3D Extrusion Using the Frenet-Serret System

3D space curve parameters

{cos(n t), r sin(m t), z cos(t)}

n

1

2

3

4

5

m

1

2

3

4

5

z

1.75

cross section parameters

type

circle

polygon

rose

circumradius

0.2

number of vertices

3

4

5

6

axial rotation

0.25

cutaway section size

0.1

viewpoint

default

above

side

This Demonstration illustrates the conversion of a parametrized three-dimensional space curve into a three-dimensional surface.

This can be done by using the built-in Mathematica command FrenetSerretSystem to extract the normal and binormal unit vectors



and

ℬ

of the space curve.

The one-variable parametric equation of the space curve

c(u)

can be combined with the polar equation of the cross section

r(v)

.

The result is the two-variable parametric equation of a surface:

c(u)+r(v)((u)cosv+ℬ(u)sinv)

.

In this Demonstration, a parametric space curve

{cos(nt),rsin(mt),zcos(t)}

is defined, with the parameters

{n,m,z}

determining the shape of the curve.