Remeber your friend
Soh Cah Toa
Then you'll never see the end
Dont yah knowa
Sin() is not a sin
cos shows the spirt within
and when you want to know the hypontose
take the square and hand a noose
Pathagragram was a cool guy
too bad he had to die
so next a triangle has you all up in a bunch
open up a bag of chips and take a munch
--I'm a poet (and i dident know it)

hahahahhahahahhhaahhahahhahahhahahahaha

People always mention sohcahtoa or whatever it is but I never learned that one. I guess I learned a different one, which I haven't heard from other people:

Oscar Had A Heap Of Apples:

OH AH OA

sin = O/H
cos = A/H
tan = O/A

O = opposite edge
H = hypotenuse
A = adjacent edge

I found that when I first started programming.. 'soh cah toa' actually made things worse! I was trying to do the stuff that was already done for me, then add it here.. subtract it here... and voila! A big mess!

i guess i should learn vectors? point me in a good direction and ill go learn about em

closest i come is an angle and a velocity...

2d vector: a pair of values, eg. (x, y)
3d vector: a trio of values, eg. (x, y, z)

So all positions are vectors. Also all directions can be stored as vectors.

Given the points A and B, the direction from A to B is the vector B-A. Call this direction vector D. Then you can start doing basic algebra with them, eg. B = A+D (this is where the C++ vector class comes in handy as it lets you write such things directly).

The direction vector D contains both direction and distance information. You can extract the distance using Pythagoras:

dist = d.magnitude() = sqrt(d.x*d.x + d.y*d.y)

Once you have that, you can normalize d:

d.x /= dist;
d.y /= dist;

Or if you have good C++ classes:

d /= dist; or
d.normalize(); or
d = ~d;

This will give you a unit vector, which holds only direction information. It points in the same direction as the original D, but has length=1. Unit vectors are especially handy for using with dot products...

The dot product, a | b = a.x*b.x + a.y*b.y. If a and b are both unit vectors, this gives the cosine of the angle between them. If only a is a unit vector, this tells you how much of vector b lies along the direction of a. Which is a lot more useful than it might seem! It will be zero if the lines are at right angles, positive if they are in the same direction, negative if they are in opposite directions.

An example of a simple vector based physics system:

Vector pos(0, 0);
Vector vel(0, 0);
Vector dir(0, 0);

// add velocity
pos += vel;

// apply air resistance
vel *= 0.9;

// apply gravity
const Vector gravity(0, 0.1);
vel += gravity;

// apply thrust
if (firing thruster)
vel += dir;

// turn by angle
dir = Vector(dir.x*cos(ang) - dir.y*sin(ang),
dir.y*cos(ang) + dir.x*sin(ang));

Which is probably very similar to stuff you are already doing, as a lot of vector math is just common sense!

Let's add a bit of collision. Assume we have a wall defined by an infinitely long line (if the line has fixed ends the same principle applies but needs a few extra checks). You can store this line as a vector holding any arbitrary point on the line, and another unit vector holding the normal of the line (the normal is just a vector at 90 degrees to the direction of the line).

Say L is a point on the line, N is the normal, and you want to check for collision of a point P. Work out:

// direction from point to line
Vector d = L - P;

// dot the direction with the line normal
float dot = d | N;

// which side of the line are we on?
if (dot > 0)
// collision!

Ok, let's try bouncing off the line, rather than just stopping dead after we hit it. For that, we want to reverse the amount of our velocity that was heading towards the surface, but leave the amount of velocity that was sideways on to the surface alone. Really easy to do with dot products:

// how much velocity was towards the surface?
float dot = vel | N;

// recoil away from the surface by that amount
Vector recoil = dot * N;

// apply recoil. One of it will cancel out
// the movement towards the surface, and two
// of it makes us bounce back away from it.
vel -= recoil*2;

You can really easily extend this to support non-elastic collisions, where the surface absorbs some amount of your energy. Either change the last line above to:

vel -= recoil*1.5;

to absorb half the energy, or for a surface that also absorbs sideways motion:

vel = (vel-recoil*2) * 0.5;

The really great thing about vectors is that they work pretty much the same in 2d and 3d. Working with angles is ok but sometimes a bit complicated in 2d, but becomes a million times worse if you try to extend it into 3d, while vectors are equally simple in either.

Yup - I normally store the velocity, the position and the front vector. If you're in 3D then also a right vector. You can then work out an up vector by crossing the front and right.

Be sure that you keep your direction vectors normalised!

Cross products only really make sense in 3D:

vc is the cross product of v1 and v2:

vc.x = v1.y * v2.z - v1.z * v2.y
vc.y = v1.z * v2.x - v1.x * v2.z
vc.z = v1.x * v2.y - v1.y * v2.x

I think it is quite standard to use the ^ operator in C++ for this.

So, (v1 ^ v2) gives you the vector perpendicular to v1 and v2. This only works if v1 and v2 are both normalised.

A whole post of pettiness for you:

It depends what you want to do. Even in general case engines, you will usually [numerically] integrate acceleration to get velocity, and integrate that to get position. Note to those scared of maths - this :

vel += acc;
pos += vel;

is actually an approximate integration using the same simple observation as Euler made many centuries ago about graphs, but taking the 'step size', often called h, to be 1.

However, my childishness makes me point out - there is no point storing velocity if you don't intend to make your objects move, and if they are moving you are probably storing a list of the forces acting upon the body also. So thats something else you want to store.

wow, im really confused... id probably get it if you showe dme an example of these few things( all i do is 2-D im not concerned with 3d yet, all it does is confuse me :

if i have position vector A and velocity vector B, to move him would be A+B, the result of which is my new position?

how would i go about rotating a vector? say i have a sprite pointing straight up and i want him to rotate 128 (fixed numbers ) degress ?

how would i clip a vector agaisnt a vector? like tell where, and when a collsion occured?

what does normalizing do, why would i want to do it, and how is it done, and how does it effect the numbers stored in the vectors?

tehre wre more quations .. confusions i mean? that i may need answers for thanks for he hwelp though

I'll answer the easy one :-) Normalising is turning a vector into a unit vector, i.e. a vector of length 1. (divide each component of vector v with the magnitude of v) If that's wrong, I'll look like a fool.

This thread has been really useful. If I had read this last year maybe I would have paid attention in class.

wow, that stuff seems to be over my head

Thomas Harte writes:
> It depends what you want to do. Even in
> general case engines, you will usually
> [numerically] integrate acceleration to
> get velocity, and integrate that to get
> position.

But acceleration is often a constant, so it's not part of the data you'd bother storing. Even when it is variable, acceleration rarely has state that needs to persist from frame to frame, eg. in MotoGP the acceleration force is a function of the engine revs, torque, current gear, current speed, and so on.

> Note to those scared of maths - this :
>
> vel += acc;
> pos += vel;
>
> is actually an approximate integration

I hate the term 'integration' because it makes the issue seem far more complicated than it really is! It's pretty rare for you to need anything more than the simple Euler integration you give above, and although this is of course just a linear approximation of a proper calculus integration, it's a lot easier to visualise if you just think of it as adding a load of forces together!

> However, my childishness makes me point
> out - there is no point storing velocity
> if you don't intend to make your objects
> move

Very true :-)

----------------

Damyan Pepper writes:
> BTW, are you sure that cupid is frame rate
> compensated?

Haha.

(to explain: about halfway through Saturday George complained that the ship turned more slowly if you were near the bottom of the level. After trying to blame Damyan for it, it turned out that I had very carefully adjusted the render function to run at a fixed framerate, while the update just ran as fast as possible in whatever time was left. Oops!)

----------------

Peter Wang writes:
> This thread has been really useful. If
> I had read this last year maybe I would
> have paid attention in class.

Nobody seems to teach vectors properly. Or at least not in ways that make them seem useful for solving real world problems.

When someone says "a dot b gives the cosine of the angle between the two vectors, as long as they are unit vectors", that doesn't seem especially handy. When I first heard this I just filed it away as "I'll remember that in case I ever need to know the cosine of the angle between two unit vectors". And I never needed to know that.

It is only when you realise that dot product tells you how much of one vector lies in the direction of another that it starts to be useful. It turns out that about 90% of what you do in games and graphics can be expressed in terms of dot products, which is why things like the NVidia vertex shader processor or PS2 vector units are basically just really fast dot product evaluators!

When I first started working at Probe, I was doing some flags in the crowd for an N64 football game, and spent about three days working out loads of trig calls to rotate a polygon so it would always face towards the camera. I got it working in the end, but it required about 10 sin/cos calls, two tangents, one inverse tangent, multiple square roots, and a huge amount of head scratching. I couldn't decide whether to be pissed off or incredibly relieved when I started on ExtremeG, and Ash (the lead coder) explained how you can do the same thing using just a couple of dot products and one cross product!