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
Best,

tiny-tim

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. 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

yes
(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

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

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 [itex](a_1, a_2, ..., a_n)[/itex] has equation [itex](x_1- a_1)^2+ (x_2- a_2)^2+\cdot\cdot\cdot+ (x_n- a_n)^2= R^2[/itex]. The line through the origin and point [itex](b_1, b_2, ..., b_n)[/itex] is given by the parametric equations [itex]x_1= b_1t[/itex], [itex]x_2= b_2t[/itex], ..., [itex]x_n= b_nt[/itex]. Replacing [itex]x_1[/itex], 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.