In article <13ksfe1hpg78j6b@corp.supernews.com>,
 "Jon G." <jon8338@peoplepc.com> wrote:

> I can't help it if the news reader doesn't accept Rich Text format.  This was 
> done in Courier.
> 
> Equation of plane of n dimensions
> 
>  n
> SUM a[p]x[p]=0
> p=0

Only for planes though origin.
> 
> Normal to plane
> 
> N=(a[0],a[1],a[2],...,a[n])
> 
> Equation of space curve of n dimensions
> 
> T=(t,t^2,t^3,...,t^n)
> 
> Point on plane
> 
> P=(1,1,1,...,(-a[0]-a[1]-a[2]-...-a[n-1])/a[n] )
> 
> Shortest vector from origin to plane
> 
>     P*N
> Q= ------N = (q[1],q[2],q[3],...,q[n])
>    |N|^2
> 
Which  in this case will have length 0.

> Vector in plane
> 
> P[1]=P-Q
> 
> Projection
> 
>        (T-Q)*P[1]
> T[1] = ----------P[1] 
>          |P[1]|
> 
> Equation of circle on plane
> 
> |P-Q|^2 = (x[1]-q[1])^2 + (x[2]-q[2])^2 + (x[3]-q[3]^2 + .... 
> ... + (-a[1]x[1]-a[2]x[2]-a[3]x[3]- ... -a[n-1]x[n-1] )/a[n] - q[n] )^2
> 
> Other vectors in plane
> 
> n=odd
> 
>   P[1]*P[b]
> ------------ = cos(2pib/n) && P[b]*N=0  b=2,3,4,...,n-1
> |P[1]||P[b]|
> 
> n=even
> 
>   P[1]*P[b]
> ------------ = cos(2pib/(n-1) ) && P[b]*N=0  b=2,3,4,...,n
> |P[1]||P[b]|
> 
> Projections
> 
>        (T-Q)*P[r]
> T[r] = ----------P[r]   r=1,2,3,...,n
>         |P[r]|^2
> 
> Solution
> 
> T = Q + T[1] + T[2] + T[3] + ... + T[n]