|
Möller–Trumbore Intersection Algorithm
The Möller–Trumbore ray-triangle intersection algorithm, named after its inventors Tomas Möller and Ben Trumbore, is a fast method for calculating the intersection of a ray and a triangle in three dimensions without needing precomputation of the plane equation of the plane containing the triangle. Among other uses, it can be used in computer graphics to implement ray tracing computations involving triangle meshes. Calculation Definitions The ray is defined by an origin point O and a direction vector \vec. Every point on the ray can be expressed by \vec(t) = O + t\vec, where the parameter t ranges from zero to infinity. The triangle is defined by three vertices, named v_1, v_2, v_3. The plane that the triangle is on, which is needed to calculate the ray-triangle intersection, is defined by a point on the plane, such as v_1, and a vector that is orthogonal to every point on that plane, such as the cross product between the vector from v_1 to v_2 and the vector from v_1 to v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (Euclidean Geometry)
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines, which either is one point (sometimes called a '' vertex'') or does not exist (if the lines are parallel). Other types of geometric intersection include: * Line–plane intersection * Line–sphere intersection * Intersection of a polyhedron with a line * Line segment intersection * Intersection curve Determination of the intersection of flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be solved numerically, for example using Newton iteration. Intersection problems between a line and a conic section (circle, ellipse, parabola, etc.) or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Combination
In convex geometry and Vector space, vector algebra, a convex combination is a linear combination of point (geometry), points (which can be vector (geometric), vectors, scalar (mathematics), scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the ''count'' of the weights as in a standard weighted average. Formal definition More formally, given a finite number of points x_1, x_2, \dots, x_n in a real vector space, a convex combination of these points is a point of the form : \alpha_1x_1+\alpha_2x_2+\cdots+\alpha_nx_n where the real numbers \alpha_i satisfy \alpha_i\ge 0 and \alpha_1+\alpha_2+\cdots+\alpha_n=1. As a particular example, every convex combination of two points lies on the line segment between the points. A set is convex set, convex if it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Algorithms
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations. With the increasing automation of services, more and more decisions are being made by algorithms. Some general examples are; risk assessments, anticipatory policing, and pattern recognition technology. The following is a list of well-known algorithms. Automated planning Combinatorial algorithms General combinatorial algorithms * Brent's algorithm: finds a cycle in function value iterations using only two iterators * Floyd's cycle-finding algorithm: finds a cycle in function value iterations * Gale–Shapley algorithm: solves the stable matching problem * Pseudorandom number generators (uniformly dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Badouel Intersection Algorithm
The Badouel ray-triangle intersection algorithm, named after its inventor Didier Badouel, is a fast method for calculating the intersection of a ray and a triangle in three dimensions without needing precomputation of the plane equation of the plane Plane most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface * Plane (mathematics), generalizations of a geometrical plane Plane or planes may also refer to: Biology * Plane ... containing the triangle. External links Ray-Polygon IntersectionAn Efficient Ray-Polygon Intersection by Didier Badouel from Graphics Gems I Computational geometry {{Algorithm-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java 3D
Java 3D is a scene graph-based 3D application programming interface (API) for the Java platform. It runs on top of either OpenGL or Direct3D until version 1.6.0, which runs on top of Java OpenGL (JOGL). Since version 1.2, Java 3D has been developed under the Java Community Process. A Java 3D scene graph is a directed acyclic graph (DAG). Compared to other solutions, Java 3D is not only a wrapper around these graphics APIs, but an interface that encapsulates the graphics programming using a true object-oriented approach. Here a scene is constructed using a scene graph that is a representation of the objects that have to be shown. This scene graph is structured as a tree containing several elements that are necessary to display the objects. Additionally, Java 3D offers extensive spatialized sound support. Java 3D and its documentation are available for download separately. They are not part of the Java Development Kit (JDK). History Intel, Silicon Graphics, Apple, and Sun al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java (programming Language)
Java is a High-level programming language, high-level, General-purpose programming language, general-purpose, Memory safety, memory-safe, object-oriented programming, object-oriented programming language. It is intended to let programmers ''write once, run anywhere'' (Write once, run anywhere, WORA), meaning that compiler, compiled Java code can run on all platforms that support Java without the need to recompile. Java applications are typically compiled to Java bytecode, bytecode that can run on any Java virtual machine (JVM) regardless of the underlying computer architecture. The syntax (programming languages), syntax of Java is similar to C (programming language), C and C++, but has fewer low-level programming language, low-level facilities than either of them. The Java runtime provides dynamic capabilities (such as Reflective programming, reflection and runtime code modification) that are typically not available in traditional compiled languages. Java gained popularity sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rust (programming Language)
Rust is a General-purpose programming language, general-purpose programming language emphasizing Computer performance, performance, type safety, and Concurrency (computer science), concurrency. It enforces memory safety, meaning that all Reference (computer science), references point to valid memory. It does so without a conventional Garbage collection (computer science), garbage collector; instead, memory safety errors and data races are prevented by the "borrow checker", which tracks the object lifetime of references Compiler, at compile time. Rust does not enforce a programming paradigm, but was influenced by ideas from functional programming, including Immutable object, immutability, higher-order functions, algebraic data types, and pattern matching. It also supports object-oriented programming via structs, Enumerated type, enums, traits, and methods. It is popular for systems programming. Software developer Graydon Hoare created Rust as a personal project while working at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cramer's Rule
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations. It is named after Gabriel Cramer, who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748, and possibly knew of it as early as 1729. Cramer's rule, implemented in a naive way, is computationally inefficient for systems of more than two or three equations. In the case of equations in unknowns, it requires computation of determinants, while Gaussian elimination produces the result with the same (up to a constant factor independent of ) computational complexity as the computation of a single determinant. Moreo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Invertible
In mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is a right inverse of . (An identity element is an element such that and for all and for which the left-hand sides are defined.) When the operation is associative, if an element has both a left inverse and a right inverse, then these two inverses are equal and unique; they are called the ''inverse element'' or simply the ''inverse''. Often an adjective is added for specifying the operation, such as in additive inverse, multiplicative inverse, and functional inverse. In this case (associative operation), an invertible element is an element that has an inverse. In a ring, an ''invertible element'', also called a unit, is an element that is invertible under multiplication (this is not ambiguous, as every element is invertible under ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric Coordinate System
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or ''barycenter'') of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex. Every point has barycentric coordinates, and their sum is never zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity. Barycentric coordin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ray (geometry)
In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word ''line'' may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points (its ''endpoints''). Euclid's ''Elements'' defines a straight line as a "breadthless length" that "lies evenly with respect to the points on itself", and introduced several postulates as basic unprovable properties on which the rest of geometry was established. ''Euclidean line'' and ''Euclidean geometry'' are terms introduced to avoid confusion with generalizations introduced since the end of the 19th century, such as non-Euclidean, projective, and affine geometry. Properties In the Greek deductive geometry of Euclid's ''Elements'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a Scalar (mathematics), scalar as a result". It is also used for other symmetric bilinear forms, for example in a pseudo-Euclidean space. Not to be confused with scalar multiplication. is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors), and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two Euclidean vector, vectors is widely used. It is often called the inner product (or rarely the projection product) of Euclidean space, even though it is not the only inner product that can be defined on Euclidean space (see ''Inner product space'' for more). It should not be confused with the cross product. Algebraically, the dot product is the sum of the Product (mathematics), products of the corresponding entries of the two sequences of numbers. Geometrically, it is the product of the Euc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
