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# Serie VIII, Nr 3

## Surface of order 12; the enveloping surface of a system of rays.

**Mathematical description**

Surface of order 12.
This model is from geometrical optics.
Rays of light from a source perpendicular to a cylinder-axis are reflected in the cylinder:

x = r cos(φ)
y = r sin(φ)
z = z,

The model shows the surface, which is enveloped by all these reflected rays.
This surface is given by the following equation:
[(4z^{2}-r^{2})(x^{2}+y^{2})-r^{2}z(2x+z)]^{3}
-27r^{4}4z^{2}y^{2}(x^{2}+y^{2}-z^{2})^{2}=0,
when *z* is the distance between the cylinder and the lightsource.
**Title model and translation**

Fläche zwölfter Ordnung, Brennfläche.

Surface of order 12; the enveloping surface of a system of rays.

**Text on Sticker and translation**

Fläche 12. Ordnung.

Surface of order twelve.

**Designer**

S. Finsterwalder under the supervision of Dr. Brill

München in 1882

**Company**

L. Brill in Darmstadt

**Original price**

Mark 17

**Dimension**

13-20-12 cm

**Material**

Plaster

**Literature**

Schilling, M., Catalog mathematischer Modelle für den höheren
mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911