Math Tricks

MATH TRICKS

This class is probably the most important one, but it is also the most
difficult to explain. I mean, there is no standard optimization method
formaths which will give good results on the general case. But it is
probably the most important one! My only little examples will be for 3D
vectors. When you rotate a point around an axis, you must do 4 MULS:

x'=x*cos(é)+y*sin(é)
y'=x*sin(é)-y*cos(é)
z'=z

So 3 rotations around 3 axes should be 12 MULS????? No, if you are an
intelligent coder! You can develop you formulaes and get something like:

x'=x*xvx+y*xvy+z*xvz+xv0
y'=x*yvx+y*yvy+z*yvz+yv0
z'=x*zvx+y*zvy+z*zvz+zv0

where x',y',z' are the 3 coordinates of you point after the rotations and
xvx,yvx,zvx,.... are some coefficients depending of the 3 rotation angles.
So you first calculate these coefficients at the beginning of your frame
code, then you have 9 MULs only for each dot! This is called the 3D matrix
method for rotating the dots, but don't panic, you don't have to know any-
thing about matrixes to use it! Another advantage of this method is that if
you decide to use 123 rotations instead of 3, or to do some gelly vectors
or whatever else, your formula remains the same, only the coefficients are
different! There is also some other methods for rotating the dots, but this
is the most precise one. Then you have the problem of calculating this 3D
matrix, i.e. calculating the 12 coefficients. Well, each of them are a sum
of products of sines and cosines of the different rotation angles, for ex.:

yvy=sin(a)*cos(b)*sin(c)-cos(a)*cos(c)

It should be interesting to eliminate the MULs instructs in this formula!
Well, it is quite possible: This is what is called linearization in mathe-
matical therms. For example you have:

cos(a)*cos(b)=(cos(a+b)+cos(a-b))/2

so, if you use this formula and some others like this, you can write yvy
(and the others coefficients) as a sum of sines and cosines, which elimi-
nates the MULs. (evidently you just get the sines and cosines in a table)
There are thousands of other mathematical tricks but I can't explain them
all, let's just say that if you are imaginative enough you can usually boost
the speed of your code by 50% using this kind of tricks. If someone knows
some very technical tips, I would be pleased to hear about.... for example
math tricks for SpaceCode or SuperComplex 3D or something like that....
Yes Mr.H. I'm talking to you.....

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Ripped from Imphobia Issue #7 by Dr. ZiXuS / TRiNiTY
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Date this article was posted to GameDev.net: 7/16/1999
(Note that this date does not necessarily correspond to the date the article was written)
See Also:
General Math