Polar2(x,y) = ( tan^-1(y/x), (x² + y²)^½) º  (f,r)

    Polar3(x,y,z) = ( cos^-1(z(x² + y² + z²)^-½,  tan^-1(y/x), (x² + y² + z²)^½ )   º  (q,f,r)

    Since Polar3(x,y,z) may be constructed as ( Polar2( Polar2(x,y)₂ , z)₁, Polar2(x,y)₁, Polar2( Polar2(x,y)₂ , z)₂ ) it is Polar2 that is fundamental.

    (d/dx)Polar2(x,y) = (-y(x²+y²)^-1 , x(x²+y²)^-½) = (-r^-1 sinf,  cosf)
    (d/dy)Polar2(x,y) = ( x(x²+y²)^-1 , y(x²+y²)^-½) = ( r^-1 cosf,  sinf)
    (d/dx)²Polar2(x,y) = ( 2xy(x²+y²)^-2 , y²(x²+y²)^-½) = ( r^-2 sin(2f) , r^-1( sinf)² )
    (d/dy)²Polar2(x,y) = ( -2xy(x²+y²)^-2 , x²(x²+y²)^-½) = ( -r^-2 sin(2f) , r^-1( cosf)² )
    (d/dx)(d/dy)Polar2(x,y) = ( (y²-x²)(x²+y²)^-2 , - xy(x²+y²)^-3/2) = -( r^-2 cos(2f) , ½r^-1 sin(2f) )

    Also note that (d/dx)r² = 2x ; (d/dy)r² = 2y ; (d/dx)² r² = (d/dy)² r² = 2 ; (d/dx)(d/dy) r² = 0 .
    One can frequently save compuations and simplfy algebra by working with (f,r²) coordinates rather than (f,r).

Euclidean Length
    The  Euclidean length or  magnitude of an N-dimensional vector x is |x| = (å_i=1^N x_i²)^½ . This corresponds to our inituitive concept of "length".  It is sometimes known as the  L₂ norm.
    The  infinity norm of an N-dimensional vector x is |x|_¥ = Max { |x_i| 1 £ i £ N}.  It is sometimes known as the  L_¥ norm.
    The  Chicago or  Manhattan distance of an N-dimensional vector x is |x|₊ = å_i=1^N |x_i|.  It is sometimes known as the  L₁ norm.

Approximations to Eulidean Length

    For xβ we can approximate |x| by a.x_ord where x_ord=(Max( |_x1|, |_x2|), Min( |_x1|, |_x2|) and a is a carefully chosen vector. a=(1,0) gives the infinity norm. a=(1,1) gives the Chicago distance.

a1	a2	Notes	Range of a.x when |x|=1
1	0	| |¥	.B504	1.0000
1	1	| |+	1.0000	1.6A08
1	3/8	Fast	.F8E8	1.1168
31/32	3/8	Fast	.F340	1.09F0
2 cos (p/8)(1+ cos (p/8))-1	a1(Ö2-1)	Optimal	.F5E0	1.0A20

    For xγ we define Mid(x,y,z) = x+y+z - Max(x,y,z) - Min(x,y,z). and approximate |x| by
a.x_ord where x_ord=(Max( |_x1|, |_x2|, |_x3|),Mid( |_x1|, |_x2|, |_x3|) ,Min( |_x1|, |_x2|, |_x3|) ).

a1	a2	a3	Notes	Range of a.x when |x|=1
1	0	0	| |¥	.93CC	1.0000
1	1	1	| |+	1.0000	1.BB68
1	15/16	15/16	Innacurate	1.0000	1.A920
15/16	7/16	1/4	Fast	.F000	1.1078
15/16	3/8	3/8	Faster	.ED98	1.13BC
2(1+Ö(9-2(Ö2+Ö6))-1	a1(Ö2-1)	a1(Ö3-Ö2)	Optimal	.F098	1.0F68

   a=(15/16,3/8,3/8) is perhaps the best, requiring only a reduced sort and fast multiplies for a euclidean approximator within ± 7.72%.   

    For xÎÂ^N with N>2, setting s_N = å_k=1^N (2k+1-2(k(k-1))^½) we have optimal accuracy attained by a₁=2(1+s_N^½)^-1 and a_i = a₁(i^½ - (i-1)^½) though at a cost of N general multiplies.
    We often want an estimator that never overestimates (eg. for (rapid range checking) in which case we normalise a by setting a1 = 2s_N^-½(1+s_N^½-1 instead.

Calculating the Polar Angle

    We have written f =  tan^-1(y/x) above but this is not strictly true since f for (-x,-y) differs by p from f for (x,y). What we require is the function C irritatingly impliments as _incode[atan2(y,x)] but which we will denote as f(x,y) and can impliment as follows:
    f(x,y) º tan^-1(y/x) if x³y> 0 ; ½p-f(x,y)  if y>x>0 ; p-f(-x,y) if x<0 ; 2p-f(x,-y) if y<0 ; p/2 if x=0 y>0 ; -p/2 if x=0, y<0 ; or 0 if x=y=0.
    This approach only requires  tan^-1() for values in the range [0,1]   (ie. angles in [0,¼p]) and returns f in [0,2p) rather than (-p,p] . We can attain the latter range by subtracting 2p if f>p.
    Consider F(x,y) º y/x  if x³y> 0 ; ¼-F(x,y)  if y>x>0 ; ½-F(-x,y)  if x<0 ; 1-F(x,-y) if y<0 ; ¼ if x=0 y>0 ; -¼ if x=0, y<0 ; or 0 if x=y=0.
    This provides a value F[x,y] in [0,1] having the same general shape as f(x,y) which we can multipluy by 2p to obtain a crude approximation to f(x,y) perfectly adequate for, say, sorting points by subtended angle as when constructing a convex hull.

Grad of distance functions
    Ñ(Ö(å_i=1^N x_i²)) = x~.
    Ñ(|x|_¥) = e_i where i : |x_i| ³ |x_j| 1£j£N.
    Ñ(|x|₊) = x_-1.

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