Does there exist a tool to construct a perfect sine wave? For example, a perfect circle can be constructed using a compass and a perfect ellipse can be constructed using two pins and a piece of string, because a circle can be defined as the locus of points equidistant from a circle point and an ellipse can be defined as the locus of all points such that the sum of the distances from that point to the two foci is constant. 

I know that the sine function can be represented by the y-coordinate of an object in uniform circular motion, but does there exist a tool which allows you to draw a perfect sine wave (i.e. drawn by a human on paper)?
 A: Take a circular cylinder and cut it by a plane not orthogonal to the axis.
As you roll the cylinder (without slipping) along the paper, the cut edge traces out a sine wave.

A: Here's an ideal mechanical device to draw a sine curve: When a disk $D$ of radius $r$ (shaded below) rolls without slipping inside a circle $C$ of radius $2r$, each point on the perimeter of $D$ traces a diameter of $C$. Place such an apparatus over a roll of paper whose lateral speed (here, left to right) is constant (possibly geared to the angular speed with which $D$ rolls inside $C$, in order to control the wavelength). The boundary point of $D$ lying on the diameter of $C$ perpendicular to the lateral motion of the paper traces a sine curve on the paper.

A: *

*Make a small cart that rolls in a straight line.

*To one of its wheels, attach a bevel gear (like the one below) which meshes with another whose axis is parallel to the motion of the cart, i.e. perpendicular to the front of the cart.


*To any point of the second gear except its middle, hang a laser so that it points downwards, and can freely swing like a gondola of a Ferris wheel. If larger amplitude is needed, add a crank and affix the laser to freely hang from it.

*Slowly roll the cart over light-sensitive paper, ensuring that the laser doesn't pendulate.


The bevel gears convert horizontal motion into rotation perpendicular to direction of travel. The free-hanging laser extracts the second gear's horizontal component which is sinusoidal. The motion of the cart translates the paper uniformly relative to the cart to trace out the sine wave.
