1. Re: Trig and the ...
Pete Eberlein wrote:
>On Wed, 6 May 1998, BABOR, JIRI wrote:
>> you can try
>>
>> vvmax = maxspeed * maxspeed -- known, can be pre-calculated
>> vv = xv*xv + yv*yv -- speed squared
>> f = (vvmax+vv) / (vv+vv)
>>
>> I shall not bore you with the mathematics of it - unless, of course,
>> you insist. Jiri
>
>
>This is pretty cool, and has an absolute error of 0 to -0.5, which I deem
>acceptable for game arithmetic. Where did you learn that trick, Jiri?
>Bore me, I insist :)
I did not learn that trick anywhere, I just derived it. And since you
insist, it goes something like this:
1. There is almost always a faster way than square root or arctan 
2. Using the same symbols as above:
f = maxspeed / speed = sqrt( maxspeed*maxspeed / speed*speed )
f = sqrt(vvmax / vv) = sqrt( 1 - (vv-vvmax)/vv )
Are you still with me? Let's make it visually a little bit simpler:
e = (vv-vvmax)/vv
Then
f = sqrt( 1-e )
Now the jump: if e is *relatively* small then, approximately
f = sqrt( 1-e ) = sqrt( (1 - e/2)(1 - e/2) ) = 1 - e/2
simply from
(1-e/2)(1-e/2) = 1 - e + e*e/4
neglecting the (small) last term. Jiri
