[Algorithms] Lots of problems...

[dga <Dio...@mi...>]

Hi all!

I have some problems with a work I have to make for school... 

Please, please, please, please (ad nauseum) take the time to read
this mail, since I'm pulling hairs already (spent the last 17 hours
trying to get this right!! And I need to deliver this in the end of the
next week... And lots of work still needing to be done)

I'm trying to do a space game (simple one), but I have problems
with several of it's components, that I've been unable to 
understand, let alone correct.

1) How to obtain the position of an object, based on it's transformation
   matrix ? I'm using DirectX, so a vector is transformed like this:

				Wv=v'

   I think I must use the (4,1), (4,2) and (4,3) positions of the matrix 
(line,column), to get the x, y and z of the object... Is this correct ?

2) How do I rotate an object around an arbitrary axis ? More exactly, given
   a matrix W that represents the objects current position, how do I built
   the matrix A such as WA gives me the transformation matrix for the object
   that rotates the object at it's current position around axis a=(ax,ay,az)
?
   Currently, I'm using the following code:

	D3DMATRIX mat;

	float c=float(cos(Deg2Rad(ang)));
	float s=float(sin(Deg2Rad(ang)));
	float c1=(1.0f-c);

	mat._11=(ax*ax)*c1+c;
	mat._12=(ax*ay)*c1-(az*s);
	mat._13=(ax*az)*c1+(ay*s);
	mat._21=(ay*ax)*c1+(az*s);
	mat._22=(ay*ay)*c1+c ;
	mat._23=(ay*az)*c1-(ax*s);
	mat._31=(az*ax)*c1-(ay*s);
	mat._32=(az*ay)*c1+(ax*s);
	mat._33=(az*az)*c1+c;
	mat._14=mat._24=mat._34=0.0f;
	mat._41=mat._42=mat._43=0.0f;
	mat._44=1.0f;

   but I think it fails in certain cases... :(

3) How do I project a 3d point into 2d ? I need to have the projection to
get my 
   lens flare to work properly... It sometimes work, in simple cases, but
when I attach
   an lens flare to an object and use it's coordinates, it sometimes go off
by a long 
   way... I'm using the following code:

	x=GetXPos();
	y=GetYPos();
	z=GetZPos();
	// v'=Cv, where v=(x,y,z) and C is the camera transformation matrix
	cameraTrans->PreMult(x,y,z,&vx,&vy,&vz);	
	if (vz<=0) return;
	// v=Pv', where v'=(vx,vy,vz) and P is the perpective transformation
matrix
	perspTrans->PreMult(vx,vy,vz,&x,&y,&z);

	// resX=resolution in X divided by 2, resY=resolution in Y divided
by 2
	x=resX*x;
	y=resY*y;

	x=resX+x;
	y=resY-y;

   I think this maybe has something to do with the first two questions,
since if I set 
   the flare position manually, it gives me the correct position, but if I
let the
   other object "drag" the flare around, it doesn't, although the
coordinates for the 
   two objects are reported correctly (but again, it can be connected to the
first).

4) How do I make an object follow another (like a missile) ? What I'm doing
is

			SCPoint3d	rotAxis;
			float		x,y,z,norm;
			float		dx,dy,dz;
			float		sAng,cAng,ang;
			float
ox=GetOrientX(),oy=GetOrientY(),oz=GetOrientZ();
			// 1. Calc orientation to axis
			x=target->GetXPos()-GetXPos();
			y=target->GetYPos()-GetYPos();
			z=target->GetZPos()-GetZPos();
			// 2. Get intended orientation projected on the
orientation vector
			dx=x*ox;
			dy=y*oy;
			dz=z*oz;
			// 3. Get the sine of the angle
			sAng=sqrtf(Sqr(x-dx)+Sqr(y-dy)+Sqr(z-dz));
			norm=sqrtf(Sqr(x)+Sqr(y)+Sqr(z));
			sAng=sAng/norm;
			// 4. Normalize vector
			x=x/norm; y=y/norm; z=z/norm;
			// 5. Get rotation axis
			rotAxis.CrossProduct(ox,oy,oz,x,y,z);
			rotAxis.Normalize();
			// 6. Get the cosine of the angle
			cAng=DotProduct(x,y,z,ox,oy,oz);
			// 7. Get the angle
			ang=GetAngleWithSinAndCos(sAng,cAng);
			if (ang>MISSILE_TURN_SPEED) ang=MISSILE_TURN_SPEED;
			else if (ang<-MISSILE_TURN_SPEED)
ang=-MISSILE_TURN_SPEED;
			// 5. Rotate missile
			RotateAroundAxis(ang,rotAxis.x,rotAxis.y,rotAxis.z);

   Again, this can be connected to the first two problems, since it relies
heavily on the 
   position of the objects and the rotation around an arbitrary axis.
   Basically, what I'm doing is, given the current orientation (normalized)
(ox,oy,oz) and 
   the intended orientation ((x,y,z) in the above example), I calculate the
sine (using
   standart trigonometry, that's the reason for 2. in the above example) and
the cosine
   (using the dotproduct of the normalized intended orientation and the
current orientation).
   Then, I use the crossproduct between those two angles and normalize it to
obtain the 
   rotation axis. Then I calculate the angle using the sine and the cosine
of it, 
   and finally, I limit the angle (to keep the missile to turn too fast).
Then I rotate it...
   On paper, this works fine (as far as I can visualize it), but when I use
the above code,
   the missile only hits the target when our ship and the target are on the
Y=0 plane.

   The track code is the most important one, since I will use it to drive
the AI of the
   enemies (they'll try to track our ship, all guns blazing, nothing too
fancy, the teacher
   it's more interested in the visual aspect of the game).

Thanks in advance

Diogo de Andrade
Dio...@mi...