# Cylindrical Surfaces from NURBS Curves

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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

## Snapshots

## Details

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:

and

.

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 .

Reference

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

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