In mathematics, a Coons patch, is a type of surface patch or manifold parametrization used in computer graphics to smoothly join other surfaces together, and in computational mechanics applications, particularly in finite element method and boundary element method, to mesh problem domains into elements.
Coons patches are named after Steven Anson Coons, and date to 1967. [1]
Given four space curves c0(s), c1(s), d0(t), d1(t) which meet at four corners c0(0) = d0(0), c0(1) = d1(0), c1(0) = d0(1), c1(1) = d1(1); linear interpolation can be used to interpolate between c0 and c1, that is
and between d0, d1
producing two ruled surfaces defined on the unit square.
The bilinear interpolation on the four corner points is another surface
A bilinearly blended Coons patch is the surface
Although the bilinear Coons patch exactly meets its four boundary curves, it does not necessarily have the same tangent plane at those curves as the surfaces to be joined, leading to creases in the joined surface along those curves. To fix this problem, the linear interpolation can be replaced with cubic Hermite splines with the weights chosen to match the partial derivatives at the corners. This forms a bicubically blended Coons patch.
A Bézier curve is a parametric curve used in computer graphics and related fields. The curves, which are related to Bernstein polynomials, are 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 field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
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.
Texture mapping is a method for defining high frequency detail, surface texture, or color information on a computer-generated graphic or 3D model. The original technique was pioneered by Edwin Catmull in 1974.
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.
In the mathematical field of differential geometry, one definition of a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface and produces a real number scalar g(v, w) in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space. In the same way as a dot product, metric tensors are used to define the length of and angle between tangent vectors. Through integration, the metric tensor allows one to define and compute the length of curves on the manifold.
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.
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.
Function Representation 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 of multidimensional geometric objects (shapes). An object as a point set in multidimensional space is defined by a single continuous real-valued function of point coordinates which is evaluated at the given point by a procedure traversing a tree structure with primitives in the leaves and operations in the nodes of the tree. The points with belong to the object, and the points with are outside of the object. The point set with is called an isosurface.
In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable; when the variates are spatial coordinates, it is also known as spatial interpolation.
The finite element method (FEM) is a widely used 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 transfer, fluid flow, mass transport, and electromagnetic potential.
In the finite element method for the numerical solution of elliptic partial differential equations, the stiffness matrix represents the system of linear equations that must be solved in order to ascertain an approximate solution to the differential equation.
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:
Isogeometric analysis is a computational approach that offers the possibility of integrating finite element analysis (FEA) into conventional NURBS-based CAD design tools. Currently, it is necessary to convert data between CAD and FEA packages to analyse new designs during development, a difficult task since the two computational geometric approaches are different. Isogeometric analysis employs complex NURBS geometry in the FEA application directly. This allows models to be designed, tested and adjusted in one go, using a common data set.
In numerical analysis, transfinite interpolation is a means to construct functions over a planar domain in such a way that they match a given function on the boundary. This method is applied in geometric modelling and in the field of finite element method.
In technical applications of 3D computer graphics (CAx) such as computer-aided design and computer-aided manufacturing, surfaces are one way of representing objects. The other ways are wireframe and solids. Point clouds 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.
