Does there exist a tool to construct a perfect sine wave?

Question

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)?

Answer 1

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.

Answer 2

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.

Answer 3

1. Make a small cart that rolls in a straight line.
2. 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.
3. 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.
4. 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.