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Polygonal Tori
This Demonstration shows a torus with a polygonal cross section.
To get a better view of its inside, open up the meridional and longitudinal gaps.
The Serret–Frenet formulas (using
Mathematica
's built-in
FrenetSerretSystem
) are used to generate the parametric equation of the torus surface.
Contributed by:
Erik Mahieu
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The parametric equation of a circular torus with a polygonal cross section is obtained as follows.
The parametric of the ring circle with radius
is:
.
The polar equation of a regular
-gon with circumradius
is:
.
Using the
Mathematica
function
FrenetSerretSystem
, we define the normal and binormal vectors
and
:
,
.
This gives the parametric equation of the elliptical torus:
,
with
vertices and
the circumradius of the cross-sectional polygon.
Expanded, with
, this becomes:
,
,
.
RELATED LINKS
Torus
(
Wolfram
MathWorld
)
Frenet Formulas
(
Wolfram
MathWorld
)
PERMANENT CITATION
Erik Mahieu
"
Polygonal Tori
"
http://demonstrations.wolfram.com/PolygonalTori/
Wolfram Demonstrations Project
Published: July 30, 2015
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