One-parameter parametrization of a unit circle in R^n

[docnet]

Homework Statement

What is the one-parameter parametrization of a unit circle (with the center as the origin) with its axis spanned by the vector $u$ in $\mathbb{R}^4$? What about the general one-variable parametrization of a unit circle in $\mathbb{R}^n$?

Relevant Equations

$\mathbb{SU}(2)$, $\mathbb{S}^3$

I tried to looking at lower-dimensional cases:
For $n=2$ we have
$$(x(t),y(t))=(cos(t),sin(t))$$
For $n=3$ we define two orthogonal unit vectors $\vec{a}$ and $\vec{b}$ that are orthogonal to $\vec{u}$, leading to
$$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$
It seems like there was a pattern for $n=2$ and $n=3$. But, there is no reason to think the pattern continues for larger values of $n$. This is a wild guess I think?

For $n=4$, we define two orthogonal unit vectors $\vec{a}$ and $\vec{b}$ that are orthogonal to $\vec{u}$, leading to
$$(x(t),y(t),z(t),w(t))=(cos(t)(a_1,a_2,a_3,a_4)+sin(t)(b_1,b_2,b_3,b_4))$$
For general $n$, we define two orthogonal unit vectors $\vec{a}$ and $\vec{b}$ that are orthogonal to $\vec{u}$, leading to
$$\vec{s}(t)=(cos(t)\vec{a}+sin(t)\vec{b})$$

 

[Orodruin]

In 4 dimensions you need two normal vectors to uniquely define a plane.

 

[docnet]

In 4 dimensions you need two normal vectors to uniquely define a plane.

I am confused. $\vec{a}$ and $\vec{b}$ should be the normal vectors that span the plane.

 

[docnet]

In 4 dimensions you need two normal vectors to uniquely define a plane.

I understand. I am sorry for the unusual wording in the problem statement. $\vec{u}$ is the vector perpendicular to the plane, while $\vec{a}$ and $\vec{b}$ are orthonormal vectors spanning the plane.

 

[fresh_42]

Why don't you use radial coordinates, and parameterize with $\varphi_1$? Then rotate your coordinate system with the element of $\operatorname{SO}(n)$ to match a possibly different given one.

 

[Orodruin]

I understand. I am sorry for the unusual wording in the problem statement. $\vec{u}$ is the vector perpendicular to the plane, while $\vec{a}$ and $\vec{b}$ are orthonormal vectors spanning the plane.

You misunderstand. In 4 dimensions, it is not sufficient to have a single vector normal to define a two-dimensional plane. You need two. In N dimensions you need N-2.

 

[docnet]

You misunderstand. In 4 dimensions, it is not sufficient to have a single vector normal to define a two-dimensional plane. You need two. In N dimensions you need N-2.

Okay, I understand. Then, we could just define the unit circle with two orthonormal vectors $\vec{a}$ and $\vec{b}$ that span the plane the circle is in, without specifying $n-2$ normal vectors right?

 

[docnet]

Why don't you use radial coordinates, and parameterize with $\varphi_1$? Then rotate your coordinate system with the element of $\operatorname{SO}(n)$ to match a possibly different given one.

hmm.. I will try. This seems like it will lead to an elegant solution.



Is it normal to want to cry because you want to visualize 4-dimensional objects and you just cannot?

 

[Orodruin]

Okay, I understand. Then, we could just define the unit circle with two orthonormal vectors $\vec{a}$ and $\vec{b}$ that span the plane the circle is in, without specifying $n-2$ normal vectors right?

Sure. It is also effectively equivalent to what @fresh_42 suggested as your vectors would be the images of the basis vectors in the original plane.