As said, i want to build a program to generate n equidistant points in an euclidian space. From what i know
- 1d : all couple of points
- 2d : all equilateral triangles
- 3d : all equilateral tetrahedra
- up to 3d : i suppose it's called an equilateral hypertriangle
So my problem is as follow, in a n-1 euclidian space, giving a defined point build the n-1 other in order to have an equilateral hypertriangle with a distant d between each points.
I suppose that we can start as follow with for example a 3d space.
- p1 = (x1,y1,z1) fixed
- p2 = (x2,y2,z2)
- p3 = (x3,y3,z3)
- p4 = (x4,y4,z4)
We start to fix p2 knowing d and p1
- $d²=(x_1-x_2)^2 + (y_1-y_2)^2 + (z_1-z_2)^2$
We have 3 variables x2,y2,z2. We can fix randomly two of them and determine the third without problem.
Then for the second point we have now 2 equations to defined it :
- $d²=(x_1-x_3)^2 + (y_1-y_3)^2 + (z_1-z_3)^2$
- $d²=(x_2-x_3)^2 + (y_2-y_3)^2 + (z_2-z_3)^2$
As previous, i presume that we can fix 2 variables to determine the third.
For the last point we now have 3 equations which defined it.
So for a n-1 dimensional space we have n-1 equation to defined the last point.
I don't know how to solve this kind of system composed of quadratic equation with one variable and if the process which consist to fix n-1 dimension to determine the last one lead to an equidistant hypertriangle. Moreover it exist maybe others methods with an smaller complexity and easier to implement.
I hope i was clear enough and i thank you for your help.