---------------- More advanced coding techniques ----------------
               ----------- By Kreator of ANARCHY UK ----------
                            ---- 3 D routines ----

  I am  going to  approach this  topic
from a mathematical point of view, the
mathematics invloved  are quite simple
but if you  are not particularly adept
in  this  area  dont  worry  it  isn't
essential to understand the underlying
theory.
  Now  suppose we  are given an object
to    transfer   into    a   wireframe
representation  on  screen.   We  must
construct a  list of coordinates which
specify the vertices of the object and
also a connection list which tells the
computer how to  connect these points
together.
 eg.    A cube  has 8  vertices,  and
12 connecting sides, the vertices are
as follows
(50,50,50)    (-50,50,50) (-50,-50,50)
(50,-50,50)   (50,50,-50) (50,-50,-50)
(-50,-50,-50) (-50,50,-50)

and if these are then labelled 1-8 we
have the connections as follows
 1-2  2-3  3-4  4-1 5-6  6-7  7-8  8-5
 1-5  2-6  3-7  4-8




  If you don't believe the example try
to   draw   the   cube   yourself  and
visualise the coordinates. Notice that
the point  (0,0,0) is at the centre of
the cube, this is important because in
our  rotations this  will  be the only
point which remains stationary.

  When rotating the object what we are
in fact doing is rotating the vertices
about the fixed ORIGIN (0,0,0).  There
is a mathematical theorem which states
that any 3 dimensional rotation can be
split into  3 individual  rotations in
only  2  dimensions,  which is  a much
simpler  thing to  calculate.  Now  in
general  it  is   quite  difficult  to
calculate  these  rotations   from  an
arbitrary  3D  rotation,  but  happily
enough   this   doesn't   matter  when
writing   demos   because   by  simply
performing  2D  rotations and  varying
the  3 Angles of  rotation we  achieve
an interesting effect.

  The formula for 2D rotation is given
as follows,

   x = X cos(s) - Y sin(s)
   y = Y cos(s) + X sin(s)

  This can easily be shown with simple
trigonometry.

  These formulae enable us to rotate a
point in just two dimensions,  but all
we now do is to rotate the point three
times in different planes.
  In  other words  if we  are given  a
general  point  (x,y,z)  and a,b,c are
the three  angles of  rotation then to
calculate the rotated point follow the
procedure below (just for interest the
angles  a,b,c  are  called  the  Euler 
angles)

   x1 = x cos(a) - y sin(a)
   y1 = y cos(a) + x sin(a)

   y2 = y1 cos(b) -  z sin(b)
   z1 = z  cos(b) + y1 sin(b)

   z2 = z1 cos(c) - x1 sin(c)
   x2 = x1 cos(c) + z1 sin(c)

  Then  (x2,y2,z2)  holds  the rotated
coordinate.  To implement  this on the
Amiga use a  Sintable which has values
from  -32768 to  32767,  this  can  be
reused  for  the  cosine  calculations
as  cos (a) = sin (a+90 degrees).  You
could code the  routine something like

    Move x,d3
    Move y,d4
    Move z,d5
    Lea  Sin,a0
    Lea  Cos,a1
    Move d3,d6
    Move d4,d7
    Move a,d0    ;a holds 2x the angle
    Move (a0,d0),d1
    Move (a1,d0),d2

    Muls d2,d6
    Muls d1,d7
    Sub  d7,d6
    Add.l d6,d6
    Swap d6      ;Calculation of x1

    Muls d2,d4
    Muls d1,d3
    Add  d3,d4
    Add.l d4,d4
    Swap d4      ;Calculation of y1
  etc......

Now up until now we haven't considered
how  the lines  will be  drawn to  the
screen, I shall assume you have access
to a blitter line draw routine, if not
there  is  one included  on  the disk,
which is  from the  System Programmers
Guide.  There are two options open now
we can  leave the  coordinates as they
are and  simply add a  displacement to
them before  plotting the lines, or go
for  the  more  realsitic technique of
perspective. This invloves scaling the
x,y coords.  according to how far into
the screen we are. A reasonable way of
doing that is as follows

     x,y,z  in d3,d4,d5

     Add    #Depth+Scale,d5
     Move.l #Scale*65536,d6
     Divu   d6,d5

     Muls   d5,d3 ;Scale the X coord
     Add.l  d3,d3
     Swap   d3

     Muls   d5,d4 ;Scale the Y coord
     Add.l  d3,d4
     Swap   d4

  Alternatively you can use a table of
scaling values.
  Now all that  remains is to plot the
lines. Dont  forget up  until  now all
vertices have been calculated with the
origin at  (0,0)  but now we must move
the origin to the centre of the screen
or where ever  else you want it.  This
means  adding a  displacement to  each
pair of coordinates.
  To  see  how  these  techniques  are
implemented  I've included some source
for you to examine.
  These routines can easily be adapted
to other purposes, eg.to create vector
bobs,  use a single point for each bob
and   before  plotting   sort   the  z
coordinates  and  plot  the   bobs  in
reverse order,  also a  simple form of
hidden line removal can be implemented
by   creating  a   list  of  surfaces,
calculating   the  normals   to  these
surfaces and if the normals point away
from  you dont  plot any  lines in the
surface. For an example of this see my
Magnetic Fields Party Demo.
   Next month I will write about sine
scrollers,   Kreator .......