WOLFRAM|DEMONSTRATIONS PROJECT

3D Extrusion Using the Frenet-Serret System

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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.