In technical applications of
3D computer graphics
3D computer graphics, or “3D graphics,” sometimes called CGI, 3D-CGI or three-dimensional computer graphics are graphics that use a three-dimensional representation of geometric data (often Cartesian) that is stored in the computer for th ...
(
CAx) such as
computer-aided design
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve co ...
and
computer-aided manufacturing
Computer-aided manufacturing (CAM) also known as computer-aided modeling or computer-aided machining is the use of software to control machine tools in the manufacturing of work pieces. This is not the only definition for CAM, but it is the most ...
, surfaces are one way of representing objects. The other ways are
wireframe (lines and curves) and solids.
Point cloud
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* Poin ...
s are also sometimes used as temporary ways to represent an object, with the goal of using the points to create one or more of the three permanent representations.
Open and closed surfaces
If one considers a local
parametrization of a surface:
:
then the curves obtained by varying ''u'' while keeping ''v'' fixed are
coordinate line
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space. The order of the coordinates is sign ...
s, sometimes called the ''u'' ''flow lines''. The curves obtained by varying ''v'' while ''u'' is fixed are called the ''v'' flow lines. These are generalizations of the ''x'' and ''y'' Cartesian coordinate lines in the
plane coordinate system and of the
meridians and
circles of latitude
A circle of latitude or line of latitude on Earth is an abstract east– west small circle connecting all locations around Earth (ignoring elevation) at a given latitude coordinate line.
Circles of latitude are often called parallels bec ...
on a
spherical coordinate system
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
.
Open surfaces are not closed in either direction. This means moving in any direction along the surface will cause an observer to hit the edge of the surface. The top of a car hood is an example of a surface open in both directions.
Surfaces closed in one direction include a cylinder, cone, and hemisphere. Depending on the direction of travel, an observer on the surface may hit a boundary on such a surface or travel forever.
Surfaces closed in both directions include a sphere and a torus. Moving in any direction on such surfaces will cause the observer to travel forever without hitting an edge.
Places where two boundaries overlap (except at a point) are called a seam. For example, if one imagines a cylinder made from a sheet of paper rolled up and taped together at the edges, the boundaries where it is taped together are called the seam.
Flattening a surface
Some open surfaces and surfaces closed in one direction may be flattened into a plane without deformation of the surface. For example, a cylinder can be flattened into a rectangular area without distorting the surface distance between surface features (except for those distances across the split created by opening up the cylinder). A cone may also be so flattened. Such surfaces are linear in one direction and curved in the other (surfaces linear in both directions were flat to begin with). Sheet metal surfaces which have flat patterns can be manufactured by stamping a flat version, then bending them into the proper shape, such as with rollers. This is a relatively inexpensive process.
Other open surfaces and surfaces closed in one direction, and all surfaces closed in both directions, can't be flattened without deformation. A hemisphere or sphere, for example, can't. Such surfaces are curved in both directions. This is why maps of the Earth are distorted. The larger the area the map represents, the greater the distortion. Sheet metal surfaces which lack a flat pattern must be manufactured by stamping using 3D dies (sometimes requiring multiple dies with different draw depths and/or draw directions), which tend to be more expensive.
Regions
Patches
A surface may be composed of one or more ''patches'', where each patch has its own U-V coordinate system. These surface patches are analogous to the multiple polynomial arcs used to build a
spline. They allow more complex surfaces to be represented by a series of relatively simple equation sets rather than a single set of complex equations. Thus, the complexity of operations such as surface intersections can be reduced to a series of patch intersections.
Surfaces closed in one or two directions frequently must also be broken into two or more surface patches by the software.
Faces
Surfaces and surface patches can only be trimmed at U and V flow lines. To overcome this severe limitation, surface ''faces'' allow a surface to be limited to a series of boundaries projected onto the surface in any orientation, so long as those boundaries are collectively closed. For example, trimming a cylinder at an angle would require such a surface face.
A single surface face may span multiple surface patches on a single surface, but can't span multiple surfaces.
Planar faces are similar to surface faces, but are limited by a collectively closed series of boundaries projected to an infinite plane, instead of a surface.
Skins and volumes
As with surfaces, surface faces closed in one or two directions frequently must also be broken into two or more surface faces by the software. To combine them back into a single entity, a skin or volume is created. A ''skin'' is an open collection of faces and a ''volume'' is a closed set. The constituent faces may have the same support surface or face or may have different supports.
Solids
Volumes can be filled in to build a
solid model (possibly with other volumes subtracted from the interior). Skins and faces can also be offset to create solids of uniform thickness.
Continuity
A surface's patches and the faces built on that surface typically have point continuity (no gaps) and tangent continuity (no sharp angles). Curvature continuity (no sharp radius changes) may or may not be maintained.
Skins and volumes, however, typically only have point continuity. Sharp angles between faces built on different supports (planes or surfaces) are common.
Visualization and display
Surfaces may be displayed in many ways:
*Wireframe mode. In this representation the surface is drawn as a series of lines and curves, without
hidden line removal
In 3D computer graphics, solid objects are usually modeled by polyhedra. A face of a polyhedron is a planar polygon bounded by straight line segments, called edges. Curved surfaces are usually approximated by a polygon mesh. Computer programs ...
. The boundaries and flow lines (isoparametric curves) may each be shown as solid or dashed curves. The advantage of this representation is that a great deal of geometry may be displayed and rotated on the screen with no delay needed for graphics processing.
Image:spoon_wf.jpg , wireframe
hidden edges
Image:spoon_uv.jpg , wireframe
uv isolines
*Faceted mode. In this mode each surface is drawn as a series of planar regions, usually rectangles. Hidden line removal is typically used with such a representation. Static hidden line removal does not update which lines are hidden during rotation, but only once the screen is refreshed. Dynamic hidden line removal continuously updates which curves are hidden during rotations.
Image:spoon_fw.jpg , Facet wireframe
Image:spoon_fs.jpg , Facet shaded
*Shaded mode. Shading can then be added to the facets, possibly with blending between the regions for a smoother display. Shading can also be static or dynamic. A lower quality of shading is typically used for dynamic shading, while high quality shading, with multiple light sources,
textures, etc., requires a delay for rendering.
Image:spoon_sh.jpg , shaded
Image:spoon_rl.jpg , reflection lines
Image:spoon_fi.jpg , reflected image
CAD/CAM representation
CAD
Computer-aided design (CAD) is the use of computers (or ) to aid in the creation, modification, analysis, or optimization of a design. This software is used to increase the productivity of the designer, improve the quality of design, improve c ...
/
CAM
Calmodulin (CaM) (an abbreviation for calcium-modulated protein) is a multifunctional intermediate calcium-binding messenger protein expressed in all eukaryotic cells. It is an intracellular target of the secondary messenger Ca2+, and the bin ...
systems use primarily two types of surfaces:
*Regular (or canonical) surfaces include surfaces of revolution such as cylinders, cones, spheres, and tori, and ruled surfaces (linear in one direction) such as surfaces of extrusion.
*
Freeform surfaces (usually
NURBS
Non-uniform rational basis spline (NURBS) is a mathematical model using B-spline, basis splines (B-splines) that is commonly used in computer graphics for representing curves and Surface (mathematics), surfaces. It offers great flexibility and pr ...
) allow more complex shapes to be represented via
freeform surface modeling.
Other surface forms such as
facet
Facets () are flat faces on geometric shapes. The organization of naturally occurring facets was key to early developments in crystallography, since they reflect the underlying symmetry of the crystal structure. Gemstones commonly have facets cut ...
and
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. I ...
are also used in a few specific applications.
CAE/FEA representation
In
computer-aided engineering
Computer-aided engineering (CAE) is the broad usage of computer software to aid in engineering analysis tasks. It includes , , , durability and optimization. It is included with computer-aided design (CAD) and computer-aided manufacturing (CAM) ...
and
finite element analysis
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat ...
, an object may be represented by a surface mesh of node points connected by triangles or quadrilaterals (
polygon mesh
In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedral object. The faces usually consist of triangles ( triangle mesh), quadrilaterals (quads), or other simple convex p ...
). More accurate, but also far more CPU-intensive, results can be obtained by using a solid mesh. The process of creating a mesh is called
tessellation
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
. Once tessellated, the mesh can be subjected to simulated stresses, strains, temperature differences, etc., to see how those changes propagate from node point to node point throughout the mesh.
VR/computer animation
In
virtual reality
Virtual reality (VR) is a simulated experience that employs pose tracking and 3D near-eye displays to give the user an immersive feel of a virtual world. Applications of virtual reality include entertainment (particularly video games), edu ...
and
computer animation
Computer animation is the process used for digitally generating animations. The more general term computer-generated imagery (CGI) encompasses both static scenes ( still images) and dynamic images ( moving images), while computer animation re ...
, an object may also be represented by a surface mesh of node points connected by triangles or quadrilaterals. If the goal is only to represent the visible portion of an object (and not show changes to the object) a solid mesh serves no purpose, for this application. The triangles or quadrilaterals can each be shaded differently depending on their orientation toward the light sources and/or viewer. This will give a rather faceted appearance, so an additional step is frequently added where the shading of adjacent regions is blended to provide smooth shading. There are several methods for performing this blending.
See also
*
Animation
Animation is a method by which still figures are manipulated to appear as moving images. In traditional animation, images are drawn or painted by hand on transparent celluloid sheets to be photographed and exhibited on film. Today, most ani ...
*
Boundary representation
In solid modeling and computer-aided design, boundary representation (often abbreviated B-rep or BREP) is a method for representing a 3D shape by defining the limits of its volume. A solid is represented as a collection of connected surface ...
*
Digital surface
Digital geometry deals with discrete sets (usually discrete point sets) considered to be digitized models or images of objects of the 2D or 3D Euclidean space.
Simply put, digitizing is replacing an object by a discrete set of its points. The i ...
*
Digital surface model
A digital elevation model (DEM) or digital surface model (DSM) is a 3D computer graphics representation of elevation data to represent terrain or overlaying objects, commonly of a planet, moon, or asteroid. A "global DEM" refers to a discret ...
*
Mesh generation
References
External links
3D-XplorMath: Program to visualize many kinds of surfaces in wireframe, patch and anaglyph mode.
{{DEFAULTSORT:Computer Representation Of Surfaces
3D computer graphics
Surfaces