Search results
Results from the WOW.Com Content Network
Affine arithmetic is potentially useful in every numeric problem where one needs guaranteed enclosures to smooth functions, such as solving systems of non-linear equations, analyzing dynamical systems, integrating functions, differential equations, etc. Applications include ray tracing, plotting curves, intersecting implicit and parametric ...
The graph of this function is shown to the right. Since the graph of an affine(*) function is a line, the graph of a piecewise linear function consists of line segments and rays. The x values (in the above example −3, 0, and 3) where the slope changes are typically called breakpoints, changepoints, threshold values or knots. As in many ...
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
See Affine space § Affine combinations and barycenter for the definition in this case. This concept is fundamental in Euclidean geometry and affine geometry , because the set of all affine combinations of a set of points forms the smallest affine space containing the points, exactly as the linear combinations of a set of vectors form their ...
In mathematics, the term linear function refers to two distinct but related notions: [1] In calculus and related areas, a linear function is a function whose graph is a straight line, that is, a polynomial function of degree zero or one. [2] For distinguishing such a linear function from the other concept, the term affine function is often used ...
An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
Therefore, the expression on the right-hand side is just the equation for the tangent line to the graph of at (, ()). For this reason, this process is also called the tangent line approximation . Linear approximations in this case are further improved when the second derivative of a, f ″ ( a ) {\displaystyle f''(a)} , is sufficiently small ...
In projective geometry, affine space means the complement of a hyperplane at infinity in a projective space. Affine space can also be viewed as a vector space whose operations are limited to those linear combinations whose coefficients sum to one, for example 2x − y, x − y + z, (x + y + z)/3, ix + (1 − i)y, etc.