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A Bézier curve is defined by a set of control points P 0 through P n, 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.
The resulting four-dimensional points may be projected back into three-space with a perspective divide. In general, operations on a rational curve (or surface) are equivalent to operations on a nonrational curve in a projective space. This representation as the "weighted control points" and weights is often convenient when evaluating rational ...
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.
Each temporary control point is written exactly once and read twice. By reversing the iteration over i {\displaystyle i} (counting down instead of up), we can run the algorithm with memory for only p + 1 {\displaystyle p+1} temporary control points, by letting d i , r {\displaystyle \mathbf {d} _{i,r}} reuse the memory for d i , r − 1 ...
A quadratic (=) Bézier triangle features 6 control points which are all located on the edges. The cubic (=) Bézier triangle is defined by 10 control points and is the lowest order Bézier triangle that has an internal control point, not located on the edges. In all cases, the edges of the triangle will be Bézier curves of the same degree.
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.
The variation diminishing property of Bézier curves is that they are smoother than the polygon formed by their control points. If a line is drawn through the curve, the number of intersections with the curve will be less than or equal to the number of intersections with the control polygon. In other words, for a Bézier curve B defined by the ...
Béziergon – The red béziergon passes through the blue vertices, the green points are control points that determine the shape of the connecting Bézier curves 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 C 0 {\displaystyle C^{0}} continuous .