There are many approaches to understanding the type of 3D math used in video +games, modelling, ray-tracing, etc. The usual is through vector algebra, matrices, and +linear transformations and, while they are not completely necesary to understand +most of the aspects of 3D game programming (from the theorical point of view), they +provide a common language to communicate with other programmers or +engineers. +

This tutorial will focus on explaining all the basic concepts needed for a +programmer to understand how to develop 3D games without getting too deep into +algebra. Instead of a math-oriented language, code examples will be given instead +when possible. The reason for this is that. while programmers may have +different backgrounds or experience (be it scientific, engineering or self taught), +code is the most familiar language and the lowest common denominator for +understanding. +

1.2 Vectors

1.2.1 Brief Introduction

When writing 2D games, interfaces and other applications, the typical convention is +to define coordinates as an x,y pair, x representing the horizontal offset and y the +vertical one. In most cases, the unit for both is pixels. This makes sense given the +screen is just a rectangle in two dimensions. +

An x,y pair can be used for two purposes. It can be an absolute position (screen +cordinate in the previous case), or a relative direction, if we trace an arrow from the +origin (0,0 coordinates) to it’s position. +

When used as a direction, this pair is called a vector, and two properties can be +observed: The first is the magnitude or length , and the second is the direction. In +two dimensions, direction can be an angle. The magnitude or length can be computed +by simply using Pithagoras theorem: +

The direction can be an arbitrary angle from either the x or y axis, and could be +computed by using trigonometry, or just using the usual atan2 function present in +most math libraries. However, when dealing with 3D, the direction can’t be described +as an angle. To separate magnitude and direction, 3D uses the concept of normal +vectors. +

1.2.2 Implementation

Vectors are implemented in Godot Engine as a class named Vector3 for 3D, and as +both Vector2, Point2 or Size2 in 2D (they are all aliases). They are used for any +purpose where a pair of 2D or 3D values (described as x,y or x,y,z) is needed. This is +somewhat a standard in most libraries or engines. In the script API, they can be +instanced like this: + +

a = Vector3()
a = Vector2( 2.0, 3.4 ) +

Vectors also support the common operators +, -, / and * for addition, +substraction, multiplication and division. + +

a = Vector3(1,2,3)
b = Vector3(4,5,6)
c = Vector3()

// writing

c = a + b

// is the same as writing

c.x = a.x + b.x
c.y = a.y + b.y
c.z = a.z + b.z

// both will result in a vector containing (5,7,9).
// the same happens for the rest of the operators. +

Vectors also can perform a wide variety of built-in functions, their most common +usages will be explored next. +

1.2.3 Normal Vectors

Two points ago, it was mentioned that 3D vectors can’t describe their direction as an +agle (as 2D vectors can). Because of this, normal vectors become important for +separating a vector between direction and magnitude. +

A normal vector is a vector with a magnitude of 1. This means, no matter where +the vector is pointing to, it’s length is always 1. +

Normal vectors have endless uses in 3D graphics programming, so it’s +recommended to get familiar with them as much as possible. +

1.2.4 Normalization

Normalization is the process through which normal vectors are obtained +from regular vectors. In other words, normalization is used to reduce the +magnitude of any vector to 1. (except of course, unless the vector is (0,0,0) +). +

To normalize a vector, it must be divided by its magnitude (which should be +greater than zero): + +

// a custom vector is created
a = Vector3(4,5,6)
// ’l’ is a single real number (or scalar) containight the length
l = Math.sqrt( a.x*a.x + a.y*a.y + a.z*a.z )
// the vector ’a’ is divided by its length, by performing scalar divide
a = a / l
// which is the same as
a.x = a.x / l
a.y = a.y / l
a.z = a.z / l + +

a = Vector3(4,5,6)
a.normalize() // in-place normalization
b = a.normalized() // returns a copy of a, normalized +

1.2.5 Dot Product

The dot product is, pheraps, the most useful operation that can be applied to 3D +vectors. In the surface, it’s multiple usages are not very obvious, but in depth it can +provide very useful information between two vectors (be it direction or just points in +space). +

The dot product takes two vectors (a and b in the example) and returns a scalar +(single real number): +

a_xb_x + a_yb_y + a_zb_z +

a = Vector3(...)
b = Vector3(...)

c = a.x*b.x + a.y*b.y + a.z*b.z

// using built-in dot() function

c = a.dot(b) +

1.2.6 Cross Product

The cross product also takes two vectors a and b, but returns another vector c that is +orthogonal to the two previous ones. +

c_x = a_xb_z - a_zb_y +

c_y = a_zb_x - a_xb_z +

c_z = a_xb_y - a_yb_x +

a = Vector3(...)
b = Vector3(...)
c = Vector3(...)

c.x = a.x*b.z - a.z*b.y
c.y = a.z*b.x - a.x*b.z
c.z = a.x*b.y - a.y*b.x

// or using the built-in function

c = a.cross(b) +

1.3 Plane

1.3.1 Theory

A plane can be considered as an infinite, flat surface that splits space in two halves, +usually one named positive and one named negative. In regular mathematics, a plane +formula is described as: +

ax + by + cz + d +

However, in 3D programming, this form alone is often of little use. For planes to +become useful, they must be in normalized form. +

A normalized plane consists of a normal vector n and a distance d. To normalize +a plane, a vector n and distance d’ are created this way: +

In any case, normalizing planes is not often needed (this was mostly for +explanation purposes), and normalized planes are useful because they can be created +and used easily. +

A normalized plane could be visualized as a plane pointing towards normal n, +offseted by d in the direction of n. +

In other words, take n, multiply it by scalar d and the resulting point will be part +of the plane. This may need some thinking, so an example with a 2D normal vector +(z is 0, so plane is orthogonal to it) is provided: +

1.3.2 Implementation

//creates a plane with normal (0,1,0) and distance 5
p = Plane( Vector3(0,1,0), 5 )
// get the distance to a point
d = p.distance( Vector3(4,5,6) ) +

1.4 Matrices, Quaternions and Coordinate Systems

It is very often needed to store the location/rotation of something. In 2D, it is often +enough to store an x,y location and maybe an angle as the rotation, as that should +be enough to represent any posible position. +

In 3D this becomes a little more difficult, as there is nothing as simple as an angle +to store a 3-axis rotation. +

The first think that may come to mind is to use 3 angles, one for x, one for y and +one for z. However this suffers from the problem that it becomes very cumbersome to +use, as the individual rotations in each axis need to be performed one after another +(they can’t be performed at the same time), leading to a problem called “gimbal +lock”. Also, it becomes impossible to accumulate rotations (add a rotation to an +existing one). +

To solve this, there are two known diferent approaches that aid in solving +rotation, Quaternions and Oriented Coordinate Systems. +

1.4.1 Oriented Coordinate Systems

Oriented Coordinate Systems (OCS) are a way of representing a coordinate system +inside the cartesian coordinate system. They are mainly composed of 3 Vectors, one +for each axis. The first vector is the x axis, the second the y axis, and the third is the + +z axis. The OCS vectors can be rotated around freely as long as they are kept the +same length (as changing the length of an axis changes its cale), and as long as they +remain orthogonal to eachother (as in, the same as the default cartesian system, +with y pointing up, x pointing left and z pointing front, but all rotated +together). +

Oriented Coordinate Systems are represented in 3D programming as a 3x3 matrix, +where each row (or column, depending on the implementation) contains one of the +axis vectors. Transforming a Vector by a rotated OCS Matrix results in the rotation +being applied to the resulting vector. OCS Matrices can also be multiplied to +accumulate their transformations. +

//create a 3x3 matrix
m = Matrix3()
//rotate the matrix in the y axis, by 45 degrees
m.rotate( Vector3(0,1,0), Math.deg2rad(45) )
//transform a vector v (xform method is used)
v = Vector3(...)
result = m.xform( v ) +

However, in most usage cases, one wants to store a translation together with the +rotation. For this, an origin vector must be added to the OCS, thus transforming it +into a 3x4 (or 4x3, depending on preference) matrix. Godot engine implements this +functionality in the Transform class: + +

t = Transform()
//rotate the transform in the y axis, by 45 degrees
t.rotate( Vector3(0,1,0), Math.deg2rad(45) )
//translate the transform by 5 in the z axis
t.translate( Vector3( 0,0,5 ) )
//transform a vector v (xform method is used)
v = Vector3(...)
result = t.xform( v ) +

Transform contains internally a Matrix3 “basis” and a Vector3 “origin” (which can +be modified individually). +

1.4.2 Transform Internals

Internally, the xform() process is quite simple, to apply a 3x3 transform to a vector, +the transposed axis vectors are used (as using the regular axis vectors will result on +an inverse of the desired transform): + +

m = Matrix3(...)
v = Vector3(..)
result = Vector3(...)

x_axis = m.get_axis(0)
y_axis = m.get_axis(1)
z_axis = m.get_axis(2)

result.x = Vector3(x_axis.x, y_axis.x, z_axis.x).dot(v)
result.y = Vector3(x_axis.y, y_axis.y, z_axis.y).dot(v)
result.z = Vector3(x_axis.z, y_axis.z, z_axis.z).dot(v)

// is the same as doing

result = m.xform(v)

// if m this was a Transform(), the origin would be added
// like this:

result = result + t.get_origin() +

1.4.3 Using The Transform

So, it is often desired apply sucessive operations to a transformation. For example, +let’s a assume that there is a turtle sitting at the origin (the turtle is a logo reference, + +for those familiar with it). The y axis is up, and the the turtle’s nose is pointing +towards the z axis. +

The turtle (like many other animals, or vehicles!) can only walk towards the +direction it’s looking at. So, moving the turtle around a little should be something +like this: + +

// turtle at the origin
turtle = Transform()
// turtle will walk 5 units in z axis
turtle.translate( Vector3(0,0,5) )
// turtle eyes a lettuce 3 units away, will rotate 45 degrees right
turtle.rotate( Vector3(0,1,0), Math.deg2rad(45) )
// turtle approaches the lettuce
turtle.translate( Vector3(0,0,5) )
// happy turtle over lettuce is at
print(turtle.get_origin()) +

As can be seen, every new action the turtle takes is based on the previous one it +took. Had the order of actions been different and the turtle would have never reached +the lettuce. +

Transforms are just that, a mean of “accumulating” rotation, translation, scale, +etc. +

1.4.4 A Warning about Numerical Precision

Performing several actions over a transform will slowly and gradually lead to +precision loss (objects that draw according to a transform may get jittery, bigger, +smaller, skewed, etc). This happens due to the nature of floating point numbers. if +transforms/matrices are created from other kind of values (like a position and +some angular rotation) this is not needed, but if has been accumulating +transformations and was never recreated, it can be normalized by calling the +.orthonormalize() built-in function. This function has little cost and calling it every +now and then will avoid the effects from precision loss to become visible. + + + + + -- cgit v1.2.3-70-g09d2

1 Introduction to 3D Math

1.1 Introduction