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Fixed Point Tutorial - By Harvey Cotton
A tutorial written by a moron for morons
Updated: 29/01/01
Mail: harvey.c@lineone.net
Http: http://www.netyaroze-europe.com/~harveyc/
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Fixed point mathematics is one of those things that
is often misunderstood. No matter how hard you look
at them or read about them, they can make no sense.

I will attempt to explain them to you. Once you know
how to use fixed point, you'll wonder how you ever
got along without them!

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Introduction
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Q: Why Fixed Point?
A: For getting precision without using floating point.

Q: Whats wrong with float?
A: Its bloody slow on the Playstation, which has NO
   floating point unit.

Q: What is a fixed point number?
A: You declare it as a long. Long is 32-bit like float.

Q: How much faster is it to use fixed point?
A: If you are making a game that relies on floats, the
   speed increase can be 200% to 400%.

Q: Is it difficult to use fixed point?
A: Nah. Once you start getting use to it, its a piece
   of cake.
 
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Representing fixed point
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- To represent a float as a fixed point, you simply
  multiply it by 4096.

- This means 4096 is equivalent to 1.

- For example the number 10 is equivalent of 40960
  because 4096 * 10.

- How do you get precision then? Simple.
  You can represent floats as well. For example you can
  represent 0.5 as 2048 because 0.5*4096=2048.

- Thus, the smallest fraction you can represent is
  0.000244140625 because 1/4096.

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What is bitshifting?
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- The value 1, bitshifted left 12 times = 4096.
- 1<<12 is the equivalent of 1*4096.
- 1>>12 is the equivalent of 1/4096.
- 2^12 = 4096.
- Bitshifting is fast. Faster than multiplying or dividing;
- Bitshifting CANNOT be used with float.

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Maths with fixed point
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MATH                            HOW                   EXAMPLE

Converting a float to fixed:    float*4096=fixed      1.5*4096=6144
Converting a fixed to float:    fixed>>12=float       4096>>12=1

Adding fixed to fixed:          fixed+fixed           4096+4096=8192
Subtracing fixed from fixed:    fixed-fixed           4096-4096=0
Multiplying fixed from fixed:   (fixed*fixed)>>12     (4096*4096)>>12=4096
Dividing fixed by fixed:        (fixed<<12)/fixed     (4096<<12)/4096=4096

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Example of using fixed point in your games
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GsSPRITE ship;			// Player's sprite

long x=100<<12,y=100<<12;	// Position of your player at 100,100 represented
                                   as fixed point (409600,409600)

x--;				// Move your left player really, really slowly

y-=4096;			// Move player up 1 pixel

ship.x=x>>12;			// Translate your player's x position to sprite

ship.y=y>>12;			// Translate your player's y position to sprite

DrawSprite(&ship);		// Draw your ship (function not included!) at 100,99

Now say you want your ship to move even slower, 1/2 a pixel at a time.
You change y-=4096; to y-=2048; Obviously theres no such thing as half
a pixel, but you will percieve the ship moving at half the speed! Its
like moving the ship at 0.5 pixel at a time, without using floating point
numbers!!