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

• docnet
In summary, the author is trying to figure out a pattern for how many orthogonal unit vectors are needed to uniquely define a plane in four dimensions, but finds that there is no pattern for larger values of 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
$$\vec{s}(t)=(cos(t)\vec{a}+sin(t)\vec{b})$$

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

Orodruin said:
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.

Orodruin said:
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.

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.

docnet
docnet said:
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.

FactChecker and docnet
Orodruin said:
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?

FactChecker
fresh_42 said:
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?

docnet said:
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.