Is there a generalization of the helix from $mathbb{R}^3$ to $mathbb{R}^4$?

Question

The helix is a curve $x(t) in mathbb{R}^3$ defined by:
$$
x(t) = begin{bmatrix}
sin(t) \
cos(t) \
t
end{bmatrix}
$$
and it takes the classic shape:

Does this have a natural extension from $mathbb{R}^3$ to $mathbb{R}^4$?  (Or even $mathbb{R}^n$?)

What I've tried so far:
The classic $mathbb{R}^3$ helix curve above has two nice properties:

$x(t)$ has constant distance from the axis of propagation $hat{e}_3$, where $hat{e}_3 = begin{bmatrix} 0 \ 0 \ 1 end{bmatrix}$
$x(t)$ has constant angular velocity when projected onto the plane normal to $hat{e}_3$.  i.e. the vector $(x_1(t), x_2(t))$ has polar coordinates $(r, theta) = (1, t)$, so $dot{theta} equiv 1$.

The classic helix can be viewed as a parametric walk of a circle in $mathbb{R}^2$, with the parameter $t$ added as the third dimension.  A natural extension to a helix in $mathbb{R}^n$ would be a parametric walk of a curve on a hypersphere in $mathbb{R}^{n-1}$, with parameter $t$ added as the nth dimension.  So for $mathbb{R}^4$, one could choose a spherical spiral to walk the sphere in $mathbb{R}^3$, and use parameter t as the 4th dimension:
$$
x(t) = begin{bmatrix}
sin(t) cos(ct) \
sin(t) sin(ct) \
cos(t) \
t
end{bmatrix}
$$
The first three components are rendered on wikipedia as:

This construction matches the two properties I listed:

$x(t)$ has constant distance from the axis of propagation $hat{e}_4$, where $hat{e}_4 = begin{bmatrix} 0 \ 0 \ 0 \ 1 end{bmatrix}$
When $c=1$, $x(t)$ has constant angular velocity when projected onto the 3-plane normal to $hat{e}_4$.  i.e. the vector $(x_1(t), x_2(t), x_3(t))$ has spherical coordinates $(r, theta, phi) = (1, t, t)$, so $dot{theta} = dot{phi} equiv 1$.

It's technically a direct extension of the $mathbb{R}^3$ helix, since $c=0$ induces an identical curve (up to a projection.)  But it still feels a little arbitrary, and the closed form will be quite ugly in higher dimensions.
Is there a generally accepted extension of the classical circular helix in $mathbb{R}^3$ to $mathbb{R}^4$?  (Or even $mathbb{R}^n$?)  And do its properties or construction at all resemble the above?

After some research, I've learned that there are interesting generalizations of helices in $mathbb{R}^n$, defined in terms of derivative constraints, Frenet frames, etc. such that even polynomial curves can behave as helices. [Altunkaya and Kula 2018].  However, that's much more general than I'm seeking, since those are aperiodic, and may have unbounded distance from the axis of propagation.  But the existence of such work is promising - I just don't know how to search this space well.

Mike F · Answer

Any answer to this question is necessarily going to be a bit arbitrary, but here are a few thoughts:

We have an interesting map $theta mapsto (cos theta, sin theta) : mathbb{R} to S^1$. The helix is the graph of this map.
In this spirit, we might consider that the graph of a parametrization of a manifold is a generalized helix. For example, we have the spherical coordinates parametrization of the 2-sphere $(theta, phi) mapsto (costheta sin phi,sin theta sin phi, cos phi) : mathbb{R}^2 to S^2$. We could consider its graph, a subset of $mathbb{R}^2 times mathbb{R}^3 = mathbb{R}^5$ to be a generalized helix.
We could also concentrate on the fact that $mathbb{R}$ is the universal cover of $S^1$. So maybe, given a submanifold $M subset mathbb{R}^n$, we should consider the graph of the projection $widetilde M to M$ to be a generalized helix. Since $S^2$ is its own universal cover, we just get another copy of $S^2$ back in this case.

kdbanman · Answer

After a few hours of digging around and thinking, I've found a way to more naturally express the spherical spiral idea in my question.
I'm still not sure if my construction or properties make sense though, so I won't mark my own answer as correct here.  Someone else with broader geometry knowledge should weigh in instead of me.

One can write the classic $mathbb{R}^3$ helix in cylindrical coordinates $(rho, phi, z)$:
$$
begin{bmatrix}
x_1(t) \
x_2(t) \
z(t)
end{bmatrix}
= 
begin{bmatrix}
sin t \
cos t \
t
end{bmatrix}
implies 
begin{bmatrix}
rho(t) \
phi(t) \
z(t)
end{bmatrix}
= 
begin{bmatrix}
1 \
t \
t
end{bmatrix}
$$
Cylindrical coordinates are a hybrid of $mathbb{R}^2$ polar coordinates $(r, theta)$, plus an additional cartesian coordinate $(z)$.  In the diagram below, the helix would propagate vertically, winding around the $L$ axis.

So we can apply the same kind of hybrid using $mathbb{R}^3$ spherical coordinates $(r, theta, phi)$ with $(z)$ to get the "hypercylindrical" coordinates $(rho, phi_1, phi_2, z)$ and write the $mathbb{R}^4$ helix from the question just as easily.
$$
begin{bmatrix}
x_1(t) \
x_2(t) \
x_3(t) \
z(t)
end{bmatrix}
= 
begin{bmatrix}
sin t cos t \
sin t sin t \
cos t \
t
end{bmatrix}
implies
begin{bmatrix}
rho(t) \
phi_1(t) \
phi_2(t) \
z(t)
end{bmatrix}
= 
begin{bmatrix}
1 \
t \
t \
t
end{bmatrix}
$$
and the pattern naturally extends for the general $mathbb{R}^n$ helix.  We use $mathbb{R}^{n-1}$ hyperspherical coordinates to write the helix in $mathbb{R}^n$ hypercylindrical coordinates
$$
begin{bmatrix}
rho \
phi_1 \
phi_2 \
... \
phi_{n-3} \
phi_{n-2} \
z
end{bmatrix}
= 
begin{bmatrix}
1 \
t \
t \
... \
t \
t \
t
end{bmatrix}
$$
This trivially meets my listed properties, because

$rho=1$ means constant (unit) distance from the axis of propagation $hat{e}_n$.
$phi_k = t implies dot{phi_k} = 1$, so angular velocity is also constant in all angular coordinate dimensions.

Like I've said, though, I'm not sure those properties actually make sense for $mathbb{R}^n$ helices.

Is there a generalization of the helix from $mathbb{R}^3$ to $mathbb{R}^4$?

What I’ve tried so far:

2 Answers

Add your own answers!

Ask a Question