term ''isoline'' is also sometimes used for domains of more than 3 dimensions.
Applications
Isosurfaces are normally displayed using
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
, and are used as data visualization methods in
computational fluid dynamics
Computational fluid dynamics (CFD) is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems that involve fluid flows. Computers are used to perform the calculations required to simulate th ...
(CFD), allowing engineers to study features of a
fluid flow (gas or liquid) around objects, such as aircraft
wing
A wing is a type of fin that produces lift while moving through air or some other fluid. Accordingly, wings have streamlined cross-sections that are subject to aerodynamic forces and act as airfoils. A wing's aerodynamic efficiency is expres ...
s. An isosurface may represent an individual
shock wave
In physics, a shock wave (also spelled shockwave), or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a med ...
in
supersonic
Supersonic speed is the speed of an object that exceeds the speed of sound ( Mach 1). For objects traveling in dry air of a temperature of 20 °C (68 °F) at sea level, this speed is approximately . Speeds greater than five times ...
flight, or several isosurfaces may be generated showing a sequence of pressure values in the air flowing around a wing. Isosurfaces tend to be a popular form of visualization for volume datasets since they can be rendered by a simple polygonal model, which can be drawn on the screen very quickly.
In
medical imaging
Medical imaging is the technique and process of imaging the interior of a body for clinical analysis and medical intervention, as well as visual representation of the function of some organs or tissues (physiology). Medical imaging seeks to rev ...
, isosurfaces may be used to represent regions of a particular
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
in a three-dimensional
CT scan, allowing the visualization of internal
organs,
bone
A bone is a Stiffness, rigid Organ (biology), organ that constitutes part of the skeleton in most vertebrate animals. Bones protect the various other organs of the body, produce red blood cell, red and white blood cells, store minerals, provid ...
s, or other structures.
Numerous other disciplines that are interested in three-dimensional data often use isosurfaces to obtain information about
pharmacology
Pharmacology is a branch of medicine, biology and pharmaceutical sciences concerned with drug or medication action, where a drug may be defined as any artificial, natural, or endogenous (from within the body) molecule which exerts a biochemica ...
,
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
,
geophysics
Geophysics () is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis. The term ''geophysics'' som ...
and
meteorology
Meteorology is a branch of the atmospheric sciences (which include atmospheric chemistry and physics) with a major focus on weather forecasting. The study of meteorology dates back millennia, though significant progress in meteorology did not ...
.
Implementation algorithms
Marching cubes
The
marching cubes
Marching cubes is a computer graphics algorithm, published in the 1987 SIGGRAPH proceedings by Lorensen and Cline, for extracting a polygonal mesh of an isosurface from a three-dimensional discrete scalar field (the elements of which are sometim ...
algorithm was first published in the 1987 SIGGRAPH proceedings by Lorensen and Cline, and it creates a surface by intersecting the edges of a
data
In the pursuit of knowledge, data (; ) is a collection of discrete values that convey information, describing quantity, quality, fact, statistics, other basic units of meaning, or simply sequences of symbols that may be further interpreted ...
volume grid with the volume contour. Where the surface intersects the edge the algorithm creates a vertex. By using a table of different triangles depending on different patterns of edge intersections the algorithm can create a surface. This algorithm has solutions for implementation both on the CPU and on the GPU.
Asymptotic decider
The
asymptotic decider
In scientific visualization the asymptotic decider is an algorithm developed by Nielson and Hamann in 1991 that creates isosurfaces from a given scalar field. It was proposed as an improvement to the marching cubes algorithm, which can produce so ...
algorithm was developed as an extension to
marching cubes
Marching cubes is a computer graphics algorithm, published in the 1987 SIGGRAPH proceedings by Lorensen and Cline, for extracting a polygonal mesh of an isosurface from a three-dimensional discrete scalar field (the elements of which are sometim ...
in order to resolve the possibility of ambiguity in it.
Marching tetrahedra
The
marching tetrahedra
Marching tetrahedra is an algorithm in the field of computer graphics to render implicit surfaces. It clarifies a minor ambiguity problem of the marching cubes algorithm with some cube configurations. It was originally introduced in 1991.
While ...
algorithm was developed as an extension to
marching cubes
Marching cubes is a computer graphics algorithm, published in the 1987 SIGGRAPH proceedings by Lorensen and Cline, for extracting a polygonal mesh of an isosurface from a three-dimensional discrete scalar field (the elements of which are sometim ...
in order to solve an ambiguity in that algorithm and to create higher quality output surface.
Surface nets
The Surface Nets algorithm places an intersecting vertex in the middle of a volume voxel instead of at the edges, leading to a smoother output surface.
Dual contouring
The
dual contouring algorithm was first published in the 2002 SIGGRAPH proceedings by Ju and Losasso, developed as an extension to both
surface nets and marching cubes. It retains a
dual vertex within the
voxel
In 3D computer graphics, a voxel represents a value on a regular grid in three-dimensional space. As with pixels in a 2D bitmap, voxels themselves do not typically have their position (i.e. coordinates) explicitly encoded with their values. Ins ...
but no longer at the center. Dual contouring leverages the position and
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
of where the surface crosses the edges of a voxel to interpolate the position of the dual vertex within the voxel. This has the benefit of retaining sharp or smooth surfaces where surface nets often look blocky or incorrectly beveled. Dual contouring often uses surface generation that leverages
octree
An octree is a tree data structure in which each internal node has exactly eight children. Octrees are most often used to partition a three-dimensional space by recursively subdividing it into eight octants. Octrees are the three-dimensional ana ...
s as an optimization to adapt the number of triangles in output to the complexity of the surface.
Manifold dual contouring
Manifold
dual contouring includes an analysis of the octree neighborhood to maintain continuity of the manifold surface
Examples
Examples of isosurfaces are '
Metaballs
In computer graphics, metaballs are organic-looking ''n''-dimensional isosurfaces, characterised by their ability to meld together when in close proximity to create single, contiguous objects.
In solid modelling, polygon meshes are commonly ...
' or 'blobby objects' used in 3D visualisation. A more general way to construct an isosurface is to use the
function representation Function Representation (FRep or F-Rep) is used in solid modeling, volume modeling and computer graphics. FRep was introduced in "Function representation in geometric modeling: concepts, implementation and applications" as a uniform representation ...
.
See also
*
Triangulation (geometry)
In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it int ...
*
Implicit surface
In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
: F(x,y,z)=0.
An ''implicit surface'' is the set of zeros of a function of three variables. ''Implicit'' means that the equation is not solved for o ...
*
Volume rendering
In scientific visualization and computer graphics, volume rendering is a set of techniques used to display a 2D projection of a 3D discretely sampled data set, typically a 3D scalar field.
A typical 3D data set is a group of 2D slice imag ...
References
* {{cite book, author1=Charles D. Hansen, author2=Chris R. Johnson, title=Visualization Handbook, url=https://books.google.com/books?id=ZFrlULckWdAC&pg=PA7, year=2004, publisher=Academic Press, isbn=978-0-12-387582-2, pages=7–11
External links
Isosurface PolygonizationThe contourist package for numeric pythongenerates triangulations of isosurfaces for arbitrary 3 dimensional functions which can be displayed using HTML5 as illustrated in thi
jsfiddle
3D computer graphics
Medical imaging
Surfaces