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Inside–outside Test
In computational geometry, the point-in-polygon (PIP) problem asks whether a given point in the plane lies inside, outside, or on the boundary of a polygon. It is a special case of point location problems and finds applications in areas that deal with processing geometrical data, such as computer graphics, computer vision, geographic information systems (GIS), motion planning, and computer-aided design (CAD). An early description of the problem in computer graphics shows two common approaches (ray casting and angle summation) in use as early as 1974. An attempt of computer graphics veterans to trace the history of the problem and some tricks for its solution can be found in an issue of the ''Ray Tracing News''. Ray casting algorithm One simple way of finding whether the point is inside or outside a simple polygon is to test how many times a ray, starting from the point and going in any fixed direction, intersects the edges of the polygon. If the point is on the outside of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Polygon
In geometry, a simple polygon is a polygon that does not Intersection (Euclidean geometry), intersect itself and has no holes. That is, it is a Piecewise linear curve, piecewise-linear Jordan curve consisting of finitely many line segments. These polygons include as special cases the convex polygons, star-shaped polygons, and monotone polygons. The sum of external angles of a simple polygon is 2\pi. Every simple polygon with n sides can be polygon triangulation, triangulated by n-3 of its diagonals, and by the art gallery theorem its interior is visible from some \lfloor n/3\rfloor of its vertices. Simple polygons are commonly seen as the input to computational geometry problems, including point in polygon testing, area computation, the convex hull of a simple polygon, triangulation, and Euclidean shortest paths. Other constructions in geometry related to simple polygons include Schwarz–Christoffel mapping, used to find conformal maps involving simple polygons, polygonalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even–odd Rule
The even–odd rule is an algorithm implemented in vector-based graphic software, like the PostScript language and Scalable Vector Graphics (SVG), which determines how a graphical shape with more than one closed outline will be filled. Unlike the nonzero-rule algorithm, this algorithm will alternatively color and leave uncolored shapes defined by nested closed paths irrespective of their winding. The SVG defines the even–odd rule by saying: The rule can be seen in effect in many vector graphic programs (such as Freehand or Illustrator), where a crossing of an outline with itself causes shapes to fill in strange ways. On a simple curve, the even–odd rule reduces to a decision algorithm for the point in polygon problem. The SVG computer vector graphics standard may be configured to use the even–odd rule when drawing polygons, though it uses the non-zero rule by default. Implementation Below is a partial example implementation in Python, by using a ray to the right o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Trigonometric Functions
In mathematics, the inverse trigonometric functions (occasionally also called ''antitrigonometric'', ''cyclometric'', or ''arcus'' functions) are the inverse functions of the trigonometric functions, under suitably restricted Domain of a function, domains. Specifically, they are the inverses of the sine, cosine, tangent (trigonometry), tangent, cotangent, secant (trigonometry), secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. Notation Several notations for the inverse trigonometric functions exist. The most common convention is to name inverse trigonometric functions using an arc- prefix: , , , etc. (This convention is used throughout this article.) This notation arises from the following geometric relationships: when measuring in radians, an angle of radians will correspond to an circular arc, arc whose length is , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtended Angle
In geometry, an angle subtended (from Latin for "stretched under") by a line segment at an arbitrary vertex is formed by the two rays between the vertex and each endpoint of the segment. For example, a side of a triangle ''subtends'' the opposite angle. More generally, an angle subtended by an arc of a curve is the angle subtended by the corresponding chord of the arc. For example, a circular arc ''subtends'' the central angle formed by the two radii through the arc endpoints. If an angle is subtended by a straight or curved segment, the segment is said to ''subtend'' the angle. Sometimes the term "subtend" is applied in the opposite sense, and the angle is said to ''subtend'' the segment. Alternately, the angle can be said to ''intercept'' or ''enclose'' the segment. The above definition of a subtended plane angle remains valid in three-dimensional space (3D), as one vertex and two endpoints (assumed non-collinear) define an Euclidean plane in 3D. For example, an ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonzero-rule
In two-dimensional computer graphics, the non-zero winding rule is a means of determining whether a given Point (geometry)#Points in Euclidean geometry, point falls within an enclosed curve. Unlike the similar even-odd rule, it relies on knowing the direction of stroke for each part of the curve. For a given curve C and a given point P: construct a ray (a straight line) heading out from P in any direction towards infinity. Find all the intersections of C with this ray. Score up the winding number as follows: for every clockwise intersection (the curve passing through the ray from left to right, as viewed from P) subtract 1; for every counter-clockwise intersection (curve passing from right to left, as viewed from P) add 1. If the total winding number is zero, P is outside C; otherwise, it is inside. The winding number is effectively a count of how many full counter-clockwise revolutions ('windings') the curve makes around P without doubling back on itself. (If P were a nail and C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Winding Number
In mathematics, the winding number or winding index of a closed curve in the plane (mathematics), plane around a given point (mathematics), point is an integer representing the total number of times that the curve travels counterclockwise around the point, i.e., the curve's number of turns. For certain open plane curves, the number of turns may be a non-integer. The winding number depends on the curve orientation, orientation of the curve, and it is negative number, negative if the curve travels around the point clockwise. Winding numbers are fundamental objects of study in algebraic topology, and they play an important role in vector calculus, complex analysis, geometric topology, differential geometry, and physics (such as in string theory). Intuitive description Suppose we are given a closed, oriented curve in the ''xy'' plane. We can imagine the curve as the path of motion of some object, with the orientation indicating the direction in which the object moves. Then the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Robustness
In computer science, robustness is the ability of a computer system to cope with errors during execution1990. IEEE Standard Glossary of Software Engineering Terminology, IEEE Std 610.12-1990 defines robustness as "The degree to which a system or component can function correctly in the presence of invalid inputs or stressful environmental conditions" and cope with erroneous input. Robustness can encompass many areas of computer science, such as robust programming, robust machine learning, and Robust Security Network. Formal techniques, such as fuzz testing, are essential to showing robustness since this type of testing involves invalid or unexpected inputs. Alternatively, fault injection can be used to test robustness. Various commercial products perform robustness testing of software analysis. Introduction In general, building robust systems that encompass every point of possible failure is difficult because of the vast quantity of possible inputs and input combinations. Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vertex (geometry)
In geometry, a vertex (: vertices or vertexes), also called a corner, is a point (geometry), point where two or more curves, line (geometry), lines, or line segments Tangency, meet or Intersection (geometry), intersect. For example, the point where two lines meet to form an angle and the point where edge (geometry), edges of polygons and polyhedron, polyhedra meet are vertices. Definition Of an angle The ''vertex'' of an angle is the point where two Line (mathematics)#Ray, rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place. :(3 vols.): (vol. 1), (vol. 2), (vol. 3). Of a polytope A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection (Euclidean geometry), intersection of Edge (geometry), edges, face (geometry), faces or facets of the object. In a polygon, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tolerance (engineering)
Engineering tolerance is the permissible limit or limits of variation in: # a physical dimension; # a measured value or physical property of a material, manufactured object, system, or service; # other measured values (such as temperature, humidity, etc.); # in engineering and safety, a physical distance or space (tolerance), as in a truck (lorry), train or boat under a bridge as well as a train in a tunnel (see structure gauge and loading gauge); # in mechanical engineering, the space between a bolt and a nut or a hole, etc. Dimensions, properties, or conditions may have some variation without significantly affecting functioning of systems, machines, structures, etc. A variation beyond the tolerance (for example, a temperature that is too hot or too cold) is said to be noncompliant, rejected, or exceeding the tolerance. Considerations when setting tolerances A primary concern is to determine how wide the tolerances may be without affecting other factors or the outcome of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt to find approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences like economics, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Program
A computer program is a sequence or set of instructions in a programming language for a computer to Execution (computing), execute. It is one component of software, which also includes software documentation, documentation and other intangible components. A ''computer program'' in its human-readable form is called source code. Source code needs another computer program to Execution (computing), execute because computers can only execute their native machine instructions. Therefore, source code may be Translator (computing), translated to machine instructions using a compiler written for the language. (Assembly language programs are translated using an Assembler (computing), assembler.) The resulting file is called an executable. Alternatively, source code may execute within an interpreter (computing), interpreter written for the language. If the executable is requested for execution, then the operating system Loader (computing), loads it into Random-access memory, memory and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Verification
In the context of hardware and software systems, formal verification is the act of proving or disproving the correctness of a system with respect to a certain formal specification or property, using formal methods of mathematics. Formal verification is a key incentive for formal specification of systems, and is at the core of formal methods. It represents an important dimension of analysis and verification in electronic design automation and is one approach to software verification. The use of formal verification enables the highest Evaluation Assurance Level ( EAL7) in the framework of common criteria for computer security certification. Formal verification can be helpful in proving the correctness of systems such as: cryptographic protocols, combinational circuits, digital circuits with internal memory, and software expressed as source code in a programming language. Prominent examples of verified software systems include the CompCert verified C compiler and the seL ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
