In a roundabout, you can drive around and around in a circle:
If you drive up or down in a cylindrical parking-garage ramp, you drive along a helix:
In a nutshell: the helix is a curve that combines a circular motion with a simultaneous upward (or downward) movement.
[About the pictures: roundabout
(CC 3.0; added lines); garage (CC SA 2.0; added lines).]
Definition of Cylindrical Helices
A cylindrical helix is a curve on a cylinder that is the trajectory of a point moving around the cylinder at a constant speed while simultaneously ascending (or descending) at a constant speed.
On an infinite cylinder, a helix can extend indefinitely in one or both directions, making it a curve of infinite length.
Choose a $xyz$-coordinate system such that the cylinder’s axis aligns with the $z$-axis, and let $r$ be the cylinder’s radius. Consider a point $P$ moving along the helix at a constant speed.
Consider the projection onto the $xy$-plane. Suppose that the projection $P'$ of $P$ rotates counterclockwise around the origin (at a constant speed). If $P'$ starts at the coordinates $(r,0)$ at time $t=0$ and completes one full rotation at $t = 2\pi$, then its coordinates at time $t$ are:
$$
(x,y) = (r\cos(t),r\sin(t))
$$
where we see the trigonometric functions sine and cosine. The polar angle describing the position of $P'$ on the circle constantly grows and equals the time $t$.
The point $P$ also moves vertically at a constant speed. If $P$ starts at height $z = 0$ when $t = 0$, then
$$
z = c\,t
$$
where $c$ is a constant: positive if the point ascends, negative if it descends.
Thus the point $P$ at time $t$ has coordinates:
$$
(x,y, z) = (r\cos(t),r\sin(t),c\,t)
$$
A geometric description of helices
A cylindrical helix can also be described geometrically:
A cylindrical helix is a curve that lies on a cylinder that forms a constant acute angle with the cylinder’s axis at every point. This property concerns the curve's current direction at a given point.
If we allowed a zero angle, the point would move parallel to the cylinder's axis without rotating. If we allowed an angle of $90^\circ$, the point would only rotate around the axis without ascending or descending.
The direction of a curve at a point is given by the tangent vector at that point.
If we consider a curve as the trajectory of a point $P$, the tangent vector at time $t$ points in the direction in which $ P $ is moving. Its length depends on the speed of $ P $.
We compute the tangent vector by taking the derivative of the functions expressing the coordinates at the time $t$. For the helix
$$
(x,y, z) = (r\cos(t),r\sin(t),c\,t)
$$
this is
$$
\begin{pmatrix}
x'\\
y'\\
z'\\
\end{pmatrix}=\begin{pmatrix}
-r\sin(t) \\
r\cos(t) \\
c
\end{pmatrix}
$$
The angle $ \phi $ between the helix and the $z$-axis satisfies
$$
\tan( \phi) = \dfrac{\text{norm of the horizontal part of the tangent vector}}{\text{norm of the vertical part of the tangent vector}}
$$
This ratio must remain constant.
The horizontal part of the tangent vector is $(-r\sin(t), r\cos(t))$, and has a norm of $r$, while the vertical part has a norm of $|c|$. Hence
$$
\tan( \phi) = \dfrac{r}{|c|},
$$
which is constant. Therefore, the helix makes a constant angle with the cylinder's axis.
