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For quadratic Bézier curves one can construct intermediate points Q 0 and Q 1 such that as t varies from 0 to 1: Point Q 0 (t) varies from P 0 to P 1 and describes a linear Bézier curve. Point Q 1 (t) varies from P 1 to P 2 and describes a linear Bézier curve. Point B(t) is interpolated linearly between Q 0 (t) to Q 1 (t) and describes a ...
If the curve is required to be in a particular sub-category of n-th degree polynomial equations, then fewer than n(n + 3) / 2 points may be necessary and sufficient to determine a unique curve. For example, three (non-collinear) points determine a circle: the generic circle is given by the equation () + = where the center is located at (a, b ...
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.
For Bézier curves, it has become customary to refer to the -vectors in a parametric representation of a curve or surface in -space as control points, while the scalar-valued functions , defined over the relevant parameter domain, are the corresponding weight or blending functions.
Given the two red points, the blue line is the linear interpolant between the points, and the value y at x may be found by linear interpolation. 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.
With the n-th polynomial normalized to give P n (1) = 1, the i-th Gauss node, x i, is the i-th root of P n and the weights are given by the formula [3] = [′ ()]. Some low-order quadrature rules are tabulated below (over interval [−1, 1] , see the section below for other intervals).
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A line will connect any two points, so a first degree polynomial equation is an exact fit through any two points with distinct x coordinates. If the order of the equation is increased to a second degree polynomial, the following results: = + +. This will exactly fit a simple curve to three points.