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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 ...
We may approximate a circle of radius from an arbitrary number of cubic Bézier curves. Let the arc start at point A {\displaystyle \mathbf {A} } and end at point B {\displaystyle \mathbf {B} } , placed at equal distances above and below the x-axis, spanning an arc of angle θ = 2 ϕ {\displaystyle \theta =2\phi } :
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. Repeat the process until you arrive at the single point – this is the point of the curve corresponding to the parameter .
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.
Graphs showing the relationship between the roots, and turning, stationary and inflection points of a cubic polynomial, and its first and second derivatives by CMG Lee. Thanks to en:user:GalacticShoe for an algorithm to exactly draw a cubic polynomial segment with a cubic Bezier. Source: Own work: Author: Cmglee: Other versions
The information that seems to be missing are the visual factors determining the enpoints, Q 1 and Q 2, of the green line segment. Without that info, the choice of Q 1 and Q 2 seems to be arbitrary for any particluar value of the parameter t, and we could trace-out any curve at all in the absence of this info.
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.
Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set. Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m i {\displaystyle m_{i}} modified to ensure the monotonicity of the resulting Hermite spline.