Cylindrical Surfaces from NURBS Curves

Initializing live version
Download to Desktop

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

This Demonstration shows how a cylinder-like surface over a curve can be generated using a nonuniform rational basis spline (NURBS).

Contributed by: Shutao Tang (November 2015)
(Northwestern Polytechnical University, Xi'an City, China)
Open content licensed under CC BY-NC-SA



Let be a vector of unit length and be a NURBS curve of degree on the knots vector with weights . We want an equation for the general cylinder obtained by sweeping a distance along . Denoting the parameter for the sweep direction by , , clearly must satisfy two conditions:

1. For fixed , is a straight line segment from to , where is some real number.

2. For fixed , .

From the translational invariance property, the desired representation is


using knot vectors and , where and is the knot vector of The control points are given by and , and the weights are . In matrix form, these are:



In addition, denotes the rational basis function of degree , that is,


This Demonstration takes the initial control points to be and the spline weights to be .


[1] L. Piegl and W. Tiller, The NURBS Book, 2nd ed., Berlin: Springer-Verlag, 1997 pp. 334–336.

Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.