As this illustration of a sliced poultry sausage shows — and as we see it every day “at the butcher’s” — sausages are very often cut open at an angle. Geometry teaches us that then the cut surface is not bounded by a circle, but by an ellipse. If you cut the sausage shell parallel to the “main axis” of the sausage and place it on a flat surface, the initially “spatial cut” becomes a flat curve, which represents a part of a so-called sine curve, i.e. a harmonic oscillation.
The associated exhibit in Mathematics Adventure Land (see the following Figure 2) shows how the plane, oblique section of a straight (circular) cylinder is mapped onto an endless foil by means of a hand crank. The image on the foil turns out to be a harmonic oscillation. It is mathematically described by an angular function (on the right-angled) triangle, the so-called sine.
And now … the mathematics of it:
1. definition of the sine function
According to the following figure 3 the sine (also: sine function) of an angle is the length of the so-called opposite cathetus in a right triangle with a hypotenuse of length one.
By means of a unit circle (radius ) one assigns to each angle the length of the arc over this angle . Considering that the circumference of the unit circle is equal to , the corresponding radian is :
The sine function is defined by . The length of the so-called adjacency in a right triangle with a hypotenuse of length 1 over the angle with arc length is called cosine (also: cosine function) with .
2. unwinding of the plane section
A (straight) circular cylinder with radius for the base circle (in the adventure land of mathematics ) is described by the following equations because of the equation (Pythagorean theorem at the unit circle):
Here and are the so-called polar coordinates as shown in the following figure 4:
A plane section of the circular cylinder (at the angle of ) is given by the bisector in the -plane
given. Thus, from equations (1) and (2) it follows
for . This sine function is visible with on the foil to be unrolled at the corresponding exhibit in Mathematics Adventure land.