A Bézier curve is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. Bézier curve is named after French engineer Pierre Bézier, who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of computer fonts and animation. Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces. The Bézier triangle is a special case of the latter.
In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data.
In geometric modelling and in computer graphics, a composite Bézier curve is a piecewise Bézier curve that is at least continuous. In other words, a composite Bézier curve is a series of Bézier curves joined end to end where the last point of one curve coincides with the starting point of the next curve. Depending on the application, additional smoothness requirements may be added.
In vector computer graphics, CAD systems, and geographic information systems, geometric primitive is the simplest geometric shape that the system can handle. Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segment, which were all that early vector graphics systems had.
Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling. As with Bézier curves, a Bézier surface is defined by a set of control points. Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central control points; rather, it is "stretched" toward them as though each were an attractive force. They are visually intuitive, and for many applications, mathematically convenient.
Non-uniform rational basis spline (NURBS) is a mathematical model using basis splines (B-splines) that is commonly used in computer graphics for representing curves and surfaces. It offers great flexibility and precision for handling both analytic and modeled shapes. It is a type of curve modeling, as opposed to polygonal modeling or digital sculpting. NURBS curves are commonly used in computer-aided design (CAD), manufacturing (CAM), and engineering (CAE). They are part of numerous industry-wide standards, such as IGES, STEP, ACIS, and PHIGS. Tools for creating and editing NURBS surfaces are found in various 3D graphics and animation software packages.
In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees.
In the field of 3D computer graphics, a subdivision surface is a curved surface represented by the specification of a coarser polygonal mesh and produced by a recursive algorithmic method. The curved surface, the underlying inner mesh, can be calculated from the coarse mesh, known as the control cage or outer mesh, as the functional limit of an iterative process of subdividing each polygonal face into smaller faces that better approximate the final underlying curved surface. Less commonly, a simple algorithm is used to add geometry to a mesh by subdividing the faces into smaller ones without changing the overall shape or volume.
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called differentiability class. At the very minimum, a function could be considered smooth if it is differentiable everywhere. At the other end, it might also possess derivatives of all orders in its domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function.
In solid modeling and computer-aided design, boundary representation 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 elements, which define the boundary between interior and exterior points.
Steven Anson Coons was an early pioneer in the field of computer graphical methods. He was a professor at the Massachusetts Institute of Technology in the Mechanical Engineering Department. He was also a professor at Syracuse University after leaving MIT. Steven Coons had a vision of interactive computer graphics as a design tool to aid the engineer.
Freeform surface modelling is a technique for engineering freeform surfaces with a CAD or CAID system.
A parametric surface is a surface in the Euclidean space which is defined by a parametric equation with two parameters :\mathbb {R} ^{2}\to \mathbb {R} ^{3}} . Parametric representation is a very general way to specify a surface, as well as implicit representation. Surfaces that occur in two of the main theorems of vector calculus, Stokes' theorem and the divergence theorem, are frequently given in a parametric form. The curvature and arc length of curves on the surface, surface area, differential geometric invariants such as the first and second fundamental forms, Gaussian, mean, and principal curvatures can all be computed from a given parametrization.
In optics, a caustic or caustic network is the envelope of light rays reflected or refracted by a curved surface or object, or the projection of that envelope of rays on another surface. The caustic is a curve or surface to which each of the light rays is tangent, defining a boundary of an envelope of rays as a curve of concentrated light. Therefore, in the photo to the right, caustics can be seen as patches of light or their bright edges. These shapes often have cusp singularities.
Thin plate splines (TPS) are a spline-based technique for data interpolation and smoothing. They were introduced to geometric design by Duchon. They are an important special case of a polyharmonic spline. Robust Point Matching (RPM) is a common extension and shortly known as the TPS-RPM algorithm.
In kinematics, the motion of a rigid body is defined as a continuous set of displacements. One-parameter motions can be defined as a continuous displacement of moving object with respect to a fixed frame in Euclidean three-space (E3), where the displacement depends on one parameter, mostly identified as time.
In geometry, an isophote is a curve on an illuminated surface that connects points of equal brightness. One supposes that the illumination is done by parallel light and the brightness is measured by the following scalar product:
Geometrical design (GD) is a branch of computational geometry. It deals with the construction and representation of free-form curves, surfaces, or volumes and is closely related to geometric modeling. Core problems are curve and surface modelling and representation. GD studies especially the construction and manipulation of curves and surfaces given by a set of points using polynomial, rational, piecewise polynomial, or piecewise rational methods. The most important instruments here are parametric curves and parametric surfaces, such as Bézier curves, spline curves and surfaces. An important non-parametric approach is the level-set method.
Solid Modeling Solutions is a company who has an implementation of a mathematical representation of NURBS, 3D geometry, and Solid modeling technology which emerged in the 1980s and 1990s into a commercial implementation known as SMLib. This article will provide the background and history of this implementation into a commercial product line from Solid Modeling Solutions (SMS). SMS is an independent supplier of source code for a powerful suite of 3D geometry kernels. SMS provides advanced NURBS-based geometry libraries, SMLib, TSNLib, GSNLib, NLib, SDLib, VSLib, and PolyMLib, that encompass extensive definition and manipulation of NURBS curves and surfaces with the latest fully functional non-manifold topology.
Computational anatomy is an interdisciplinary field of biology focused on quantitative investigation and modelling of anatomical shapes variability. It involves the development and application of mathematical, statistical and data-analytical methods for modelling and simulation of biological structures.
