This month, we begin writing a computer program to simulate the physics
of racing. Such a program is quite an ambitious one. A simple racing video game, such as
``Pole Position,'' probably took an expert programmer several months to write. A big,
realistic game like ``Hard Drivin''' probably took three to five people more than a year
to create. The point is that the topic of writing a racing simulation is one that we will
have to revisit many times in these articles, assuming your patience holds out. There are
many `just physics' topics still to cover too, such as springs and dampers, transients,
and thermodynamics. Your author hopes you will find the computer programming topic an
enjoyable sideline and is interested, as always, in your feedback. We will use a
computer programming language called Scheme. You have probably encountered BASIC, a
language that is very common on personal computers. Scheme is like BASIC in that it is *interactive*.
An interactive computer language is the right kind to use when inventing a program as you
go along. Scheme is better than BASIC, however, because it is a good deal simpler and also
more powerful and modern. Scheme is available for most PCs at very modest cost (MIT Press
has published a book and diskette with Scheme for IBM compatibles for about $40; I have a
free version for Macintoshes). I will explain everything we need to know about Scheme as
we go along. Although I assume little or no knowledge about computer programming on your
part, we will ultimately learn some very advanced things.
The first thing we need to do is create a *data structure* that contains the
mathematical state of the car at any time. This data structure is a block of computer
memory. As simulated time progresses, mathematical operations performed on the data
structure simulate the physics. We create a new instance of this data structure by typing
the following on the computer keyboard at the Scheme prompt:
`
(new-race-car)`
This is an example of an *expression*. The expression includes the parentheses.
When it is typed in, it is `evaluated` immediately. When we say that Scheme is an
interactive programming language, we mean that it evaluates expressions immediately. Later
on, I show how we *define* this expression. It is by defining such expressions that
we write our simulation program.
Everything in Scheme is an expression (that's why Scheme is simple). Every expression
has a value. The value of the expression above is the new data structure itself. We need
to give the new data structure a name so we can refer to it in later expressions:
`
(define car-161 (new-race-car))`
This expression illustrates two Scheme features. The first is that expressions can
contain sub-expressions inside them. The inside expressions are called *nested*.
Scheme figures out which expressions are nested by counting parentheses. It is partly by
nesting expressions that we build up the complexity needed to simulate racing. The second
feature is the use of the special Scheme word `define`. This causes the immediately
following word to become a stand-in synonym for the value just after. The technical name
for such a stand-in synonym is *variable*. Thus, the expression `car-161`,
wherever it appears after the `define` expression, is a synonym for the data
structure created by the nested expression `(new-race-car)`.
We will have another data structure (with the same format) for `car-240`,
another for `car-70`, and so on. We get to choose these names to be almost anything
we like . So, we would
create all the data structures for the cars in our simulation with expressions like the
following:
`
(define car-161 (new-race-car))
(define car-240 (new-race-car))
(define car-70 (new-race-car))`
The state of a race car consists of several numbers describing the physics of the car.
First, there is the car's position. Imagine a map of the course. Every position on the map
is denoted by a pair of coordinates, and . For elevation changes, we add a height
coordinate, . The
position of the center of gravity of a car at any time is denoted with expressions such as
the following:
`
(race-car-x car-161)
(race-car-y car-161)
(race-car-z car-161)`
Each of these expressions performs *data retrieval* on the data structure `car-161`.
The value of the first expression is the coordinate of the car, *etc.*
Normally, when running the Scheme interpreter, typing an expression simply causes its
value to be printed, so we would see the car position coordinates printed out as we typed.
We could also store these positions in another block of computer memory for further
manipulations, or we could specify various mathematical operations to be performed on
them.
The next pieces of state information are the three components of the car's velocity.
When the car is going in any direction on the course, we can ask ``how fast is it going in
the direction,
ignoring its motion in the and directions?'' Similarly, we want to know how
fast it is going in the direction, ignoring the and directions, and so on. Decomposing an
object's velocity into separate components along the principal coordinate directions is
necessary for computation. The technique was originated by the French mathematician
Descartes, and Newton found that the motion in each direction can be analyzed
independently of the motions in the other directions at right angles to the first
direction.
The velocity of our race car is retrieved via the following expressions:
`
(race-car-vx car-161)
(race-car-vy car-161)
(race-car-vz car-161)`
To end this month's article, we show how velocity is computed. Suppose we retrieve the
position of the car at simulated time and save it in some variables, as follows:
`
(define x1 (race-car-x car-161))
(define y1 (race-car-y car-161))
(define z1 (race-car-z car-161))`
and again, at a slightly later instant of simulated time, :
`
(define x2 (race-car-x car-161))
(define y2 (race-car-y car-161))
(define z2 (race-car-z car-161))`
We have used `define` to create some new variables that now have the values of
the car's positions at two times. To calculate the average velocity of the car between the
two times and store it in some more variables, we evaluate the following expressions:
`
(define vx (/ (- x2 x1) (- t2 t1))) (define vy (/ (- y2 y1) (- t2 t1)))
(define vz (/ (- z2 z1) (- t2 t1)))`
The nesting of expressions is one level deeper than we have seen heretofore, but these
expressions can be easily analyzed. Since they all have the same form, it suffices to
explain just one of them. First of all, the `define` operation works as before,
just creating the variable `vx` and assigning it the value of the following
expression. This expression is
`
(/ (- x2 x1) (- t2 t1))`
In normal mathematical notation, this expression would read and in most computer languages, it would look
like this:
`
(x2 - x1) / (t2 - t1)`
We can immediately see this is the velocity in the direction: a change in position divided by
the corresponding change in time. The Scheme version of this expression looks a little
strange, but there is a good reason for it: consistency. Scheme requires that all
operations, including everyday mathematical ones, appear in the first position in a
parenthesized expression, immediately after the left parenthesis. Although consistency
makes mathematical expressions look strange, the payback is simplicity: all expressions
have the same form. If Scheme had one notation for mathematical expressions and another
notation for non-mathematical expressions, like most computer languages, it would be more
complicated. Incidentally, Scheme's notation is called Polish notation. Perhaps you have
been exposed to Hewlett-Packard calculators, which use reverse Polish, in in which the
operator always appears in the *last* position. Same idea, and advantages, as
Scheme, only reversed.
So, to analyze the expression completely, it is a division expression
`
(/ ...)`
whose two arguments are nested subtraction expressions
`
(- ...) (- ...)`
The whole expression has the form
`
(/ (- ...) (- ...))`
which, when the variables are filled in, is
`
(/ (- x2 x1) (- t2 t1))`
After a little practice, Scheme's style for mathematics becomes second nature and the
advantages of consistent notation pay off in the long run.
Finally, we should like to store the velocity values in our data structure. We do so as
follows:
`
(set-race-car-vx! car-161 vx)
(set-race-car-vy! car-161 vy)
(set-race-car-vz! car-161 vz)`
The `set` operations change the values in the data structure named `car-161`.
The exclamation point at the end of the names of these operations doesn't do anything
special. It's just a Scheme idiom for operations that change data structures. |