How to draw a sine wave by slicing a cylinder?

Question

Found the following answer in quora

https://www.quora.com/At-what-specific-angle-do-you-need-to-slice-a-hollow-cylinder-in-order-to-produce-a-perfect-sine-wave/answer/David-Joyce-11?ch=10&oid=353388410&share=ef8411da&srid=oHoH&target_type=answer

Can anyone explain why it works?

Answer 1

Define a coordinate system in the plane of the cut such that the $x$-axis is parallel to the major axis of the ellipse and the $y$-axis is parallel to the minor axis of the ellipse. Some fairly simple geometry shows that we can parameterize the ellipse as $$x(t)=\sqrt 2 \sin(\theta) \qquad y(t)=\cos(\theta),$$ where $\theta$ is considered as an angle in one of the bases of the cylinder (the general idea is that slicing at an angle essentially "stretches out" the circular base of the cylinder, and spinning around at a constant angle still interacts nicely with this stretching out).

Once we have this, we can simply unwrap the cylinder to get a new plane with a horizontal axis acting like $\theta$ and a vertical axis acting like $\frac{1}{\sqrt 2}x$, so the cylinder becomes the graph of $y=\sin(x)$.

Answer 2

To visualize the 3D formation of the sine curve by planar section experimentally take a thick paper width c and height $2A$.

Roll the paper so that generators are parallel to height $2A$ into a cylinder form so that perimeter $c=2 \pi a$.

Imagine a plane cutting/slicing the cylinder. It passes through points $(x,y,z)=(a,0,A),(a,0,-A)$ so that its normal is co-planar to the cylinder axis. It makes an ellipse intersection whose whose base projection is a circle of radius $a$, so that the base is a circle

$$ x= a \cos \theta, y= a \sin \theta,\quad x^2+y^2=a^2 $$

The slope of the cutting plane to base is given by

$$ \tan \alpha=\frac{A}{a}$$

In the spread out development we have a sine wave for a distance $u$ along arc of circumference $ c= 2 \pi a$ and $ u=a \theta. $

$$ z= A \cos \frac{2 \pi u }{c}=A \cos \frac{2 \pi u }{2 \pi a}= A \cos \theta. $$

Answer 3

Imagine a cylinder whose base is centered at the origin, and extending upward (in the positive $z$ axis direction).

It Cartesian equation is

$ x^2 + y^2 = a^2 $

where $a$ is the radius of the circular base.

Now pass a plane with equation

$ z = z_0 + \tan(\phi) x $

Parameterization:

The parameterization of $x$ and $y$ is as follows

$ x = a \cos(\theta) , y = a \sin(\theta) $

It follows that

$ z = z_0 + \tan(\phi) \cos(\theta) $

Thus $z$ has a sinusoidal variation with the azimuthal angle $\theta$.