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A Bézier curve is defined by a set of control points P 0 through P n, where n is called the order of the curve (n = 1 for linear, 2 for quadratic, 3 for cubic, etc.). The first and last control points are always the endpoints of the curve; however, the intermediate control points generally do not lie on the curve.
For example, given a = f(x) = a 0 x 0 + a 1 x 1 + ··· and b = g(x) = b 0 x 0 + b 1 x 1 + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba ...
Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints. [ 4 ] [ 5 ] Curve fitting can involve either interpolation , [ 6 ] [ 7 ] where an exact fit to the data is required, or smoothing , [ 8 ] [ 9 ] in which a "smooth ...
The curve is named after Edwin Catmull and Raphael Rom. The principal advantage of this technique is that the points along the original set of points also make up the control points for the spline curve. [7] Two additional points are required on either end of the curve. The uniform Catmull–Rom implementation can produce loops and self ...
The blue curve shows the function whose definite integral on the interval [−1, 1] is to be calculated (the integrand). The trapezoidal rule approximates the function with a linear function that coincides with the integrand at the endpoints of the interval and is represented by an orange dashed line.
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.
In computer graphics, the centripetal Catmull–Rom spline is a variant form of the Catmull–Rom spline, originally formulated by Edwin Catmull and Raphael Rom, [1] which can be evaluated using a recursive algorithm proposed by Barry and Goldman. [2]
The envelope of the normals of the tractrix (that is, the evolute of the tractrix) is the catenary (or chain curve) given by y = a cosh x / a . The surface of revolution created by revolving a tractrix about its asymptote is a pseudosphere. The tractrix is a transcendental curve; it cannot be defined by a polynomial equation.