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

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.

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 $(\theta, \phi) \mapsto (\cos\theta \sin \phi,\sin \theta \sin \phi, \cos \phi) : \mathbb{R}^2 \to S^2$. 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.

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.