BSplineCurve
BSplineCurve[{pt_{1},pt_{2},…}]
is a graphics primitive that represents a nonuniform rational Bspline curve with control points pt_{i}.
Details and Options
 BSplineCurve is also known as basis spline curve or nonuniform rational Bspline (NURBS) curve.
 BSplineCurve can be used in both Graphics and Graphics3D (two‐ and three‐dimensional graphics).
 The positions of control points can be specified either in ordinary coordinates as {x,y} or {x,y,z}, or in scaled coordinates as Scaled[{x,y}] or Scaled[{x,y,z}].
 In two dimensions, Offset and ImageScaled can be used to specify coordinates.
 The following options can be given:

SplineDegree Automatic degree of polynomial basis SplineKnots Automatic knot sequence for spline SplineWeights Automatic control point weights SplineClosed False whether to make the spline closed  By default, BSplineCurve uses cubic splines.
 The option setting SplineDegree>d specifies that the underlying polynomial basis should have maximal degree d.
 By default, knots are chosen uniformly in parameter space, with additional knots added so that the curve starts at the first control point and ends at the last one.
 With an explicit setting for SplineKnots, the degree of the polynomial basis is determined from the number of knots specified and the number of control points.
 With the default setting SplineWeights>Automatic, all control points are chosen to have equal weights, corresponding to a polynomial Bspline curve.
 Curve thickness can be specified using Thickness or AbsoluteThickness, as well as Thick and Thin.
 Curve dashing can be specified using Dashing or AbsoluteDashing, as well as Dashed, Dotted, etc.
 Curve shading or coloring can be specified using CMYKColor, GrayLevel, Hue, Opacity, or RGBColor.
 Individual coordinates and lists of coordinates in BSplineCurve can be Dynamic objects.
Examples
open allclose allBasic Examples (1)
Scope (12)
Curve Specification (5)
Curve Styling (4)
Coordinate Specification (3)
Generalizations & Extensions (4)
Knot Sequence (3)
By default, knots are generated in such a way that the curve is smooth overall:
By repeating knots, you can decrease the smoothness of the curve:
"Unclamped" generates uniform knots, and the curve will not go through the endpoints:
Unclamped knots combined with SplineClosed will make a uniform periodic Bspline curve:
Applications (5)
Interpolation (2)
Choose six points to be interpolated:
Compute distances between control points:
Compute normalized parameters wrt the distances (chord length parametrization):
A cubic Bspline curve with clamped knots will be used:
Set up the square basis matrix to solve:
Solve the linear system to get control points:
Show the interpolating curve with the original data:
Choose 3D points to be interpolated:
Compute distances between control points:
Compute normalized parameters wrt the distances (chord length parametrization):
A cubic Bspline curve with clamped knots will be used:
Set up the square basis matrix to solve:
Least Square Fitting (1)
Sample a list of points to be approximated with random noise:
Define a function to generate clamped knots for a given number of control points and degrees:
Define the basis matrix for least squares:
A cubic Bspline curve with 12 control points will be used for fitting:
Properties & Relations (6)
A Bspline curve with degree 1 is equivalent to a line:
A Bspline curve is affine invariant:
A Bspline curve lies in the union of convex hulls of subsets of control points:
In 3D, a Bspline curve with planar control points lies in the plane:
BSplineBasis can be used to build up Bspline curve objects:
The individual basis functions have bounded support:
Changing the knots affects the basis functions just as it does the BSplineCurve:
A Bspline curve generated from the average of two sets of control points:
Text
Wolfram Research (2008), BSplineCurve, Wolfram Language function, https://reference.wolfram.com/language/ref/BSplineCurve.html.
CMS
Wolfram Language. 2008. "BSplineCurve." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BSplineCurve.html.
APA
Wolfram Language. (2008). BSplineCurve. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BSplineCurve.html