Bézier curves are a particular family of curves in that parameterize a smooth path given points. They are often used to model or approximate general curved shapes because of their algebraic simplicity.
Bézier curves were originally developed by Paul de Casteljau in 1959. But it became a famous shape only when Pierre Bézier, a French engineer at Renault, used it to automobiles in the 1970’s. They are now widely used in many such as industrial and computer-aided , vector-based drawing, font design, and 3D modeling.
3 Special cases
A Bézier curve between two control points is simply the line segment between the two points:
For three control points, we have the quadratic Bézier curve:
For four control points, we have the cubic Bézier curve:
These are the most commonly employed versions of Bézier curves.
The curve is tangent to the segment of line at the start and at the end.
It is easy to keep until the first order continuity when chaining curves. Moreover, every Bézier curve can be cut at any point into two new Bézier curves. In fact, the industry uses series of Bézier curves with only 4 control points (a bicubic version of the equations) to draw complex shapes.
The curves never do fancy stuff and always remain within the convex hull of the control points.
For one and two points the definition is still valid and the result became a point and a segment of line, respectively.
But it is impossible to draw circles or arcs of circle.
If the first and the last point are the same, the curve is closed. And if and are aligned with then the curve is closed with a first order continuity.
5 Curve Cut
Let be a Bezier curve, defined by its control points to for .
Let be such as Let be such as
is a part of the initial bezier curve with new control points to whose will be determined.
For control points is a -order polynom.
Let take more points, evenly spaced in [a,b]
For a 4 control points bezier curve If we take in , hence , , , respectively equal to , , ,
|Date of creation||2015-01-23 16:33:51|
|Last modified on||2015-01-23 16:33:51|
|Last modified by||yota (10184)|