Raytrace of a Star Sapphire

Date: 12/11/96 at 21:04:33
From: Charles
Subject: Angle from one 3-dimensional vector to another in terms of 
rotating on x,y,and z axis.

Hello,  

I am a HS graduate with no college math who is _very_ rusty on pre-
calc and geometry/trig.  My problem comes from an attempt to create a 
raytrace of a star sapphire that acts like a real star sapphire, which  
always faces the camera.  To do this, I need to find the degrees of 
rotation to use to face the star towards the camera position no matter 
where it is.  If memory serves, (which is questionable) the following 
formula _should_ apply:

rotation on x axis = degrees (atan((CamY-ObjY)/(CamZ-ObjZ)))
rotation on y axis = degrees (atan((CamZ-ObjZ)/(CamX-ObjX)))
rotation on z axis = degrees (atan((CamX-ObjX)/(CamY-ObjY)))

Where:
CamX,Y,Z = 3-dimensional vector of Camera
ObjX,Y,Z = 3-dimensional vector of Object
Rotation refers to star on the sapphire.

I just can't seem to get the results I want from this.  I remember 
something about the tangent of 90 causing problems, but I'm not too 
sure.  I will really appreciate it if you can help clear this up for 
me.

					Thanks...

Date: 12/12/96 at 09:39:01
From: Doctor Jerry
Subject: Re: Angle from one 3-dimensional vector to another in terms 
of rotating on x,y,and z axis.

Hi Charles,

I may be able to help you, but I've got to get the problem clear in 
terms I can understand and visualize.  From your comments above, here 
are my impressions and, such as they are, my understanding:

You have a camera at, say, (a,b,c), which can be thought of as a point 
or a position vector from the origin.  There is an object at the point 
(A,B,C).  There is a plane (the plane of the star) going through the 
point (A,B,C). 

My understanding starts to fade as I think about the plane.  A plane 
containing (A,B,C) can have any orientation.  The orientation of a 
plane is usually specified by its normal vector.  This is a vector 
that is perpendicular (normal) to the plane.  You appear to be saying 
that the orientation of the plane, the normal vector to the plane, is 
the vector {A,B,C}.

In any case, if {n1,n2,n3} is a normal to a plane through (A,B,C), do 
you want to know how to aim the camera so that the film plane is 
parallel to the plane with normal {n1,n2,n3}?  Specifically, do you 
want the angles the normal vector makes with the x-, y-, and z-axes?


-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 12/12/96 at 12:51:42
From: Anonymous
Subject: Re: Angle from one 3-dimensional vector to another in terms 
of rotating on x,y,and z axis.

Sorry for being unclear.  The object is originally created at (0,0,0) 
with a surface normal for the star being positive y.  The object is 
then translated to any other position, (1,0,3) for example, after 
which, I need the star texture on it to rotate in such a way that the 
surface normal now faces directly at the camera, at location (2,0,-1).  
I am trying to find out what formula to use to determine the rotations 
necessary (in degrees) first on the x, then the y, and finally the z 
axis in order to face the _texture_ towards the camera at all times, 
no matter the position of the object or the camera.  I hope this 
clears things up a bit.  I am very grateful for your assistance.

					Charles

Date: 12/14/96 at 13:24:54
From: Doctor Jerry
Subject: Re: Angle from one 3-dimensional vector to another in terms 
of rotating on x,y,and z axis.

Hi Charles,

I solved a sample problem (see below), which may help you decide if 
I'm on the right track and, if I am, whether you want to pursue this 
approach.  Before we get to this, I need to  make a comment or two.  
First, although I haven't solved this kind of problem before and don't 
find a full solution in the books I own, I'm certain that the problem 
has been solved many times before.  There are individuals who 
could help you solve your problem more rapidly and efficiently than I.  
To act on this, I'll post a memo on the Math Forum.  Someone may offer 
to help.

I think that your problem boils down to this: 

If u and v are unit vectors based at the origin, you want to:
 
(i) rotate v about the x-axis, giving a new vector p 
(ii) rotate p about the y-axis, giving a vector q
(iii) rotate q about the z-axis, giving a vector r, such that r = u. 

I took 

u = {1/Sqrt[6],1/Sqrt[6],-2/Sqrt[6]}

v = {1/Sqrt[21],4/Sqrt[21],2/Sqrt[21]}

Since u and v are unit vectors, their third components give the 
cosines of the angles these vectors make with the z-axis.  I 
visualized the vector u as lying on a cone whose vertex is at the 
origin and vertical axis. The third component of u determines the 
cone.  I first rotate v about the x-axis until it lies on the upper 
nappe (in this case) of the cone.  This gives p.  Next, I rotate p 
about the y-axis until it lies on the lower nappe of the cone.  This 
gives q.  Finally, I rotate q about the z-axis until it coincides with 
u.  Each rotation is done using a rotation matrix. These matrices are:

M1(a) =    1     0      0
           0   cos a  -sin a
           0   sin a   cos a
         
M2(b) =    cos b     0    -sin b
             0       1       0
           sin b     0     cos b
          
M3(c) =    cos c   -sin c    0
           sin c    cos c    0
              0       0      1

I choose a so that the third component of M1(a)*v is equal to the 
absolute value of the third component of u.  I got  a = 0.527509 in 
radians.  Then p = {0.218218,0.534523,0.816496}.

I choose b so that the third component of M2(b)*p is equal to the 
third component of u.  I got b = pi.  
Then q = {-0.218217,0.534523,-0.816497}.

Finally, I choose c so that the first component of M3(c)*q is equal to 
the first component of u.  I got c = 3.53939.  
Then r={0.408248,-0.408249,-0.816497}, which is u.

There were several decision points in this procedure, requiring some 
visualization.  I graphed the vectors as I went.  I used Mathematica 
for my calculations and graphs.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/