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A Bézier curve is defined by a set of control points P0 through Pn, 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.
De Casteljau's algorithm. In the mathematical field of numerical analysis, De Casteljau's algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau. De Casteljau's algorithm can also be used to split a single Bézier curve into two Bézier curves at an arbitrary ...
Paul de Casteljau (19 November 1930 – 24 March 2022) was a French physicist and mathematician. In 1959, while working at Citroën, he developed an algorithm for evaluating calculations on a certain family of curves, which would later be formalized and popularized by engineer Pierre Bézier, leading to the curves widely known as Bézier curves.
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of Bernstein basis polynomials. The idea is named after mathematician Sergei Natanovich Bernstein. Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Weierstrass approximation theorem.
Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling. As with Bézier curves, a Bézier surface is defined by a set of control points. Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central ...
Cubic Bézier triangle. 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. [2][1] Alternatively, a cubic ...
The quarter of Viviani's curve that lies in the all-positive quadrant of 3D space cannot be represented exactly by a regular Bezier curve of any degree. However, it can be represented exactly by a 3D rational Bezier segment of degree 4, and there is an infinite family of rational Bezier control points generating that segment.
Composite Bézier curve. In geometric modelling and in computer graphics, a composite Bézier curve or Bézier spline is a spline made out of Bézier curves that is at least continuous. In other words, a composite Bézier curve is a series of Bézier curves joined end to end where the last point of one curve coincides with the starting point of ...