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

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.
  • #1
docnet
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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
$$\vec{s}(t)=(cos(t)\vec{a}+sin(t)\vec{b})$$
 
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  • #2
In 4 dimensions you need two normal vectors to uniquely define a plane.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
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  • #6
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.
 
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  • #7
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?
 
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  • #8
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?
 
  • #9
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.
 

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

What is a one-parameter parametrization?

A one-parameter parametrization is a way of representing a mathematical object, such as a unit circle in n-dimensional space, using a single parameter. This parameter can be thought of as a variable that changes as we move along the object, allowing us to describe the object's points in a systematic way.

How is a unit circle parametrized in n-dimensional space?

A unit circle in n-dimensional space can be parametrized using a single parameter t, where t ranges from 0 to 2π. The parametrization can be written as (cos(t), sin(t), 0, ..., 0), where the number of zeros corresponds to the dimension of the space.

Why is a one-parameter parametrization useful?

A one-parameter parametrization allows us to describe a mathematical object in a simpler and more organized way. It also allows us to easily manipulate and analyze the object using techniques from calculus and differential geometry.

Can a unit circle be parametrized in multiple ways?

Yes, a unit circle can be parametrized in multiple ways using different parameters. For example, in 2-dimensional space, a unit circle can also be parametrized using the parameter t as (sin(t), cos(t)). However, both parametrizations describe the same object.

How is a one-parameter parametrization related to a unit circle's arc length?

A one-parameter parametrization can be used to calculate the arc length of a unit circle in n-dimensional space. By integrating the magnitude of the derivative of the parametrization with respect to the parameter t, we can find the length of the curve traced out by the parametrization. In the case of a unit circle, this length is equal to 2π.

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