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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.

enter image description here 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)?

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    $\begingroup$ Humans can't draw anything perfectly. $\endgroup$ – Robert Israel Oct 15 '15 at 21:50
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    $\begingroup$ Not quite what you want, but ... Take a transparent $2\pi \times 2\pi$ sheet, and draw one of its diagonals. Wrap the sheet around a clear cylinder of radius $1$. With an appropriately-placed light, the (now-helical) diagonal casts a shadow in the shape of a perfect sinusoid. (See this answer for a picture.) $\endgroup$ – Blue Oct 15 '15 at 21:55
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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.

enter image description here

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    $\begingroup$ +1. Relatedly, you can wrap the cylinder in paper, make the cut as described above, and then unwrap: the paper will have a sinusoidal edge. (If you wrap the paper around more than once, the thickness of the paper will introduce imperfections that get worse and worse with each layer. A roll of paper towels, for instance, would be very bad for this kind of thing.) $\endgroup$ – Blue Oct 15 '15 at 22:36
  • $\begingroup$ A further question has been asked in math.stackexchange.com/questions/1482727/… . It is not asked in here because I cannot add a picture in the comment. $\endgroup$ – Mick Oct 16 '15 at 6:08
  • $\begingroup$ Sneaky ! ;-$)$ $\endgroup$ – Lucian Oct 16 '15 at 20:09
  • $\begingroup$ I just stumbled upon this interesting constriction and have been trying to work on a proof; not sure if I should post this as a separate question but how can it be analytically proved that the graph is indeed sinusoidal? $\endgroup$ – E.Nole Mar 29 '19 at 12:06
  • $\begingroup$ To transfer the cut to paper, coat the cylinder near the cut with ink and roll it on the paper. $\endgroup$ – Gnubie Jul 8 '19 at 4:08
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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 rolling wheel traces a sine curve on a roll of paper

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  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. enter image description here
  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.

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