╓────────────────┤% VLA Proudly Presents %├────────────────╖
        ║                                                          ║

                Three Dimensional Shading In Computer Graphics
                              By Lithium /VLA

    Hopefully you have read the companion document 3DROTATE.DOC, as this one
will build apon the concepts presented in my attempt to teach some of the
math need to make 3D graphics a reality.  This file will cover such important
topics as the Dot Product and how routines are best constructed for real-time 
3D rotations and planar shading.

                        Our Friend, The Dot Product

    The Dot Product is a neat relation that will allow you to quickly find
the angle between any two vectors.  It's easiest to explain graphicly, so
I will exercise my extended-ASCII keys.

Two Vectors A & B

A (Xa, Ya, Za)     │A│ = √( (Xa)² + (Ya)² + (Za)² )

B (Xb, Yb, Zb)     │B│ = √( (Xb)² + (Yb)² + (Zb)² )

Where Xa, and the others coorispond to some value on their respective Axis's

   \ Θ   <-- Angle Theta between vector A and B

 Cos(Θ) =  Xa * Xb + Ya * Yb + Za * Zb


A (1,2,3)         │A│ = √( 1² + 2² + 3²) = √(14) = 3.7417

B (4,5,6)         │b│ = √( 4² + 5² + 6²) = √(77) = 8.7750

 Cos(Θ) =  1 * 4 + 2 * 5 + 3 * 6   =  4 + 10 + 18    =    32    =  0.9746
           ─────────────────────     ───────────────    ─────
             (3.7417)*(8.7750)           32.8334       32.8334

 ArcCos (.9746) = 12.9° 

    So, your wondering how this revolutionizes you code, huh?  Well, remember
our other friend, the Normal vector?  You use Normal vectors that define
the directions of everything in our 3D world.  Let's say that vector A was
the Normal vector from my plane, and B is a vector that shows the direction
that the light in my scene is pointing.  If I do the Dot Product of them,
you will get the angle between them, if that angle is >= 90° and <= 270°
then no light falls on the visible surface and it doesn't need to be 

Also notice, the way the values of the Cosine orient themselves


           90°                  Cos 000° =  1
                                Cos 090° =  0
            │                   Cos 180° = -1 
  Negative  │  Positive         Cos 270° =  0
180° ───────┼───────  0°        An angle between a light and a plane that
            │                   is less than 90° or greater than 270° will
            │                   be visible, so you can check if the Cos(Θ)
  Negative  │  Positive         is greater than 0 to see if it is visible.


                How Do You Implement The Code?  Easy As π.

Examples in ASM structures

We will define our points like this

        Xpos    dd  ?
        Ypos    dd  ?
        Zpos    dd  ?
        Dist    dd  ?       
    ENDS  XYZs              ;size is 16 bytes

The X,Y,Zpos define a point in 3D space, Dist is the distance from the origin

Dist = √( X² + Y² + Z² )

Precalculate these values and have them handy in your data area

Our planes should look something like this

    STRUC PlaneSt
        NumPts      db      ?               ;3 or 4
        NormIndex   dw      ?
        PtsIndex    dw      ?
                    dw      ?
                    dw      ?
                    dw      ?
    ENDS  PlaneSt

The number of points that in the plane depends on the number your fill
routines can handle you must have at least 3 and more than 6 is not suggested

Then we set up our data like this

MaxPoints   =           100
MaxPlanes   =           100

PointList   XYZs        MaxPoints DUP()
PlaneList   PlaneSt     MaxPlanes DUP()
NormalList  XYZs        <0,0,0, 10000h> , MaxPlanes DUP()

    Non-ASM User Note:   
            I set up points in a structure that had an X,Y,Z and Distance
        value.  I set up a plane structure that had the number of points
        the index number of the normal vector for that plane and the index
        numbers for the points in the plane.

            The next lines set up arrays of these points in PointList, and
        the number of points was defined as MaxPoints.  An array of planes
        was created as PlaneList with MaxPlanes as the total number of 
        plane structures in the array.  NormalList is an array of the vectors
        that are normal to the planes, one is set up initally (I'll explain 
        that next) and then one for each possible plane is allocated.

You'll notice that I defined the first Normal and then created space for 
the rest of the possible normals.  I'll call this first normal, the 
Zero Normal.  It will have special properties for planes that don't shade 
and are never hidden.

    Well, before I start telling all the tricks to the writting code, let me
make sure a couple of points are clear.

-       In the 3DROTATE.DOC I said that you could set your view point on the 
    Z-Axis and then figure out if planes were visible by the post-rotation
    Normal vectors, if their Z was > 0 then display, if not, don't
        That is an easy way to set up the data, and I didn't feel like going
    into the Dot Product at the time, so I generalized.  So, what if you
    don't view your plane from the Z-Axis, the answer is you use the...
    Dot Product!  
    that's right.  The angle will be used now to figure wheither or not to
    display the plane.

-       I have been mentioning lights and view points as vectors that I can
    use with the Normal vector from my plane.  To work correctly, these 
    vectors for the lights and view should point in the direction that you
    are looking or the direction that the light is pointing, *NOT* a vector 
    drawn from the origin to the viewer position or light position.

-       True Normal vectors only state a direction, and should therefore have
    a unit distance of 1.  This will have the advantage of simplifying the
    math involved to figure you values.  Also, for God's sake, pre-compute
    your normal, don't do this everytime.  Just rotate them when you do your
    points and that will update their direction.

        If the Normal's have a length of 1 then │A│*│B│ = 1 * 1 = 1

        Cos(Θ) = Xa * Xb + Ya * Yb + Za * Zb

    Is Reduced To:    
        Cos(Θ) = Xa * Xb + Ya * Yb + Za * Zb

    We eliminated a multiply and a divide!  Pat yourself on the back.

-       You ASM users might be wondering why I defined my Zero Normal as:
    <0,0,0,10000h>  How does 10000h = a length of 1 ?

    Well, this is a trick you can do in ASM, instead of using floating point
    values that will be slow on computers without math co-processors, we can
    use a double word to hold our value.  The high word holds the integer
    value, and the low word is our decimal.  You do all of your computations
    with the whole register, but only pull the high word when you go to 
    display the point.  So, with that under consideration, 10000h = 1.00000
    Not bad for integers.

-       How does the Zero Normal work?  Since the X,Y,and Z are all 0, the
    Cos(Θ) = 0, so if you always display when Cos(Θ) = 0, then that plane
    will always be seen.

            So, Beyond The Babble...  How To Set Up Your Code    

Define Data Points, Normals, and Planes
    Pre-Calculate as many values as possible

 Rotate Points and Normals

 Determin Visible Planes With Dot Product
    (Save this value if you want to shade)

    Sort Visible Planes Back to Front

         (Determin Shade From Dot Product)

         Clip Plane to fit scene

         Draw to the screen

 Change Angles

 Goto Rotation

        A quick way to figure out which color to shade your plane if you are
    using the double word values like I described before is to take the
    Dot Product result, it will lie between 10000h - 0h if you would like
    say 16 shades over the angles, then take that value and shr ,12 that will
    give you a value from 0h - 10h (0-16, or 17 colors)  if you make 10h into
    0fh, add that offset to a gradient in your palette, then you will have
    the color to fill your polygon with.

        Note also that the Cosine function is weighted toward the extremes.
    If you want a smooth palette change as the angles change, your palette
    should weight the gradient accordingly.

        A useful little relation for depth sorting is to be able to find the
    center of a triangle.

        E         The center C = (D + E + F)/3
       / \        Divide each cooridinate by (Xd + Xe + Xf)/3 = Xc
      / C \         and do the same for the Y's and Z's if you 
     /     \        choose to sort with this method.  Then rotate
   D─────────F      that point and use it to depth sort the planes

Phong and Goraud Shading

    Recently, someone asked me about the practiblity of real-time phong and
goraud shading.  The technique is common to ray-tracers and requires a great
deal of calculation when working with individual rays cast from each pixel,
but when only using this for each plane, it is possible.  This type of shading
involves taking into account the reduced luminousity of light as distance
increases.  For each light, you define a falloff value.  This value should be
the distance a which the light will be at full intensity.  Then at 2*FallOff 
you will have 1/2 intensity, 3*FallOff will yeild 1/3 and so on.  To implement
this type of shading, you will need to determin the distance from the light
to the center of the plane.  If distance < FallOff, then use the normal
intensity.  If it is greater, divide the FallOff value by the distance.  This
will give you a scalar value that you can multiple by the shading color that
the plane should have.  Use that offset and it will be darker since it is
further away from the light source.
    However, to determin the distance form the light to each plane, you must
use a Square Root function, these are inherently slow unless you don't care
about accuracy.  Also, it would be difficult to notice the use of this 
technique unless you have a relatively small FallOff value and your objects
move about in the low intesity boundries.

Well, that's all that I feel like doing tonight, and besides, Star Trek is on!
So, see VLA.NFO for information about contacting myself or any of the other 
members of VLA.

                            Happy Coding!