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The points P i are called control points for the Bézier curve. The polygon formed by connecting the Bézier points with lines, starting with P 0 and finishing with P n, is called the Bézier polygon (or control polygon). The convex hull of the Bézier polygon contains the Bézier curve.
Consider a Bézier curve with control points , …,. Connecting the consecutive points we create the control polygon of the curve. Subdivide now each line segment of this polygon with the ratio : and connect the points you get. This way you arrive at the new polygon having one fewer segment.
The process of degree elevation for Bézier curves can be considered an instance of piecewise linear interpolation. Piecewise linear interpolation can be shown to be variation diminishing. [4] Thus, if R 1, R 2, R 3 and so on denote the set of polygons obtained by the degree elevation of the initial control polygon R, then it can be shown that
Spline curve drawn as a weighted sum of B-splines with control points/control polygon, and marked component curves. In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition.
The gray polygonal chain connecting the control points is called the control polygon. In computer-aided geometric design, smooth curves are often defined by a list of control points, e.g. in defining Bézier curve segments. When connected together, the control points form a polygonal chain called a control polygon.
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.
The geometry of a single bicubic patch is thus completely defined by a set of 16 control points. These are typically linked up to form a B-spline surface in a similar way as Bézier curves are linked up to form a B-spline curve. Simpler Bézier surfaces are formed from biquadratic patches (m = n = 2), or Bézier triangles.
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 ...