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  2. Bézier curve - Wikipedia

    en.wikipedia.org/wiki/Bézier_curve

    The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years later when mathematician Paul de Casteljau in 1959 developed de Casteljau's algorithm, a numerically stable method for evaluating the curves, and became the first to apply them to computer-aided design at French automaker Citroën ...

  3. Bézier triangle - Wikipedia

    en.wikipedia.org/wiki/Bézier_triangle

    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.

  4. Control point (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Control_point_(mathematics)

    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.

  5. Variation diminishing property - Wikipedia

    en.wikipedia.org/wiki/Variation_diminishing_property

    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.

  6. Paul de Casteljau - Wikipedia

    en.wikipedia.org/wiki/Paul_de_Casteljau

    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.

  7. De Casteljau's algorithm - Wikipedia

    en.wikipedia.org/wiki/De_Casteljau's_algorithm

    Note that the intermediate points that were constructed are in fact the control points for two new Bézier curves, both exactly coincident with the old one. This algorithm not only evaluates the curve at , but splits the curve into two pieces at , and provides the equations of the two sub-curves in Bézier form.

  8. List of curves - Wikipedia

    en.wikipedia.org/wiki/List_of_curves

    Main page; Contents; Current events; Random article; About Wikipedia; Contact us; Pages for logged out editors learn more

  9. File:Quadratic Bezier parabola equivalence.svg - Wikipedia

    en.wikipedia.org/wiki/File:Quadratic_Bezier...

    Equivalence of a quadratic Bezier curve and a segment of a parabola by CMG Lee. Tangents to the parabola at the end-points of the curve (A and B) intersect at its control point (C). If D is the mid-point of AB, the tangent to the curve which is perpendicular to CD (dashed cyan line) defines its vertex (V).

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