----- Original Message ----- 
From: "Denny Esterline" <firmware@tds.net>
To: "Microcontroller discussion list - Public." <piclist@mit.edu>
Sent: Friday, April 01, 2005 5:07 PM
Subject: Re: [EE] Math formula help.


> It's not an easy one...
>
> I don't know about mathematical crescent wrenches, but my crowbar (ti-89
> calculator) spits out this for the sin/cos version:
>
> (ad-bc)cos(theta)-(af-ce)sin(theta)-(bf+de)=0

Your calculator got partway there. Unfortunately it doesn't appear to be as 
bright as Scott Dattalo!

> The equation as written is a monster, but with a better understanding of 
> the
> underlying problem, we might be able to help you find a better solution.


The underlying problem was trying to determine the correct angle to fire a 
gun of a certain length (gunLength) so that the projectile, travelling at a 
given velocity (bulletSpeed) would intercept an object that starts at 
location (x0,y0) relative to the gun and moves at velocity (deltaX,deltaY).

Here is where the formula came from:

where will the bullet be at time t relative to shoot time?

 sin(theta)*(gunLength + bulletSpeed*t)
 cos(theta)*(gunLength + bulletSpeed*t)

where will the enemy be at time t relative to shoot time?

 x0+deltaX*t
 y0+deltaY*t

so, the bullet hits the enemy when the two are at the same point:

 sin(theta)*(gunLength + bulletSpeed*t) = x0+deltaX*t
 cos(theta)*(gunLength + bulletSpeed*t) = y0+deltaY*t

then solve each of the two equations above for t:

       (sin(theta)*gunLength-x0)
t  =  -------------------------------
       (deltaX-sin(theta)*bulletSpeed)

and:

      (cos(theta)*gunLength-y0)
t = --------------------------------
     (deltaY-cos(theta)*bulletSpeed)

Now use the above two substitutions for t to create an equation that has 
theta as the only unknown:

cos(theta)*gunLength - y0                 sin(theta)*gunLength-x0
-------------------------------     =    -------------------------------
deltaY-cos(theta)*bulletSpeed           deltaX - sin(theta)*bulletSpeed

Doing a little judicious juggling and sign changing I substituted single 
letters for the funny variable names to create my post:

a = gunLength
c = -bulletSpeed
b = -x0
d = deltaX
e = -y0
f  = deltaY

Oh, and don't ask what I needed to know this for.. :-)

Bob Ammerman
RAm Systems


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