[tuhopuu-devel] UV-LSCM

[bjornmose]

Hi Brecht and all

Did a refresh to the math i learned some years ago.
The least square conform mapping algo has it's root in complex functions=20
therory. Knowing this the whole math boils down to poor some ten lines in=
=20
Bronstein ( the math bible with me )
A mapping from a subset in the complex plane to some other subset in the=20
complex plane is conform (preserving angles ; ABF :-) math is beautifull =
!! )=20
when it satisfies the Cauchy Riemann equations.
The other way around : If we can not match the C-R equations, we'd like t=
o=20
come as close as possible. And we can't, since our mapping wants to have =
the=20
edges of triangles in the image to be straight lines. If we would drop th=
is=20
Riemann gives a proof for the existace of such a mapping. So we can hope =
to=20
get close to it and as Bruno Levy shows the solution will be unique with =
a=20
proper definition of distance.

So what i want to say
1. i do understand LSCM down to the details now.
2. i am not quite sure if building up the matrix and solving the minimiza=
tion=20
with the "old workin horse" LU decomposition ( ya i am into numerics quit=
e=20
while )  fits the needs we have in blender/tuhopuu. it looks a bit oversi=
zed=20
to me.=20

I am working an a "local relaxing" solution, like you (brecht) built in =20
tuhopuu, minimizing the error in respect of the C-R equation. this should=
=20
meet the blender needs more closely and would be far less waste of CPU th=
an=20
the "one hammer fits all screws" aproach in 'Graphite'. ( hoping not to=20
offend anyone, but definitly setting variables to zero and still happly=20
multipling with (knowing it is zero) is waste of CPU)

my 2 cents so far
ole / bjornmose