Back to school: 3D Vectors

Foreword
This (beginners) tutorial is written for Engine22 users, but also if you just want to know more about making games, engines, and 3D graphics. More tutorials will follow for sure, and can be found on Engine.Fuel22.net/Help/ (and at the time of writing, the link isn't up yet!). Have fun!



Remember Vectors from physics classes? Don't worry, neither do I. No but seriously, I'm a loser when it comes to math. Some parents say their kid is like a sponge. I'm more like bowling ball, without holes. If I don't apply it immediately, my brain does a toilet flush and the gained knowledge exits the body through farts or nose pickles.

“Apply it immediately”... what healthy 16 year old applies Pythagoras or Fresnel lens formula's for his daily problems? Only Newton's gravity law seems to apply sometimes, when falling drunk of your bike. E=MC2 in your face, stupid. So, there we are. In a classroom, thinking about the next part in Half life(1), staring at girls boobs, drawing idiotic doodles on a piece of paper, waiting until you can finally leave that sweaty moldy reeking place. Somewhere in the background, an old teacher with glasses is drawing "arrows" on the chalkboard. Whatever man.

But who would have thought that those arrows are called "Vectors", and that I would use them on a daily base, 17 years later? Just as realistic as using the leapfrog you learned at gymnastics.


As said, I'm just not that good at math so I won't be able to explain all the fine details about the math behind vectors. Then again, not being a professor, I can hopefully teach you a few things in human-language. While I'll put on my glasses and walk to the chalkboard, please stop chewing bubble-gum, stop doodling on your note block, and pay attention class. It's not that hard really, and if you want to make a game, you'd better stop watching your neighbour’s boobs, and focus on the chalkboard. Ahum:

Points

Earth is flat, and coordinates in space are made of 3 components: x,y and z. Don't believe me? Look at your finger. Move it to the left. Now it moved into a negative X direction; your finger's X-coordinate decreased. Move up. Your finger's Y coordinate increased. And how about that Z component then? Simple, just move your finger backwards, away from you. Z, not from Zorro but "depth", coordinate increased. From your perspective at least. In your bedroom door opening, your mom is looking at you now, and thinking what the hell you're doing. From her perspective, that finger moved into different directions, and she is about to call an ambulance.



So, a position, also called "point", can be defined as {x,y,z}. A 3-component struct in programming terms, like this:

// pseudo Engine22 notation

eVec3 = record

       x,y,z : eFloat;
end;

And a point can be in different "spaces". In general, we have "local space" (or "model space") and "world space". The finger example explained the coordinates, relative to your own position. Move it up, and Y increases. If you were the centre of the universe -as your mom always says-, your body would be at {x:0, y:0, z:0}. But you're not. For a Chinese on the other side of the world, your finger is going down instead of up.

That is... if the Earth was the centre of the universe. Which isn't either. In real-life, coordinate systems are always sort of local. It really depends what you take as a centre point, and also what direction is accounted for "forward", "up" and "left". If you hang upside-down like a bat the whole day, "up" might get a different definition.

Games are a bit easier. We just take a random point as the centre. Or well, random... On a GTA map, you may decide the bottom/left corner, at sea level, would be {0,0,0}. For Tower22, the centre of the ground floor is the centre of the "world". Game-world. When going up, Y goes higher. But there also games/3D programs that take Z as "up". It's arbitrary, and it doesn't really matter just as long your 3D models and programming math all accept the same rules.

In a game-context, "local" coordinates usually apply on/within a 3D model. So we make a large outdoor area. And we place assets in them. Cars, light-posts, garbage containers, trees, cats, et cetera. Each object gets a "world coordinate". However, earlier while creating these objects (say we were modelling a car), we didn't know if/where and how these objects are placed within the world. We just pretend the world is gone, and now the centre of the car itself becomes the "centre of the world". So, probably the handbrake would be somewhere near that centre. The front bumper, is placed forward from the centre, thus having a larger Z coordinate. The rear bumper, "behind" us, would get a negative Z coordinate.

Now when we place this car in our world, we can rotate it 180 degrees. In local-car-space, the handbrake is still the centre, and the bumper is still forward at a higher Z coordinate. But in world-coordinates, you'll get a whole different picture.

So, in practice, every object in your world gets an "absolute", world-coordinate. Physics,  movement, placement, and all that kind of stuff will be calculated with world-positions, world-vectors, and world-matrices (more about that later). But sometimes, you also want to do something in local-space. Or convert from local- to world space, or vice-versa. Example. Your NPC has a sniper rifle, and wants to blow off your head. That happens. Your "head" is a sub-part of your "player-body" object. The NPC knows your player world-coordinate. But next, the exact head-position depends on what the player is doing. If he is in "prone" pose, the head will be low at the ground. If he is licking his own balls, his head will be somewhere in the middle of the whole body-object. The body-object, which drives such animations, knows the local head position. In order to make a good shot, blowing of your head while you were walking on hands with your legs around your neck, the local head coordinate needs to be transformed to world space.

Jesus, that sounds hard. Yeah, it kinda is, and I won't explain the math now. But to do so, you'll need vector & matrix transformation calculations. Which are fortunately very common in any game engine, so you don’t have to reinvent the wheel. For now I'll just illustrate what can be done

with points. And I think... that I'm pretty much done. But to finish this story, let's throw some random (Engine22) code examples to get an idea:

                // Misc. Point code

                function makeYourPointPlease( const x,y,z : eFloat ) : eVec3;

                begin

                               result.x := x;

                               result.y := y;

                               result.z := z;

                end;

               

                procedure movePointToTheLeft( var pnt : eVec3; const distance : eFloat );

                begin

                               pnt.x := pnt.x - distance;

                end;

                              

                function distanceBetween2Points( const A, B : eVec3) : eFloat;

                var x,y,z : eFloat;

                begin

                          x      := A.x - B.x;

                          y      := A.y - B.y;

                          z      := A.z - B.z;

                         result := Sqrt(x*x+y*y+z*z);

                end;

Vectors

A point is just a... point, somewhere in a space. And you mainly need them for

- A.I. navigation

- Physics (movement, jumping, collision detection, ...)

- Tell video-card WHERE to draw

- Animations

Now to the Vector. In terms of programming, we can notate it the same. In fact, points and vectors are often the same struct or class in engines. In Engine22, a "vector" can be a vector(duh), but also a point or RGB colour. Anyhow, vectors can be visualized as arrows. Yes, that old man with glasses on the chalkboard wasn't fooling after all. A vector defines one or two things:

                - Direction: arrow point to the left, upwards, a bit backwards, ...

                - Force: The vector length. The longer the vector-arrow, the faster it moves into that direction

Note that "normalized" vectors do not define a Force or Strength, only a direction. A normalized

vector is made in such a way that the Force/Strength/Length is always 1. So a vector pointing to the left, would be noted as {x:-1; y:0 z:0}. An upwards vector would be {x:0 y:+1 z:0}. Polygon “Normals” (the direction its facing), is an example of a normalized vector.

The Matrix

Normalized vectors are used a lot for defining directions in drawing, physics, and rays in graphics(shaders). You see, objects usually don't only have a point, but also a certain direction. Looking from helicopter perspective, you could rotate a car 360 degrees. This angle can be encoded into a direction vector. Although you would need 2 more vectors for the full picture, as you could also roll and flip this car, in case it crashes or something. This is where Matrices are used for. A (4x4) matrix defines 5 things:

                turn / roll / pitch  rotation vectors

                position

                scale

Basically a matrix is constructed from multiple vectors. In Engine22, every asset has a matrix. When you place a new object into your world, you define its position, you can rotate it, and eventually scale (shrink / enlarge) it. In other words, you'll be adjusting its matrix while moving it around your mouse or keyboard.

This same matrix is fed into the physics system, so it knows where everything is, and also

shaders will get this matrix so they can position your object correctly on the screen (if visible

at all).

And back to Vectors

Matrix-math is quite a bit harder than vector-math, so let's spare that for another time, and continue this tutorial with some practical programming examples that show how these direction vectors can be used. Let's move bitch:

                procedure movePointUpwards( var targetPoint : eVec3; const unitsUp : eFloat );

                begin

                               targetPoint.y := targetPoint.y + unitsUp;

                               // Note it would move down if you give a negative number

                end;

                procedure movePoint( var targetPoint : eVec3; const forceVector : eVec3 );

                begin

                               // Assume the forceVector is NOT normalized,

                               // thus having a length/strength as well

                               targetPoint.x := targetPoint.x + forceVector.x;

                               targetPoint.y := targetPoint.x + forceVector.y;

                               targetPoint.z := targetPoint.x + forceVector.z;

                end;

               

                procedure movePoint( var targetPoint : eVec3; const direction : eVec3; const speed : eFloat );

                begin

                               // Direction vector is normalized here, multiply with speed

                               targetPoint.x := targetPoint.x + direction.x * speed;

                               targetPoint.y := targetPoint.x + direction.y * speed;

                               targetPoint.z := targetPoint.x + direction.z * speed;

                end;

Units in 3D space

So... if we move a point "20" to the left... then how far did it move exactly? My old physics teacher would outrage when writing a number without its corresponding unit. The mass of this block would be "10" mister. "10 what?! 10 ounce? 10 grams? 10 donkeys?!". "Kilograms mister". "Then say so". And of course, he was right. But still an asshole.

The guys at OpenGL or DirectX didn't listen very well to their teachers though; there is no

pre-defined unit for 3D coordinates. They basically say: "You figure out.". So, if we move

that point "20" to the left, we should decide ourselves if those are inches, meters,

nautical sea miles, or donkeys. In Engine22, "1" = "1 meter". But in another program it could

just as well be millimeters. So, it's important to pick 1 standard for your unit system, and eventually scale up/down imported models that come from a different 3D package. Lightwave also uses meters, but 3D Max centimetres if I'm not mistaking, so an imported model would be 100x bigger if you forget to downsize.

                procedure checkInput( );

                var movementSpeed : eFloat;

                begin

                               movementSpeed := 3.0;  // meters

                              

                               // Move our player with the arrows, over the X and Z axis

                               if ( keyboard.leftArrowPressed ) then

                                               movePoint( player.position,  vec3(-movementSpeed, 0,0 ) );

                               if ( keyboard.rightArrowPressed ) then

                                               movePoint( player.position,  vec3(+movementSpeed, 0,0 ) );

                               if ( keyboard.downArrowPressed ) then

                                               movePoint( player.position,  vec3(0,0, -movementSpeed ) );

                               if ( keyboard.upArrowPressed ) then

                                               movePoint( player.position,  vec3(0,0, +movementSpeed ) );

                end;

Stop. DeltaTime!

The procedure above has a little problem. Assuming that your game runs at 60 frames per second. And you'll be checking input every cycle. Thus "checkInput" would be called 60 times per second. Since we move "3.0 meters" every time, your player will go in warp-drive; holding the key for 1 second would move him 3.0 * 60 = 180 meters per second! That's a bit too much, don't you think.

Now we could simply reduce the "movementSpeed", to avoid super-human speeds. 0.03 for example would lead to 0.03 x 60 = 1.8 meters per second.

But this still stinks. What if you run your game on slower computer, that can only reach a lousy 20 FPS at times? 0.03 x 20 = 0.6 meters per second. In the worst case, your FPS fluctuates, because you're downloading internet porn at the same time, causing hitches. Or how about this? You'll buy a quantum computer that easily reaches 1000 FPS. Now suddenly your guy moves with 30 meters per second!

Ever tried old DOS games on an emulator? You may have noticed that some games seem to play in fast-forward. That's because they didn't guard their speeds. One trick is to introduce a maximum FPS. Another (better) trick is to calculate "DeltaTime". In Engine22, most "update()" functions will get a "DeltaSecs" argument with them; the elapsed time in seconds between the current and previous cycle. So if we're at a steady 60 FPS, DeltaSecs would be 1/60 = 0.0166667. Now check this:

                procedure checkInput( const deltaSecs : eFloat );

                var movementSpeed : eFloat;

                begin

                               movementSpeed := 3.0 * deltaSecs;     // meters PER SECOND

                              

                               // Move our player with the arrows, over the X and Z axis

                               if ( keyboard.leftArrowPressed ) then

                                               movePoint( player.position,  vec3(-movementSpeed, 0,0 ) );

                               if ( keyboard.rightArrowPressed ) then

                                               movePoint( player.position,  vec3(+movementSpeed, 0,0 ) );

                               if ( keyboard.downArrowPressed ) then

                                               movePoint( player.position,  vec3(0,0, -movementSpeed ) );

                               if ( keyboard.upArrowPressed ) then

                                               movePoint( player.position,  vec3(0,0, +movementSpeed ) );

                end;

Problem fixed. We multiply our original "3" with the elapsed time, so it gets a small number, 0.05 if the FPS was 60. And how much is 60 times 0.05? Exactly, 3. We moved 3 meters over a second.

Vectors in Engine22

You saw some of the basics. How to set a point, apply some movement, a bit about local versus world-space. There is a lot more you can do with vectors and matrices though. And therefore one of the basic elements each game-engine should have, is a Vector Library. And so does Engine22 of course. You already saw the "eVec3" type a few times above, which stands for "eVector3". That thing is being used all over the place, and has a bunch of handy help functions.

eVec3 is just a record with 3 floats. Note there is also an integer/byte/float32 variant, as well as a eVec4 type that has an extra W component. One of the reasons Engine22 won't work anymore in Delphi7, is because I made thankful use of record operators & functions, which wasn't possible in the old days. The means you can call functions and do math with vectors:

                var v1, v2, vResult : eVec3;

                begin

                               vResult := v1 + v2;           // Sums up (v1.x + v2.x, y+y, z+z)

                               vResult := v1 - v2;            // Subtracts (v1.x - v2.x, ...)

                               vResult := v1 * v2;           // Multiplies (not CROSS!)

                               vResult := vec3( 10, -4.2, 90);     // Constructor

                               vResult := v1.norm();    // Get normalized vector

                               string  := v1.toString();

                               vResult.fromString( "10 -4,2 90" );

                               x             := v1.dot( v2 );// dot product

                               distance:= v1.dist( v2 ); // Distance between v1 and v2

                               lookDir := v1.lookat( targetPosition );    // Gets direction towards target

                               length  := v1.len();          // Get vector length/strength

                               if ( v1 = v2 ) then vectorsAreEqual;

                               if ( v1 > v2 ) then v1IsStrongerThanV2; // Compare forces

                               ...

                end;

Another nice feature is that you can use the same vectors for RGB (or eVec4 for RGBA) colors. Instead of typing "vector.x", you can type "vector.r" as well. It does the exact same thing, but makes more sense in terms of writing. Vector colors are often used as input parameters for shaders and graphics. Note that 0 stands for black, and 1 for 255 (white) here, as we use a floating point notation instead of bytes.

And about 3D models, did you know that...

3D models (the stuff you make in 3D Max, Lightwave, Blender, Maya, ...) are basically just arrays of vectors? A polygon is made of (typically 3) "vertices". Where a vertex could have a position, a texture coordinate, a normal-direction, maybe a tangent, maybe a weight? So, writing it down:

                Vertex = record

                               position               : eVec3;

                               textureCoord    : eVec2;               // Only 2 coordinates (U and V)

                               normal                 : eVec3;

                               tangent                               : eVec3;

                               weight                 : eVec3;

                end;

               

                ModelMesh = class

                               vertexCount      : eInt;

                               vertices               : array of Vertex;

                end;

It's not exactly how the Engine22 VertexArray classes look like, but it's not far away from that either. When loading a 3D model, you'll be making arrays of vectors. When rendering the model, you'll be pushing those arrays to your videocard. Just keep that in mind.

It's not exactly how the Engine22 VertexArray classes look like, but it's not far away from that either. When loading a 3D model, you'll be making arrays of vectors. When rendering the model, you'll be pushing those arrays to your videocard. Just keep that in mind.

So much stuff you can do with them. The E22_Util.Base.Vector unit is one of the most common included units. And even if you're not a math genious, you'll get comfortable with them quickly.

To conclude, our deadly sniper headshot:

                procedure sniperHeadShot( targetEntity : eES_Entity );

                var         localHeadPos    : eVec3;

                               worldHeadPos : eVec3;

                               localGunPos      : eVec3;

                               worldGunPos    : eVec3;

                              

                        targetDirection : eVec3;

                        targetDistance  : eFloat;

                        bulletPos    : eVec3;

                        traveledDist              : eFloat;

                begin

                               // Get absolute head position

                               localHeadPos    := targetEntity.getLocalPoint( HEAD );

                               worldHeadPos  := targetEntity.getMatrix().localPointToAbsolute( localHeadPos );

                              

                               // Get our gun position

                               localGunPos      := sniper.getLocalPoint( GUN_NOZZLE );

                               worldGunPos    := sniper.getMatrix().localPointToAbsolute( localGunPos );

                              

                               // Get the (world) direction and distance between our gun and the head

                               worldGunPos.distAndDirectionTo( worldHeadPos, targetDirection, targetDistance );

                              

                               // Simulate bullet path

                               bulletPos            := worldGunPos; // Start here

                               traveledDist       := 0.0;                   // Traveled distance so far

                               while ( not collided ) and ( traveledDist < targetDistance ) do

                               begin   

                                               // Check if our path is clear

                                               collided := world.rayTrace( bulletPos, targetDirection, bulletSpeedMS * deltaSecs );

                                              

                                               // Move forward, with the speed of a bullet

                                               bulletPos := bulletPos + targetDirection * bulletSpeedMS * deltaSecs;

                                               traveledDist := traveledDist + bulletSpeedMS * deltaSecs;

                               end; // while

                              

                               if ( traveledDist >= targetDist ) then begin

                                               // Reached the target

                                               targetEntity.dieMotherFucker();

                               end;

                end;

               

Of course, that's bullocks code, but it does give an impression of how to work with vectors. The vector-related functions shown are in E22 at least.