Writing Games in Perl - Part 6 - Math for dummies

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Following
posts 1,
2,
3,
4 and
5 on
the subject of writing games in Perl, now we are going to fix the math
in the game.

In the first post, I used a very naive simplification of the
movement calculation. I simply considered that the velocity was
constant during the time of the frame and recalculated the final
velocity after the frame so it would affect the next calculation.

I have to confess that I didn't do it just for the simplification
of the code. I did it because of my lack of good understanding of
math. Some people have noticed that I should've used
a Runge-Kutta
method to solve the problem, but, honestly, the math language is
something that really requires a level of practice I simply don't have
(I've been working on Information Systems for 12 years, now it's the
first time I really miss calculus knowledge).

The problem I was trying to solve is: Considering I have a ball
that is falling at a speed of 3 m/s with a gravity of 9.8 m/s², how
far would it fall after 25 miliseconds (about 40 FPS). I'm strongly
visually-oriented, so let me try to represent in some ascii-art what I
was trying to find out.


position | .
| I
| .
|
| .
|
|
| F
|
|
| .
0-------------------------
time

I was considering I had defined the position I (initial) and I
wanted to know which was the position F (final).

It was only after I shared the problem with Edilson (a colleague
that works in the same place as I do), and after he present me a sheet
full of math calculations which I simply ignored, since I couldn't
understand, and then he said me: "You're looking at the wrong graphic,
this graphic is derived from another graphic, which is velocity vs
time".

This was a very important realization for me, bear with me: Let's
simplify the problem a bit, let's consider we have a constant
velocity. The graphic of velocity vs time would be something like:


velocity |
|
|
|
|
|
|.......I..........F.....
|
|
|
|
0-------------------------
time

You probably remember that in order to find out how much an object
moved in a given time-frame, the formula would be:


ΔS = Δt * v

As I said before, I'm a very visually-oriented person, and at that
point I figured out the following:


velocity |
|
|
|
|
|
| I..........F
| | |
| | |
| | |
| | |
0-------------------------
time

Wait, that's a rectangle, its width is Δt and it's height is v,
so the distance travelled is the area of the rectangle.

WAIT! That's the definition of Integral I've been reading in math
books for a while and that never really meant anything to me because
of all the math blabbering that really require consistent math
practice to actually understand anything.

So now that I feel a lot less dumb, let's proceed to the problem at
hand. The velocity in our game is lineary-variable, which means that
its graphic over time will look like:


velocity | .
| .
| .
| F
| .
| .
| .
| I
| .
|.
|
0-------------------------
time

The intial grahic on the position over time at the beggining of
this post is derived from this graphic -- and this is actually the
meaning of derivative -- so the distance travelled in a given time
frame is the area of the trapezoid representing that time frame:


velocity |
|
|
| F
| . |
| . |
| . |
| I |
| | |
| | |
| | |
0-------------------------
time

So, the answer to my initial question is just a matter of
calculating that area:


Δs = ((vI + vF) * Δt)/2

It looks pretty easy now, and, in fact, I feel quite dumb for
taking so long to realize that. But anyway, that is probably all the
required math for a lot of games. I hope I wasn't the only one who had
a hard time understading all that, and, anyway, now I can start to
understand more complex integral and derivative calculations.

So, let's apply that to the code in our game, which happens to be
at the Ball.pm file.


sub time_lapse {
my ($self, $old_time, $new_time) = @_;
my $elapsed = ($new_time - $old_time)/1000; # convert to seconds...

my $vf_h = $self->vel_h + $self->acc_h * $elapsed;
my $vf_v = $self->vel_v + ($self->acc_v - g) * $elapsed;

my $ds_h = (($self->vel_h + $vf_h) * $elapsed) / 2;
my $ds_v = (($self->vel_v + $vf_v) * $elapsed) / 2;

$self->vel_h($vf_h);
$self->vel_v($vf_v);
$self->cen_h($self->cen_h + $ds_h);
$self->cen_v($self->cen_v + $ds_v);
}

I also fixed the code in the main loop that was re-calculating that
instead of calling time_lapse.


foreach my $wall (@{$self->walls}) {
if (my $coll = collide($ball, $wall, $frame_elapsed_time)) {
# need to place the ball in the result after the bounce given
# the time elapsed after the collision.
$ball->time_lapse($oldtime, $oldtime + (($coll->time)*1000) - 1);

if (defined $coll->axis &&
$coll->axis eq 'x') {
$ball->vel_h($ball->vel_h * -1);
} elsif (defined $coll->axis &&
$coll->axis eq 'y') {
$ball->vel_v($ball->vel_v * -1);
} elsif (defined $coll->axis &&
ref $coll->axis eq 'ARRAY') {
my ($xv, $yv) = @{$coll->bounce_vector};
$ball->vel_h($xv);
$ball->vel_v($yv);
} else {
warn 'BAD BALL!';
$ball->vel_h($ball->vel_h * -1);
$ball->vel_v($ball->vel_v * -1);
}
return $self->handle_frame($oldtime + ($coll->time*1000), $now);
}
}

I'm not going to post any video for this post, since there's no
visual difference. But I hope the ascii-art graphics are good enough.

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