Calculate the arc length between two points over a hyper-sphere

[7toni7]

Good morning,

I'm trying to compute the arclength (geodesic distance) between two n-dimensional points over a n-dimensional sphere (hypersphere). Do you know if it is possible? If yes, please, I'd be very pleased if you, as experts, provide me this knowledge.

Thank you very much

 

[tiny-tim]

welcome to pf!

hi 7toni7! welcome to pf!

won't it just be the radius times the angle between them?

(which you can get from the dot-product)

 

[7toni7]

Hello tiny-tim,

Thank you very much for your answer, and I'm pleased to be in this forum.
Yes, I think the same.
In 2D and 3D is just: (arclength = S, radius = R (in radians), angle between points= ω)

S = R*ω.

Then, I have 3 doubts:
1 - when dealing with dimensions greater than 3...the order of hundreds, could we do the same computation than 2 and 3 Dimensions? Could we extend this equation to higher dimensions?

2 - The dot product in N dimensions...is just the same than 2 and 3 dimensions?

3 - This formula is in an euclidean space, isn't it?

Thank you very much,
Best regards.

 

[tiny-tim]

hello 7toni7!

1 - when dealing with dimensions greater than 3...the order of hundreds, could we do the same computation than 2 and 3 Dimensions? Could we extend this equation to higher dimensions?

2 - The dot product in N dimensions...is just the same than 2 and 3 dimensions?

3 - This formula is in an euclidean space, isn't it?

1. yes

2. yes: (a₁,a₂,…a_n).(b₁,b₂,…b_n) = a₁b₁ + a₂b₂ + …a_nb_n

(don't forget that the dot product gives you R²cosω, so you'll have to divide by R², and then use the cos^-1 button ! )

3. yes

 

[7toni7]

Thank you.
Then, the arclength on a n-sphere can be computed as follows:

S = R*acos(a.b/R²).

I think it is correct. Isn't it?


A last question, do you know how to compute the intersection point between a n-vector and a n-sphere?

Thank you so much again.
Best

 

[tiny-tim]

Then, the arclength on a n-sphere can be computed as follows:

S = R*acos(a.b/R²).

yes

A last question, do you know how to compute the intersection point between a n-vector and a n-sphere?

(this is from your other thread, isn't it?)

do you mean an n-vector starting from the origin (the centre of the n-sphere)?

if not, how are you defining the n-vector and the n-sphere?

 

[7toni7]

Hello,

Yes, suppose that we have one n-sphere. Inside it, we have a n-point (this point different of the origin, it is another point named H).

So, I have to compute the intersection of the line (that goes from the origin of the n-sphere passing from H) with the n-sphere. do you understand? is it possible?

Thank you in advance again,
Best.

 

[tiny-tim]

So, I have to compute the intersection of the line (that goes from the origin of the n-sphere passing from H) with the n-sphere. do you understand? is it possible?

ah, so the line is a diameter of the n-sphere?

then yes, it's easy …

the n-vector to the intersection will be a scalar multiple of the n-vector to H, such that the magnitude of the n-vector (ie, the square-root of the dot-product with itself) equals the radius

 

[7toni7]

Well,
This is how I do it in 2 dimensions. See image.

Now, my question is: could this development be extended to N dimensions?

Thank you

 

[tiny-tim]

Now, my question is: could this development be extended to N dimensions?

yes, the same formula (radius times the unit vector in the P direction) works in n dimensions …

Q = R*(P/|P|) ​

 

[HallsofIvy]

In n-dimensional Euclidean space, the (hyper)sphere with radius R and center at $(a_1, a_2, ..., a_n)$ has equation $(x_1- a_1)^2+ (x_2- a_2)^2+\cdot\cdot\cdot+ (x_n- a_n)^2= R^2$. The line through the origin and point $(b_1, b_2, ..., b_n)$ is given by the parametric equations $x_1= b_1t$, $x_2= b_2t$, ..., $x_n= b_nt$. Replacing $x_1$, etc. in the equation of the sphere with those gives a single quadratic equation for t. Finding the two solutions to that equation gives the two points at which the line crosses the sphere.