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 angle describing the position of \(P'\) on the circle constantly grows and it equals the time \(t\).
The point \(P\) also moves vertically at a constant speed. If its \(z\)-coordinate starts at \(t = 0\), then at time \(t\):
\[
z = c\,t
\]
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 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 would allow a zero angle, the point would move parallel to the cylinder's axis without rotating. If we would allow a \(90^\circ\) angle, the point would only rotate around the axis wihout ascending or descending.
The tangent line to a curve at a point is a line whose direction is the direction matches the curve's direction at that point.
If we consider the curve as the trajectory of a point \(P\), the tangent vector (at the time \(t\)) is a vector whose direction aligns with the curve and whose length depends on \(P\)'s speed.
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 of the helix with the \(z\)-axis is the arctangent of the ratio:
\[
\dfrac{\text{norm of horizontal part of the tangent vector}}{\text{norm of vertical part of the tangent vector}}
\]
This ratio must remain constant.
The horizontal part of the tangent vector is \((-r\sin(t), r\cos(t))\), has a norm of \(r\), while the vertical part has a norm of \(|c|\) (namely \(c\), or \(-c\) in case c is negative). Thus, the ratio is always \(r/|c|\), ensuring it is constant.
