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Intersection (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 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 a quadric (sphere, cylinder, hyperboloid, et ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line–sphere Intersection
In analytic geometry, a line and a sphere can intersect in three ways: # No intersection at all # Intersection in exactly one point # Intersection in two points. Methods for distinguishing these cases, and determining the coordinates for the points in the latter cases, are useful in a number of circumstances. For example, it is a common calculation to perform during ray tracing. Calculation using vectors in 3D In vector notation, the equations are as follows: Equation for a sphere :\left\Vert \mathbf - \mathbf \right\Vert^2=r^2 :*\mathbf : points on the sphere :*\mathbf : center point :*r : radius of the sphere Equation for a line starting at \mathbf :\mathbf=\mathbf + d\mathbf :*\mathbf : points on the line :*\mathbf : origin of the line :*d : distance from the origin of the line :*\mathbf : direction of line (a non-zero vector) Searching for points that are on the line and on the sphere means combining the equations and solving for d, involving the dot product of vectors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line–line Intersection
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point, or another line. Distinguishing these cases and finding the intersection have uses, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines are not in the same plane, they have no point of intersection and are called skew lines. If they are in the same plane, however, there are three possibilities: if they coincide (are not distinct lines), they have an infinitude of points in common (namely all of the points on either of them); if they are distinct but have the same slope, they are said to be parallel and have no points in common; otherwise, they have a single point of intersection. The distinguishing features of non-Euclidean geometry are the number and locations of possible intersections between two lines and the number of possible lines with no intersections (parallel lines) with a given line. Formulas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton Iteration
In numerical analysis, Newton's method, also known as the Newton–Raphson method, named after Isaac Newton and Joseph Raphson, is a root-finding algorithm which produces successively better approximations to the roots (or zeroes) of a real-valued function. The most basic version starts with a single-variable function defined for a real variable , the function's derivative , and an initial guess for a root of . If the function satisfies sufficient assumptions and the initial guess is close, then :x_ = x_0 - \frac is a better approximation of the root than . Geometrically, is the intersection of the -axis and the tangent of the graph of at : that is, the improved guess is the unique root of the linear approximation at the initial point. The process is repeated as :x_ = x_n - \frac until a sufficiently precise value is reached. This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution (algebra)
In algebra, the operation of substitution can be applied in various contexts involving formal objects containing symbols (often called variables or indeterminates); the operation consists of systematically replacing occurrences of some symbol by a given value. Substitution is a basic operation of computer algebra. It is generally called "subs" or "subst" in computer algebra systems. A common case of substitution involves polynomials, where substitution of a numerical value for the indeterminate of a (univariate) polynomial amounts to evaluating the polynomial at that value. Indeed, this operation occurs so frequently that the notation for polynomials is often adapted to it; instead of designating a polynomial by a name like ''P'', as one would do for other mathematical objects, one could define :P(X)=X^5-3X^2+5X-17 so that substitution for ''X'' can be designated by replacement inside "''P''(''X'')", say :P(2) = 13 or :P(X+1) = X^5 + 5X^4 + 10X^3 + 7X^2 + 4X - 14. Substitution c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant. The distance between any point of the circle and the centre is called the radius. Usually, the radius is required to be a positive number. A circle with r=0 (a single point) is a degenerate case. This article is about circles in Euclidean geometry, and, in particular, the Euclidean plane, except where otherwise noted. Specifically, a circle is a simple closed curve that divides the plane into two regions: an interior and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a '' disc''. A circle may also be defined as a special ki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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