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

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7toni7
#1
Dec13-12, 04:22 AM
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,
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tiny-tim
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Dec13-12, 05:21 PM
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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
#3
Dec14-12, 04:48 AM
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.

tiny-tim
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Dec14-12, 05:23 AM
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Calculate the arc length between two points over a hyper-sphere

hello 7toni7!
Quote Quote by 7toni7 View Post
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
7toni7
#5
Dec14-12, 07:03 AM
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
tiny-tim
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Dec14-12, 07:58 AM
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Quote Quote by 7toni7 View Post
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
#7
Dec14-12, 08:44 AM
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.
tiny-tim
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Dec14-12, 08:52 AM
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Quote Quote by 7toni7 View Post
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
#9
Dec14-12, 09:39 AM
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
tiny-tim
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Dec14-12, 11:02 AM
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Quote Quote by 7toni7 View Post
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
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Dec14-12, 11:11 AM
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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.


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