Math Help Forum - View Single Post

aidan · #2 July 3rd, 2009, 03:41 PM

Quote:

Originally Posted by iluci

Hello!

I'm trying to solve this problem:

Having k points in an n-dimentional space, I want to find a new point, any point P that is at distance 1 from each of them, or find out if it is impossible.
So, the system of equations would be

for each i = 1..k
Sum, for j = 1..n of (Xij - Pj) ^2 = 1

but Xij are constants, so

for each i = 1..k
Ci + (Sum, for j = 1..n of Pj ^2 - 2XijPj) = 1

With Ci = Sum, for j=1..n of Xij^2

Is there an easy, computational way of calculating this?

Sorry for my not-standard notation, I'm new to the forum.

Typically for a n-dimensional space, you will have a matrix with n dimensions.

suppose you have four points in a 5-dimensional space:

Point(x1,y1,z1,w1,v1)
Point(x2,y2,z2,w2,v2)
Point(x3,y3,z3,w3,v3)
Point(x4,y4,z4,w4,v4)

Then for your AnyPoint(x0,y0,z0,w0,v0)

for i= 1 to 5

Distance = $\sqrt{(x_i-x_0)^2 + (y_i-y_0)^2 + (z_i-z_0)^2 + (w_i-w_0)^2 + (v_i-v_0)^2}$

If Distance <> 1 then EXIT: NO SUCH NUMBER POSSIBLE.

next i

''AnyPoint() is at distance 1 from all other points.

If your number of points, exceeds the number of dimensions in your n-dimensional space, and NONE of the points are coincident with any other point then you will have an impossible condition to meet.

Just wondering, but how are you using more than three dimensions to define your points.