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Dupin's Indicatrix of a Torus
This Demonstration shows how Dupin's indicatrix changes at a variable point on a torus.
In the tangent plane at a point
of a surface
,
Dupin's indicatrix is given by the equation
,
where the
,
axes coincide with the principal directions at
and
,
are the principal curvatures of
at
.
• If
is an elliptical point, the indicatrix consists of two ellipses (one real and one imaginary).
• If
is a hyperbolic point, the indicatrix consists of two conjugate hyperbolas with asymptotes through the asymptotic directions at
.
• If
is a parabolic point, the indicatrix degenerates into a pair of parallel lines.
Contributed by:
Sonja Gorjanc
and
Desana Štambuk
THINGS TO TRY
Rotate and Zoom in 3D
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SNAPSHOTS
RELATED LINKS
Torus
(
Wolfram
MathWorld
)
Dupin's Indicatrix
(
Wolfram
MathWorld
)
Gaussian Curvature
(
Wolfram
MathWorld
)
Principal Direction
(
Wolfram
MathWorld
)
Asymptotic Direction
(
Wolfram
MathWorld
)
Elliptic Point
(
Wolfram
MathWorld
)
Hyperbolic Point
(
Wolfram
MathWorld
)
Parabolic Point
(
Wolfram
MathWorld
)
PERMANENT CITATION
"
Dupin's Indicatrix of a Torus
" from
the Wolfram Demonstrations Project
http://demonstrations.wolfram.com/DupinsIndicatrixOfATorus/
Contributed by:
Sonja Gorjanc
and
Desana Štambuk
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