Conic Section as Bézier Curve
Conic Section as Bézier Curve
Any conic section can be represented as a rational Bézier curve of degree two defined by , where (t) are the Bernstein polynomials and the control points. It is always possible to write the expression in a standard form such that ==1. From such a form it is easy to determine the type of the conic section: if >1, it is a hyperbola; if =1, it is a parabola; and if <1, it is an ellipse.
C(t)=(t)+(t)+(t)(t)+(t)+(t)
B
0,2
ω
0
P
0
B
1,2
ω
1
P
1
B
2,2
ω
2
P
2
B
0,2
ω
0
B
1,2
ω
1
B
2,2
ω
2
B
i,n
P
i
ω
0
ω
2
ω
1
ω
1
ω
1