[size="5"]Fixed Point Arithmetic
It seems natural to use floating point arithmetic in graphics programs. Floating point arithmetic is easy to use and it comes closer to working the way we expect arithmetic to work than any other kind of arithmetic we can use in programs.
Interactive graphics demands speed and floating point is not usually the fastest kind of arithmetic available on personal computers. Because graphics programs do so much arithmetic we want to do the arithmetic as fast as possible. On most machines we want to use integer arithmetic instead of floating point arithmetic because it is much faster.
If you have a fast 486 DX2 or DX4, a Pentium, or any of the RISC processors with floating point hardware you might not get much by avoiding floating point arithmetic. But if you want your programs to run quickly on anything but the newest processors with floating point hardware you want to avoid floating point arithmetic as much as possible.
The alternatives to floating point arithmetic are fixed point and integer arithmetic. Fixed point arithmetic uses ordinary integer operations to do arithmetic with numbers that have both a whole part and a fractional part. A fixed point number is called fixed point because the decimal point (I'm going to use decimal numbers in the first part of this discussion) is in a fixed location in the number. (Floating point on the other hand lets the decimal point float around.) You also have a fixed number of digits to work with.
Let's say we have a machine that lets us do arithmetic on signed 4 digit decimal numbers. That is, the numbers we can work with have a range from -9999 to 9999.
Let's assume we have an addition instruction that will add two of these numbers and give us a 4 digit result and a carry. We also have a subtraction instruction that subtracts two 4 digit numbers and gives us a 4 digit result and a borrow.
Our example machine also has multiply and divide instructions. Multiplication and division are not like addition and subtraction. The multiplier takes two 4 digit numbers and gives us an 8 digit result. The divider divides an 8 digit number by a 4 digit number and gives us a 4 digit quotient and a 4 digit remainder.
Multiply and divide might seem a little odd, but take a look at the 386 multiply and divide instructions. The multiply takes two 32 bit numbers and gives a 64 bit result and the divider divides a 64 bit number by a 32 bit number and gives a 32 bit quotient and a 32 bit remainder.
The number of digits you get when you multiply 2 numbers is the sum of the number of digits in the two numbers. This means that multiplication naturally gives you numbers twice as long as the ones you started with. Once you have the double long numbers you get from multiplying you need a divide operation that can scale them back down to 4 digits.
Having a double length intermediate value makes it possible to evaluate expressions like:
x = (a * b ) / cwithout getting the wrong answer. For example.
If products weren't allowed to be longer than normal numbers then we would have:
x = 1000 * 10 / 2
x = 10000 / 2
x = 5000
because 10,000 is a five digit number and we're only allowed 4 digit numbers the 1 at the top would get lost and we would get the wrong answer.
x = 1000 * 10 / 2
x = 0000 / 2
x = 0
Ok, that gives us a nice definition of decimal integers. But what about fractions?
Just pretend that there is a decimal point in the middle of our numbers. This means that instead of writing "1" as 0001 we write one as 01.00. "01.00" is still a 4 digit decimal number. Inside our machine it is represented at "0100." The computer doesn't know or care about our imaginary decimal point.
We call this way of looking at numbers a "2.2" fixed point format because it has 2 digits on each side of the decimal point. We could also pretend that the decimal point is all the way to left giving us a 0.4 format or, if we want to, we could use a 3.1 format with one digit of fraction and 3 digits of integer. Of course, an ordinary 4 digit integer is in 4.0 fixed point format.
If we want to add 4 + 3 in 2.2 format we set it up like this:
Notice that a normal add operation that adds 2 integers also adds 2 fixed point numbers. It doesn't matter what format they are in, it just adds.
The same is true for subtraction:
Integer subtraction works on fixed point numbers just like it does on integers. Remember that we are the ones deciding where the decimal point is, the hardware neither knows nor cares.
Fixed point multiplication and division are not quite as easy as addition and subtraction.
When I learned to multiply decimal fractions I was taught to do the multiplication ignoring the decimal points and then, at the end, put the decimal point in the product so that the number of digits of fraction in the product is the sum of the number of digits of fraction in the multiplier and the multiplicand. For example:
The digits in the product depend on the digits in the multiplier and the multiplicand. They do not depend on where the decimal points are in the multiplier or the multiplicand.
12.345 * 67.89 = 838.10205
12345 * 6789 = 83810205
Here are some rules to remember when multiplying fixed point numbers:
- The maximum number of digits you can have in the product is the sum of the number of digits in the multiplier and the multiplicand.
- The number of digits of fraction in the product is the sum of the number of digits of fraction in the multiplier and the number of digits of fraction in the multiplicand.
- An integer is a fixed point number with 0 digits of fraction.
When you multiply 2 fixed point numbers in M.N format you get a result that has 2N digits of fraction. For example:
0007 * 03.00 = 21.00
07.00 * 03.00 = 21.0000To get back to an M.N format number you throw away the low order N digits of the fraction. That is, you shift the number left by N digits. Truncating the low order digits is a source of error in fixed point arithmetic. You can round the fraction instead of truncating it, but that takes longer.
These rules also imply that you can multiply numbers in different fixed points formats and get a meaningful result. For example, you might have a table of sines in 2.30 binary fixed point format (in a binary format the numbers are the number of bits, not the number of digits) that you want to multiply by numbers in an 18.14 format. The result will be numbers in a 20.44 fixed point format. You can get back to 18.14 format by throwing away the bottom 30 bits of the fraction and the top two extra bits of the integer part of the number.
When I first learned to do long division I was taught to write the answer as a quotient and a remainder. Later I learned how to continue the division to produce a decimal fraction by adding zeros to the right end of the dividend. A few years ago I realized that adding zeros is the same as shifting the dividend left. In fixed point arithmetic you shift the number left before you divide instead of adding one zero at a time as you do in long division.
Think about dividing 1 by 3. We know that the decimal fraction for 1/3 is 0.333... the threes just keep going on and on and on. But, if we divide the integer value 1 by the integer value 3 on a computer we get 0. Zero is not a very good approximation of 1/3. So, let's try adding a zero at the end of the 1. The result of dividing the integer 10 by 3 is 3. And the result of dividing 100 by 3 is 33. For every 0 we tack on to the 1 we get another 3 in the result.
In the 2.2 fixed point we've been using for our examples 1 is 01.00 which is really the integer 100 or 1 with 2 zeros tacked on the end. Dividing 01.00 by 3 is the same as dividing 0100 by 3 and we get the same 33 as a result. But, since we started with 01.00 we interpret the result as 00.33. This shows that when a fixed point number is divided by an integer the result is a fixed point number.
To divide a fixed point by a fixed point you have to tack on even more zeros to the end of the number you are dividing. This means that if we want 2 digits of fraction in the result we have to shift the dividend to the left by 2 digits before the division takes place.
When we want to use fixed point arithmetic in a program we have to decide what format to use and how to implement it within the programming language we are using.
0100 / 0300 = 0000
01000 / 0300 = 0003
010000 / 0300 = 0033
The first thing you have to do is pick a fixed point format. On 32 bit computers it is tempting to use a 16.16 format. That is, 16 bits of fraction and 16 bits of integer. My experience with that format is that it is very easy to mess up the sign of the number during multiplies when the result of a multiply uses all the available bits. I like to leave a spare bit at the top of the format to guard against overflow. I tend to use a 17.15 format or 18.14 format to avoid these problems overflow problems.
You can use as many different fixed point formats as you need. If you are working with numbers that you know are always in the range 0 to 1 then an unsigned 1.31 format might be exactly what you want. If you are doing color computation then you might use a 0.8 format and pack each number in a byte. Using the right format can prevent a lot of problems.
Your choice of fixed point formats is restricted by your choice of programming language. If you pick a 32 bit fixed point format like 16.16 then you need to be able to multiply two 32 bit numbers, get a 64 bit result, and then shift the result right to get it back into the original fixed point format. Most PC C compilers don't provide the 64 bit integer type that you need for the intermediate values you generate when you multiply 32 bit numbers.
This means that if you want to stick to "pure" high level language programming you need to restrict yourself to 16 bit fixed point formats so that the product of 2 numbers will fit in a 32 bit "long" variable. You can do 3D graphics programming using 16 bit fixed point formats in pure C. I've done it, it works, and it's a royal pain. It is especially painful when you know that with just a few lines of assembly language you can write the 32 bit fixed point multiply and divide routines you want.
My personal opinion is that you should use small amounts of assembly language in macros or in inline procedures to implement the fixed point operations and do the rest in the high level language of your choice.
When choosing a fixed point format look carefully at how many digits of fraction you need to do the job. If you only need a couple of digits of fraction then you can live with fixed point numbers with 7 bits of fraction. Two decimal digits of fraction gives you 100 steps between 0 and 1 or any two adjacent integer numbers. Seven bits of fraction gives you 128 steps between integers. If you need 3 decimal digits of fraction then 10 bits of fraction will do the job and so on.
Using more bits of fraction makes your fixed point number more precise at the expense of the range of numbers you can use. If you take 16 bits out of 32 total bits then you have limited the range of numbers you can represent to the range -32768 to 32767 for two's compliment integers or 0 to 65535 for unsigned integers.
The files fixed.cpp and fixed.h contain type definitions and in line assembly language arithmetic routines for an 18.14 fixed point format written in Watcom C including a couple of fixed point square root functions and table based sine and cosine functions.
The multiply and divide functions use assembly language to get at the hardware arithmetic capabilities that are hidden by high level programming languages.
One of the fixed point square root routines uses a standard Newton iteration to extract the root. It just happens to use fixed point arithmetic to do it. The routine is surprisingly (at least to me) accurate and fast enough to be useful, but only slightly faster than a software floating point square root and not nearly as accurate as the square root provided in the Watcom math library.
The other fixed point square root routine uses the integer square root procedure that is also in fixed.cpp. This routine uses the fact that fixed point numbers are really integers. The integer square root of a fixed point number divided by the square root of 2 to the F power where F is the number of bits of fraction in the fixed point number is the square root of the original fixed point number. If you have an even number of bits of fraction you can convert the integer square root to the fixed point square root with a shift. Otherwise you have to do one divide to recover the square root.
Since the integer square root routine is simple and fast the second fixed point routine is faster than the first one. But, it is not as accurate, especially when taking the roots of large numbers.
The file demo.cpp contains code that tests the fixed point and integer square root routines and does a timing test on each of them. There is also a timing test of the C library sqrt() function. It is interesting to compare the speed and accuracy of the double precision floating point square root and the 32 bit fixed point square roots.
On my home machine, a 386/40 with no floating point coprocessor the fast fixed point square root is about 2.5 times faster than the library floating point square root.
The sine and cosine functions are implemented as arrays of 256 fixed point sine values and 256 fixed point cosine values. The trick is to use 256 "degrees" instead of 360. This lets you use a simple masking operation to make sure that you don't over run the array boundaries.
256 degrees seems to be plenty for interactive graphics applications. If it isn't good enough for your application it is easy to generate tables with 512, 1024, 2048, or more entries. If you can live with slower functions but need better accuracy then linear interpolation between adjacent table entries is a good compromise. It you need better accuracy than that you're probably not going to be happy using fixed point arithmetic.
Code for generating the sine and cosine tables is also in demo.cpp. It's worth looking at that code if you are planning to generate tables of other functions. It is very easy to build tables that have cumulative errors built into them. Errors caused by bad tables are awfully hard to find. Also remember that the tables are only as good as the source of the values in the tables.
Fixed.cpp and fixed.h also contain a couple of routines that are handy for doing integer arithmetic. I've already mentioned the integer square root routine. This routine is based on one I picked up off the net sometime ago. I tested a number of integer square root routines and this one worked, it's simple, and it's fast. After a little study I even convinced myself that I understand how it works. It seems to be an implementation of the pencil and paper root extraction algorithm you may have learned in grade school. It takes advantage of the fact that all even powers of 2 are the squares of even powers of 2.
The other routine is mulDiv(a, b, c) this function multiplies A times B and then divides by C. The nice thing about the routine is that it multiplies 32 bit numbers and keeps around the 64 bit intermediate result. This means that you can do the common (A*B)/C operation over the entire range of 32 bit numbers and get the correct answer.
Ok, let's start using some arithmetic. Pick up your typical graphics book and you'll find a chapter or three on 2 dimensional and 3 dimensional transformations. Transformations are used to convert coordinates in the coordinate system used to describe the world into the coordinate system used by the display screen as seen from the eye point. The world tends to be described in units like feet or millimeters while the screen is measured in pixels.
The "eye point" is where you are in the virtual world. The eye "point" actually has 4 parts. The first part really is a point, an x, y, z coordinate that tells where we are. This coordinate is also known as the "position vector." Once you know where you are you need to know which direction you are looking. An eye can't see in all directions at once so you need to know which direction is forward, which way is up, and which way is to your left. You could get by with knowing which way backward, down, and right, the choice is a matter of taste and convention. These three values are called the direction vectors and each of these has 3 components just like the X, Y, and Z of the position vector.
The position vector tells you where you are relative to the 0, 0, 0 point of the coordinate system you are using to describe the world and the direction vectors tell you how far your personal forward, up, and left are rotated from the axes of the base coordinate system.
As long as you have 3 directions, all at right angles to each other, you have a coordinate system that defines how the eye sees the world. You also have the information you need to transform world coordinates into eye coordinates so that you can see the world from the location and orientation of the eye.
Traditionally this is all described using translations and rotations relative to the X, Y, and Z axes. I find that I can keep it all straight by thinking about walking around a room. Imagine that you are standing in a room with your head at the 0, 0, 0 point of the room. The Z axis is sticking out your nose, pointing in the direction you are looking. The Y axis is stick out of the top of your head, pointing up. And the X axis is sticking out of both ears pointing to the left and right.
Any kind of movement that leaves you looking in exactly the same direction, not tipping your head or turning, is a translation. If you take a step forward you have translated in the +Z direction. A step to the left is a -X translation, standing on your toes is a +Y translation. Mathematically a translation is simply increasing or decreasing the X, Y, or Z components of your location vector.
Rotations leave your location vector alone but change the direction vectors. If you already understand all this nod your head up and down for "yes." Otherwise turn your head left and right for "no." If you nodded yes you rotated your head around the X axis sticking through your years. First you rotated by a positive angle and then by a negative angle. Because you rotated about the X axis the rotation was in the YZ plane. If you didn't understand then you rotated your head about the Y axis and you did the rotation in the XZ plane.
Let's say you take a step forward, adding 1 to your Z coordinate. You have translated along the Z axis. To draw the room with the eye in the new position you subtract 1 from the coordinates of all the objects in the room before you draw them. If you rotate the eye so that it is looking to the right then you actually draw everything by first rotating them to the left. The eye is actually in the center of the universe and cannot be moved because it is out side looking at the computers screen. But we can move everything else the other way to give the impression that the eye location and viewing direction has changed.
If you look in any good book on 3D graphics you will see that each different kind of transformation can be described as a 4x4 matrix and the effect of performing several different transformations in a row can be described as the product of a sequence of individual transformation matrices. Because matrix algebra is well understood, and works for this application, most 3D graphics books talk about transformations in terms of 4x4 matrices and talk about transforming points by multiplying vectors by transformation matrices to get the transformed points.
Let's say you want to simulate walking through a mutant filled fortress. You compute each frame of the animation by first updating the location and viewing direction of the player, creating a transformation matrix from the new location information, transform the X, Y, and Z, coordinates of everything the player can see, and then drawing visible objects including the walls, floor, and ceiling. The transformation of all those coordinates can take a lot of time. In an animation you can't spend a lot of time on each frame or you lose the sense of "being there" that is so essential to a good game or a virtual reality.
The first question to ask is whether or not we really spend much time generating transformation matrices and transforming points? It turns out that you don't spend very much time creating the transformation matrices. You may well render an entire scene using a single transformation matrix. But, you have to transform every vertex of every polygon you are going to render. Which means you may spend a lot of time transforming points.
I can think of 4 different techniques (ignoring building faster hardware) that can be used to speed up building transformation matrices and transforming points. The first thing to do is to pick the fastest kind of arithmetic you can use. That's the reason the first half of this article is about fixed point arithmetic.
The second and third techniques are standard code tweaking techniques for improving speed. Namely, loop unrolling and inlining.
Loop unrolling simply says that instead of writing a loop to execute N times you should make N copies of the loop body. For example:
for (i = 0; i < 4; i++)
when unrolled becomes:
The unrolled code may be larger in both source and object form but it will run without the overhead of incrementing and testing the loop variable and without paying the cost of all those taken jump instructions. The result is that while the code is larger it is also faster. It can be very much faster.
Inlining simply means that you should copy critical pieces of code "in line" in the body of the program instead of putting it in a subroutine. The reason for doing this is that the cost of a subroutine call can have a dramatic effect on execution time.
The last technique is to take advantage of 0 elements in the transformation matrix. Since multiplying by zero gives zero and adding zero gives the original value then if you know that an element in the matrix is zero you can just skip doing any multiplies and adds that involve that element and save the time that you would have used doing useless arithmetic. The trick is to take advantage of zeros without spending more time discovering them than you save by using them.
People have found several ways to make use of zeros in the transformation matrix. A standard observation is that the fourth column in a 4x4 matrix is often [0 0 0 1]. Actually, you can arrange it so that the fourth column is always [0 0 0 1] which can save you quite a bit. What follows is a fully unrolled 4x4 matrix multiply that doesn't assume there are any zeros:
The following is also a 4x4 matrix multiply but this time the assumption is made that the fourth column of each matrix is [0 0 0 1]:
a00 = (b00*c00) + (b01*c10) + (b02*c20) + (b03*c30)
a01 = (b00*c01) + (b01*c11) + (b02*c21) + (b03*c31)
a02 = (b00*c02) + (b01*c12) + (b02*c22) + (b03*c32)
a03 = (b00*c03) + (b01*c13) + (b02*c23) + (b03*c33)
a10 = (b10*c00) + (b11*c10) + (b12*c20) + (b13*c30)
a11 = (b10*c01) + (b11*c11) + (b12*c21) + (b13*c31)
a12 = (b10*c02) + (b11*c12) + (b12*c22) + (b13*c32)
a13 = (b10*c03) + (b11*c13) + (b12*c23) + (b13*c33)
a20 = (b20*c00) + (b21*c10) + (b22*c20) + (b23*c30)
a21 = (b20*c01) + (b21*c11) + (b22*c21) + (b23*c31)
a22 = (b20*c02) + (b21*c12) + (b22*c22) + (b23*c32)
a23 = (b20*c03) + (b21*c13) + (b22*c23) + (b23*c33)
a30 = (b30*c00) + (b31*c10) + (b32*c20) + (b33*c30)
a31 = (b30*c01) + (b31*c11) + (b32*c21) + (b33*c31)
a32 = (b30*c02) + (b31*c12) + (b32*c22) + (b33*c32)
a33 = (b30*c03) + (b31*c13) + (b32*c23) + (b33*c33)
The general case matrix multiply takes 64 multiplies and 48 adds. The special case only needs 36 multiplies and 27 adds.
a00 = (b00*c00) + (b01*c10) + (b02*c20)
a01 = (b00*c01) + (b01*c11) + (b02*c21)
a02 = (b00*c02) + (b01*c12) + (b02*c22)
a03 = 0
a10 = (b10*c00) + (b11*c10) + (b12*c20)
a11 = (b10*c01) + (b11*c11) + (b12*c21)
a12 = (b10*c02) + (b11*c12) + (b12*c22)
a13 = 0
a20 = (b20*c00) + (b21*c10) + (b22*c20)
a21 = (b20*c01) + (b21*c11) + (b22*c21)
a22 = (b20*c02) + (b21*c12) + (b22*c22)
a23 = 0
a30 = (b30*c00) + (b31*c10) + (b32*c20) + c30
a31 = (b30*c01) + (b31*c11) + (b32*c21) + c31
a32 = (b30*c02) + (b31*c12) + (b32*c22) + c32
a33 = 1
When transforming points the general case vector x matrix looks like:
The special case looks like:
a0 = (x*b00) + (y*b10) + (z*b20) + b30
a1 = (x*b01) + (y*b11) + (z*b21) + b31
a2 = (x*b02) + (y*b12) + (z*b22) + b32
a3 = (x*b03) + (y*b13) + (z*b23) + b33
Not quite as dramatic a change but saving 3 multiplies and 3 adds each time you transform a point can make a serious difference when you are transforming hundreds or thousands of points.
a0 = (x*b00) + (y*b10) + (z*b20) + b30
a1 = (x*b01) + (y*b11) + (z*b21) + b31
a2 = (x*b02) + (y*b12) + (z*b22) + b32
a3 = 1
Another way to make use of zeros is to restrict the kinds of transforms that you allow so that you know that specific elements in the body of the transformation matrix are always zero. Games like DOOM from Id Software let you translate your view point and rotate it around the Y axis, but they do not let you rotate around the X or Z axes. That is, they don't let you look up or down or stand on your head. These restrictions force several zeros in the matrix which lets you transform points much faster.
The demo contains code to measure how long it would take to transform 1,000,000 points using different techniques. In all cases the matrix multiply is fully unrolled.
When you run "demo trans" it prints 4 execution times. The first one measures how much time it takes to execute an empty loop 1,000,000 times. The second number is how long it takes to transform 1,000,000 points making a subroutine call for each point. The third time is the same test with the transform inlined. And, the fourth time is inlined, unrolled, and restricted so that only translation, scaling, and rotation about the Y axis are allowed. All times are in seconds.
On my 386/40 I get:
In "Graphics Gems II" there is a very interesting chapter by Jim Arvo on classifying transformation matrices by their zero structure. Knowing the zero structure lets you use a matrix multiply optimized for a specific pattern of zeros. I haven't tested the technique, but it looks like a winner.
Empty loop 0.276
(I've tried building a matrix multiplier that has an unrolled matrix multiply optimized for all possible zero structures... (Ok, it was late at night and I had had way to much coffee...) It was pretty easy to write a program that would write the code for me. But none of the compilers I have would compile a 1.5 megabyte .c file with a switch statement that had 65536 cases. Even when I trimmed it down to 4096 cases it still croaked all my compilers.)
A very interesting chapter in the original "Graphics Gems" written by Joseph M. Cychosz talks about efficient ways to build transformation matrices without doing full matrix multiplies. This work also takes advantage of zeros in the matrices to reduce the amount of arithmetic needed to multiply a transformation matrix of a known structure by a matrix of unknown structure. I've used this technique and it works very well.
The books "Graphics Gems," "Graphics Gems II," "Graphics Gems III," "Graphics Gems IV," and "Graphics Gems V." are well worth the cost to anyone learning graphics programming or anyone doing graphics software development.
"Graphics Gems", edited by Andrew Glassner, Academic Press 1990.
"Graphics Gems II", edited by Jim Arvo, Academic Press 1991.
"Principles of Interactive Computer Graphics", William M. Newman and Robert F. Sproull, McGraw-Hill 1979.
"Software Manual for the Elementary Functions", William J. Cody and William Waite, Prentice-Hall 1980.
"Computer Graphics, Principles and Practice", second edition, James D. Foley, Andries van Dam, Steven K. Feiner, and John F. Hughes, Addison-Wesley Publishing Company 1992.
"Numerical Analysis", second edition, Richard L. Burden, J. Douglas Faires, Albert C. Reynolds, Prindle, Weber & Schmidt 1981.
Copyright (C) 1993, 1996 Robert C. Pendleton. All rights reserved.