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homogeneous cordinates
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Author:  vpachecu2 [ Mon Apr 05, 2010 4:44 pm ]
Post subject:  homogeneous cordinates
Hello again friends
I have a problem in algebraic geometry
I have a vector field in Cartesian coordinates.

I know also that the ring C ³ in homogeneous coordinates is P ²
ie (X, Y, Z) can actually be (x / z, y / z, 1)

I want to get the vector field in these new coordinates then

is there any lib who can help me?


proc transf (matrix a)
(
int n = nrows (a);
dual matrix [n] [n];

dual [1,1] = a [2,1] / / transform the field in the dual form
dual [2,1] = -1 * a [1,1];

return (dual);
)

transf (dual (a));


now need to do this function so
Author:  gorzel [ Thu Apr 08, 2010 6:14 pm ]
Post subject:  Re: homogeneous cordinates
vpachecu2 wrote:
I have a vector field in Cartesian coordinates.

I know also that the ring C ³ in homogeneous coordinates is P ²
ie (X, Y, Z) can actually be (x / z, y / z, 1)

I want to get the vector field in these new coordinates then


Not clear what this should mean.
Give an explicit mathematical example.
Author:  vpachecu2 [ Fri Apr 09, 2010 1:58 pm ]
Post subject:  Re: homogeneous cordinates
i have one field vector w= -q(x,y)dx +p(x,y)dy

that vector is in P²(polinomial ring with x,y)

then i transform P² in C³\{0} by one equivalence relation


then i have a space (x,y,z), but i need transform the ring such that the result coordinates are in the form
(x/z:y/z:1), this is in homogeneous coordinates, because i consider the class of points that pass through
(x,y,z)
z=1 is the plane that transform C³ in C²,

i need one transformation of this type:

w'= z^(m+2) * w=q(x,y)*z^m +p(x,y)*z^m
Author:  gorzel [ Fri Apr 09, 2010 3:39 pm ]
Post subject:  Re: homogeneous cordinates
If you define at the beginning already a ring in three variables
(x,y,z), where you have two polynomial p and q in the variables x,y,
then you can just set

Code:
      q*z^m + p*z^m
   



Otherwise, you have to define a new ring with an additional variable,
fetch/imap your input to this ring and proceed then.

See: extendring from ring.lib and
also the command homog

If not enough, give an explicit example.
Input: p = ... q =
Output:

And where comes m = ... from ?
Author:  vpachecu2 [ Mon Apr 12, 2010 4:41 am ]
Post subject:  Re: homogeneous cordinates
m is the max degree of monomials p, q

i need, in true,create a function for desomoneization and homegeneization of polynomials.


gorzel wrote:
If you define at the beginning already a ring in three variables
(x,y,z), where you have two polynomial p and q in the variables x,y,
then you can just set

Code:
      q*z^m + p*z^m
   



Otherwise, you have to define a new ring with an additional variable,
fetch/imap your input to this ring and proceed then.

See: extendring from ring.lib and
also the command homog

If not enough, give an explicit example.
Input: p = ... q =
Output:

And where comes m = ... from ?
Author:  gorzel [ Mon Apr 12, 2010 9:32 pm ]
Post subject:  Re: homogeneous cordinates
It would be really helpful if you could give an concrete example.
State what you have, how the basering looks like, and
what you want to get.

Is m the degree of p and q, or should this also be
an additional argument to a possible proc?

In general, dehomogenization i.e. passing from homogenous
coordinates X,Y,Z to affine coordinates, where Z = 1, is
obtained by substituting Z=1.

This is done by subst(q,Z,1).

The homogenization is done by homog(q,z).
If q has degree d, then homog(q,z) is
homogenous of degree d.

// ---
Or could you point to an article/textbook where this situation is considered?

C. Gorzel
Author:  vpachecu2 [ Fri Apr 16, 2010 3:04 pm ]
Post subject:  Re: homogeneous cordinates
thx, but i cant do the change of variables x=X/Z and y=Y/Z, that is,

i need tha t. One vector v= x²+y²
use subst(v,x,x/z)
and subst(v,y,y/z)

and the return is v= (x/z)² +(y/z)²

gorzel wrote:
It would be really helpful if you could give an concrete example.
State what you have, how the basering looks like, and
what you want to get.

Is m the degree of p and q, or should this also be
an additional argument to a possible proc?

In general, dehomogenization i.e. passing from homogenous
coordinates X,Y,Z to affine coordinates, where Z = 1, is
obtained by substituting Z=1.

This is done by subst(q,Z,1).

The homogenization is done by homog(q,z).
If q has degree d, then homog(q,z) is
homogenous of degree d.

// ---
Or could you point to an article/textbook where this situation is considered?

C. Gorzel
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