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References: http://en.wikipedia.org/wiki/B%C3%A9zier_curve
Examination of cases
A Bézier curve is defined by a set of control points P~0~ through P~n~, where n is called its order (n = 1 for linear, 2 for quadratic, etc.). The first and last control points are always the end points of the curve; however, the intermediate control points (if any) generally do not lie on the curve.
Given points P~0~ and P~1~, a linear Bézier curve is simply a straight line between those two points. The curve is given by
and is equivalent to linear interpolation.
A quadratic Bézier curve is the path traced by the function B(t), given points P~0~, P~1~, and P~2~,
,
which can be interpreted as the linear interpolant of corresponding points on the linear Bézier curves from P~0~ to P~1~ and from P~1~ to P~2~ respectively. Rearranging the preceding equation yields:
The derivative of the Bézier curve with respect to t is
from which it can be concluded that the tangents to the curve at P~0~ and P~2~ intersect at P~1~. As t increases from 0 to 1, the curve departs from P~0~ in the direction of P~1~, then bends to arrive at P~2~ in the direction from P~1~.
A quadratic Bézier curve is also a parabolic segment. As a parabola is a conic section, some sources refer to quadratic Béziers as “conic arcs”.^[2]^
Four points P~0~, P~1~, P~2~ and P~3~ in the plane or in higher-dimensional space define a cubic Bézier curve. The curve starts at P~0~ going toward P~1~ and arrives at P~3~ coming from the direction of P~2~. Usually, it will not pass through P~1~ or P~2~; these points are only there to provide directional information. The distance between P~0~ and P~1~ determines “how long” the curve moves into direction P~2~ before turning towards P~3~.
Writing B~P~i~,P~j~,P~k~~(t) for the quadratic Bézier curve defined by points P~i~, P~j~, and P~k~, the cubic Bézier curve can be defined as a linear combination of two quadratic Bézier curves:
The explicit form of the curve is:
For some choices of P~1~ and P~2~ the curve may intersect itself, or contain a cusp.
Some terminology is associated with these parametric curves. We have
where the polynomials
are known as Bernstein basis polynomials of degree n.
Note that t^0^ = 1,
(1 − t)^0^ = 1, and that the binomial
coefficient, ,
also expressed as
or
is:
The points P~i~ are called control points for the Bézier curve. The polygon formed by connecting the Bézier points with lines, starting with P~0~ and finishing with P~n~, is called the Bézier polygon (or control polygon). The convex hull of the Bézier polygon contains the Bézier curve.
Animation of a linear Bézier curve, t in [0,1]
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The t in the function for a linear Bézier curve can be thought of as describing how far B(t) is from P~0~ toP~1~. For example when t=0.25, B(t) is one quarter of the way from point P~0~ to P~1~. As t varies from 0 to 1,B(t) describes a straight line from P~0~ to P~1~.
For quadratic Bézier curves one can construct intermediate points Q~0~ and Q~1~ such that as t varies from 0 to 1:
[![Construction of a [ of a quadratic Bézier cu
rve, t in [0,1]”)
Construction of a Animation of a quadratic quadratic Bézier curve Bézier curve, t in [0,1] ———————— ———————— ————————
For higher-order curves one needs correspondingly more intermediate points. For cubic curves one can construct intermediate points Q~0~, Q~1~, and Q~2~that describe linear Bézier curves, and points R~0~ & R~1~ that describe quadratic Bézier curves:
[![Construction of a [ t in [0,1]”)
Construction of a cubic Animation of a cubic Bézier curve Bézier curve, t in [0,1] ———————— ———————— ————————
For fourth-order curves one can construct intermediate points Q~0~, Q~1~, Q~2~ & Q~3~ that describe linear Bézier curves, points R~0~, R~1~ & R~2~ that describe quadratic Bézier curves, and points S~0~ & S~1~ that describe cubic Bézier curves:
[![Construction of a [ of a quartic Bézier curv
e, t in [0,1]”)
Construction of a Animation of a quartic quartic Bézier curve Bézier curve, t in [0,1] ———————— ———————— ————————
For fifth-order curves, one can construct similar intermediate points.
Animation of a fifth order Bézier curve, t in [0,1]
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These representations rest on the process used in De Casteljau’s algorithm to calculate Bezier curves.^[3]^