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

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,
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 Blog Entries: 27 Recognitions: Gold Member Homework Help Science Advisor 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)
 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
 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?
 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
 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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 Quote by 7toni7 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|)
 Recognitions: Gold Member Science Advisor Staff Emeritus 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.

 Tags arclength, distance, geodesic, hypersphere

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