B-Spline Curve with Knots

This Demonstration illustrates the relation between B-spline curves and their knot vectors. Start with the control points and a knot vector , where the degree of the B-spline is . The knot vector satisfies and . The B-spline basis functions are defined as:
and a B-spline curve is defined as:
For nonperiodic B-splines, the first knots are equal to 0 and the last knots are equal to 1. If duplication happens at the other knots, the curve becomes times differentiable. So, by overlapping the knots, you can generate a curve with sharp turns or even discontinuities.
When the number of control points is , the basis functions are reduced to Bernstein polynomial, thus the curve becomes a Bézier curve.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


Red points indicate the knot points on the curve. Hold down the Alt key and click to add new control points (up to 12). Changes in degree and number of control points will cause the knot vector to be recomputed.
Choose "view basis functions" to show the B-spline basis functions of a given knot vector instead of the B-spline curve.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+