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Bézier curves or Bézier splines are certain polynomials first described in 1972 by the French engineer Pierre Bézier who used them to design automobile bodies. The most important Bézier curves, the cubic ones, are used in computer graphics and several imaging systems such as PostScript, Metafont and GIMP for drawing "smooth" curves when joined into bezigons. TrueType fonts use the simpler quadratic Bézier curves.
Four points A, B, C and D in the plane or in threedimensional space define a cubic Bézier curve. The curve starts at A going toward B and arrives at D coming from the direction of C. In general, it will not pass through B or C; these points are only there to provide directional information. The distance between A and B determines "how long" the curve moves into direction B before turning towards D.
The parametric form of the curve is:
Bézier curves are attractive in computer graphics for two main reasons:
A' \\ B' \\ C' \\ D'\end{vmatrix} = \begin{vmatrix}
1 & 0 & 0 & 0 \\ 1/2 & 1/2 & 0 & 0 \\ 1/4 & 2/4 & 1/4 & 0 \\ 1/8 & 3/8 & 3/8 & 1/8\end{vmatrix} \cdot \begin{vmatrix}
A \\ B \\ C \\ D\end{vmatrix}</math>
D:=(C+D)/2, C:=(B+C)/2, D:=(C+D)/2, B:=(A+B)/2, C:=(B+C)/2, D:=(C+D)/2 (No language in particular)
Generalizing the cubic case leads to higher order curves which require more than four control points; however, these do not find much use in practice. Instead, complicated curves are pieced together from cubic curves to form bezigons: the first Bézier curve has control points A, B, C, and D, the second has control points D, E, F, and G, and if G1 continuity (i.e. smoothness of the curve) at D is required, then the direction of CD needs to equal the direction of DE.
See also: Spline[?], Bernstein polynomial[?], Bézier surface[?], Bézier triangle, NURBS
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