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

by 7toni7
Tags: arclength, distance, geodesic, hypersphere
 P: 7 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,
 PF Patron HW Helper Sci Advisor Thanks P: 25,585 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)
 P: 7 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.
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## Calculate the arc length between two points over a hyper-sphere

hello 7toni7!
 Quote by 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,…an).(b1,b2,…bn) = a1b1 + a2b2 + …anbn

(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
 P: 7 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
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 Quote by 7toni7 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?
 P: 7 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.
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 Quote by 7toni7 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
 P: 7 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
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 PF Patron Sci Advisor Thanks Emeritus P: 38,450 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.