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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 control points; rather, it is "stretched" toward them as though each were an attractive force.
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Following is a list of notable software, computer programs, used to develop a mathematical representation of any three dimensional surface of objects, as 3D computer graphics, also called 3D modeling.
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.
Boundary representation has also been extended to allow special, non-solid model types called non-manifold models. As described by Braid, normal solids found in nature have the property that, at every point on the boundary, a small enough sphere around the point is divided into two pieces, one inside and one outside the object.
A biharmonic Bézier surface is a smooth polynomial surface which conforms to the biharmonic equation and has the same formulations as a Bézier surface. This formulation for Bézier surfaces was developed by Juan Monterde and Hassan Ugail .
In computer graphics, free-form deformation (FFD) is a geometric technique used to model simple deformations of rigid objects. It is based on the idea of enclosing an object within a cube or another hull object, and transforming the object within the hull as the hull is deformed.
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 ...