Iso-Optic Plane of the Regular Tetrahedron

α

This Demonstration shows the points in space that subtend an angle

α

from a regular tetrahedron with vertices

0,0,

2

3

-

1

2

6

,-

1

2

3

,-

1

2

,-

1

2

6

,-

1

2

3

,

1

2

,-

1

2

6

,

1

3

,0,-

1

2

6



.

Details

In the plane, it is easy to show those points from which a segment subtends an angle

α

because they form a circle. However, for a segment in space, the points subtended form a torus, where

R<r

(i.e., the torus intersects itself). The long derivation for a tetrahedron is not shown; only the result is used.

You can select

α

between

5°

and

90°

.

In the first snapshot you can see that if

α

is small, the figure looks like a sphere; the second snapshot shows the general case; and in the third snapshot spheres appear because

α

is

90°

.

External Links

Iso-Optic Curve of the Ellipse

Iso-Optic Curve of a Regular Polygon

Permanent Citation

Géza Csima, Jenő Szirmai, János Tóth

"Iso-Optic Plane of the Regular Tetrahedron"
http://demonstrations.wolfram.com/IsoOpticPlaneOfTheRegularTetrahedron/
Wolfram Demonstrations Project
Published: June 3, 2009