Example: a circle Non-uniform rational B-spline
nurbs have ability describe circles. here, black triangle control polygon of nurbs curve (shown @ w=1). blue dotted line shows corresponding control polygon of b-spline curve in 3d homogeneous coordinates, formed multiplying nurbs control points corresponding weights. blue parabolas corresponding b-spline curve in 3d, consisting of 3 parabolas. choosing nurbs control points , weights, parabolas parallel opposite face of gray cone (with tip @ 3d origin), dividing w project parabolas onto w=1 plane results in circular arcs (red circle; see conic section).
non-rational splines or bézier curves may approximate circle, cannot represent exactly. rational splines can represent conic section, including circle, exactly. representation not unique, 1 possibility appears below:
the order three, since circle quadratic curve , spline s order 1 more degree of piecewise polynomial segments. knot vector
{
0
,
0
,
0
,
π
/
2
,
π
/
2
,
π
,
π
,
3
π
/
2
,
3
π
/
2
,
2
π
,
2
π
,
2
π
}
{\displaystyle \{0,0,0,\pi /2,\pi /2,\pi ,\pi ,3\pi /2,3\pi /2,2\pi ,2\pi ,2\pi \}\,}
. circle composed of 4 quarter circles, tied double knots. although double knots in third order nurbs curve result in loss of continuity in first derivative, control points positioned in such way first derivative continuous. in fact, curve infinitely differentiable everywhere, must if represents circle.
the curve represents circle exactly, not parametrized in circle s arc length. means, example, point @
t
{\displaystyle t}
not lie @
(
sin
(
t
)
,
cos
(
t
)
)
{\displaystyle (\sin(t),\cos(t))}
(except start, middle , end point of each quarter circle, since representation symmetrical). impossible, since x coordinate of circle provide exact rational polynomial expression
cos
(
t
)
{\displaystyle \cos(t)}
, impossible. circle make 1 full revolution parameter
t
{\displaystyle t}
goes 0
2
π
{\displaystyle 2\pi }
, because knot vector arbitrarily chosen multiples of
π
/
2
{\displaystyle \pi /2}
.
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