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A Bézier curve of degree n can be converted into a Bézier curve of degree n + 1 with the same shape. This is useful if software supports Bézier curves only of specific degree. For example, systems that can only work with cubic Bézier curves can implicitly work with quadratic curves by using their equivalent cubic representation.
Bernstein polynomials thus provide one way to prove the Weierstrass approximation theorem that every real-valued continuous function on a real interval [a, b] can be uniformly approximated by polynomial functions over . [7] A more general statement for a function with continuous k th derivative is
The following JavaScript function applies De Casteljau's algorithm to an array of control points or poles as originally named by De Casteljau to reduce them one by one until reaching a point in the curve for a given t between 0 for the first point of the curve and 1 for the last one
In the mathematical subfield of numerical analysis, de Boor's algorithm [1] is a polynomial-time and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of de Casteljau's algorithm for Bézier curves. The algorithm was devised by German-American mathematician Carl R. de Boor. Simplified ...
Using the above points, we say that since the Bézier curve B is the limit of these polygons as r goes to , it will have fewer intersections with a given plane than R i for all i, and in particular fewer intersections that the original control polygon R. This is the statement of the variation diminishing property.
Curves with such knot vectors start and end in a control point. The values of the knots control the mapping between the input parameter and the corresponding NURBS value. For example, if a NURBS describes a path through space over time, the knots control the time that the function proceeds past the control points.
In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography. This is usually done with Bézier curves , which are a simple generalization of interpolation polynomials (having specified tangents as well as specified points).
An example Bézier triangle with control points marked. A cubic Bézier triangle is a surface with the equation (,,) = (+ +) = + + + + + + + + +where α 3, β 3, γ 3, α 2 β, αβ 2, β 2 γ, βγ 2, αγ 2, α 2 γ and αβγ are the control points of the triangle and s, t, u (with 0 ≤ s, t, u ≤ 1 and s + t + u = 1) are the barycentric coordinates inside the triangle.