Author : Grant Smith
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matrixes math at this stage, as there are
many books on the subject (it is not part of matric maths, however). To
multiply a matrix, to add the products of the row of the left matrix and
the column of the right matrix, and repeat this for all the columns of the
left matrix. I don't explain it as well as my first year maths lecturer,
but have a look at how I derived A2 and B2 above. Here are the other
matrixes :
Matrix for rotation around the Y axis :
[ Cos (a) 0 -Sin (a) 0 ] [ x ]
[ 0 1 0 0 ] . [ y ]
[ Sin (a) 0 Cos (a) 0 ] [ z ]
[ 0 0 0 1 ] [ 1 ]
Matrix for rotation around the X axis :
[ 1 0 0 ] [ x ]
[ 0 Cos (a) -Sin (a) 0 ] . [ y ]
[ 0 Sin (a) Cos (a) 0 ] [ z ]
[ 0 0 0 1 ] [ 1 ]
By putting all these matrixes together, we can translate out 3D points
around the origin of 0,0,0. See the sample program for how we put them
together.
In the sample program, we have a constant, never changing base object.
This is rotated into a second variable, which is then drawn. I am sure
many of you can thing of cool ways to change the base object, the
effects of which will appear while the object is rotating. One idea is
to "pulsate" a certain point of the object according to the beat of the
music being played in the background. Be creative. If you feel up to it,
you could make your own version of transformers ;)
Drawing a 3D point to screen
Having a rotated 3D object is useless unless we can draw it to screen.
But how do we show a 3D point on a 2D screen? The answer needs a bit of
explaining. Examine the following diagram :
| ________-------------
____|___------ o Object at X,Y,Z o1 Object at X,Y,Z2
Eye -> O)____|___
| ------________
| -------------- Field of vision
Screen
Let us pretend that the centre of the screen is the horizon of our
little 3D world. If we draw a three dimensional line from object "o" to
the centre of the eye, and place a pixel on the X and Y coordinates
where it passes through the screen, we will notice that when we do the
same with object o1, the pixel is closer to the horizon, even though
their 3D X and Y coords are identical, but "o1"'s Z is larger then
"o"'s. This means that the further away a point is, the closer to the
horizon it is, or the smaller the object will appear. That sounds
right, doesent it? But, I hear you cry, how do we translate this into a
formula? The answer is quite simple. Divide your X and your Y by your Z.
Think about it. The larger the number you divide by, the closer to zero,
or the horizon, is the result! This means, the bigger the Z, the
further away is the object! Here it is in equation form :
[Pascal]
nx := 256*x div (z-Zoff)+Xoff
ny := 256*y div (z-Zoff)+Yoff
[C++]
nx = ((256*x) / (z-Zoff)) + Xoff;
nx = ((256*y) / (z-Zoff)) + Yoff;
NOTE : Zoff is how far away the entire object is, Xoff is the objects X
value, and Yoff is the objects Y value. In the sample program,
Xoff start off at 160 and Yoff starts off at 100, so that the
object is in the middle of the screen.
The 256 that you times by is the perspective with which you are viewing.
Changing this value gives you a "fish eye" effect when viewing the
object. Anyway, there you have it! Draw a pixel at nx,ny, and viola! you
are now doing 3D! Easy, wasn't it?
Possible improvements
This program is not the most optimised routine you will ever encounter
(;-)) ... it uses 12 muls and 2 divs per point. (Asphyxia currently has
9 muls and 2 divs per point) Real math is used for all the calculations
in the sample program, which is slow, so fixed point math should be
implemented (I will cover fixed point math in a future trainer). The
line routine currently being used is very slow. Chain-4 could be used to
cut down on screen flipping times.
Color values per line should be added, base object morphing could be put
in, polygons could be used instead of lines, handling of more then one
object should be implemented, clipping should be added instead of not
drawing something if any part of it is out of bounds.
In other words, you have a lot of work ahead of you ;)
In closing
There are a lot of books out there on 3D, and quite a few sample
programs too. Have a look at them, and use the best bits to create your
own, unique 3D engine, with which you can do anything you want. I am
very interested in 3D (though EzE and Goth wrote most of ASPHYXIA'S 3D
routines), and would like to see what you can do with it. Leave me a
message through one of the means described above.
I am delving into the murky world of texture mapping. If anyone out
there has some routines on the subject and are interested in swapping,
give me a buzz!
What to do in future trainers? Help me out on this one! Are there any
effects/areas you would like a bit of info on? Leave me a message!
I unfortunately did not get any messages regarding BBS's that carry this
series, so the list that follows is the same one from last time. Give
me your names, sysops!
Aaaaargh!!! Try as I might, I can't think of a new quote. Next time, I
promise! ;-)
Bye for now,
- Denthor
[Note: things in brackets have been added by Snowman. The original text
has remained mostly unaltered except for the inclusion of C++ material]
Introduction
Hi there! ASPHYXIA is BACK with our first MegaDemo, Psycho Neurosis! A
paltry 1.3MB download is all it takes to see the group from Durbs first
major production! We are quite proud of it, and think you should see it
;)
Secondly, I released a small little trainer (a trainerette ;-)) on
RsaPROG and Connexctix BBS mail, also on the ASPHYXIA BBS as COPPERS.ZIP
It is a small Pascal program demonstrating how to display copper bars in
text mode. Also includes a check for horizontal retrace (A lot of people
wanted it, that is why I wrote the program) (ASPHYXIA ... first with the
trainer goodies ;-) aargh, sorry, had to be done ))
Thirdly, sorry about the problems with Tut 8! If you had all the
checking on, the tutorial would probably die on the first points. The
reason is this : in the first loop,
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