Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”)

1. Maths - Vectors
Like many mathematical concepts, vectors can be understood and investigated in
different ways.
There are two main complimentary ways to look at vectors:
• Algebraic - Treats a vector as set of scalar values as a single entity with
addition, subtraction and scalar multiplication which operate on the whole
vector.
• Geometric - A vector represents a quantity with both magnitude and
direction.
Geometric Properties
A vector is a quantity with both magnitude and direction, there are two operations
defined on vectors and these both have a very direct geometric interpretation. We
draw a vector as a line with an arrow, for now I will call the end without the arrow
the 'start' of the vector and the end with the arrow the 'end' of the vector.
• Vector addition: to add two vectors we take the start of the second vector
and move it to the end of the first vector. The addition of these two vectors
is the vector from the start of the first vector to the end of the second
vector.
• Scalar multiplication changes the length of a vector without changing its
direction. That is we 'scale' it by the multiplying factor. So scalar
multiplication involves multiplying a scalar (single number) by a vector to
give another number.
We can think of these two operations: vector addition and scalar multiplication as
defining a linear space (see Euclidean space).

2. So how do we get vectors in the first place? We could assume a pre-existing
coordinate system and define all our vectors in this coordinate system, or we could
start with a set of basis vectors and represent the vectors as a linier combination of
these basis vectors, that is by scalar multiplication and addition of the basis vectors
we can produce any vector in the space provided that:
• There are as many basis vectors as these are dimensions in the space.
• The basis vectors are all independent (no more than two are in any given
plane).
So any point could be identified by:
α Va + β Vb
where:
• α, β = scalar multipliers
• Va, Vb = basis vectors.
So the two scalar multipliers (α, β) can represent the position of the point in terms
of our basis vectors. This leads to a way to work with vectors in a purely algebraic
way.
Algebraic Properties
The algebraic approach and its operations are explained on this page so here we
will just give an overview.
We can think of a vector as being like the concept of an array in a computer
language, for instance,
• Vectors have a size which is the number of elements in the array.
• All elements in the vector must be of the same type.

3. 5.6
9.3
3.5
7.0
The vector may be shown as a single column
8.4 1.8 5.5 6.2 or as a row
However there is a difference from a computer array because, in the computer
case, the elements of the array can be any valid objects provided they are all of the
same type. In the case of vectors the elements must have certain mathematical
properties, in particular they must have the operations of addition and
multiplication defined on them with certain properties. The properties required of
the elements of the vector are that they must form a mathematical structure known
as a field (see box on right). In mathematical terminology this is known as a vector
over a field, in other words a vector whose elements are fields.
operation notation explanation
addition V(a+b) = V(a) + V(b)
the addition of two vectors is done by adding the
corresponding elements of the two vectors.
scalar
multiplication
V(s*a) = s * V(a)
a scalar product of a vector is done by multiplying the scalar
product with each of its terms individually.
These operations interact according to the distributivity property:
s*(b+c)=s*b+s*c
Which gives the vectors a linear property. We can now put together a set of axioms
for vectors:
axiom addition scalar multiplication
associativity (a+b)+c=a+(b+c) (s1 s2) a = s1 (s2 a)
commutativity a+b=b+a
distributivity
s*(b+c)=s*b+s*c
(s1+s2)*a=s1*a+s2*a
identity a+0 = a 1 a = a

4. 0+a = a
inverses
a+(-a) = 0
(-a)+a = 0
Where:
• a,b,c are vectors
• 0 is identity vector
• s,s1,s2 are scalars
• 1 is the identity scalar
Contrast this with the axioms for a field (on this page)
Vectors may also have additional structure defined in terms of other
multiplications defined on them such as the dot and cross products as we shall see
later. These are optional, the only compulsory operations are addition and scalar
multiplication.
Vector Notation
So far we have shown a vector as a set of values in a grid as this is more
convenient on an html web page but the usual notation for a vector is to put the
values in square brackets:
Where:
• x = the component of in the x dimension.
• y = the component of in the y dimension.
• z = the component of in the z dimension.
Sometimes, when we represent the whole vector as a symbol, we may put an arrow
above the symbol (in this case v) to emphasise that it is a vector.
Or an alternative we can use the following notation:
= x i + y j + z k

5. Where:
• i = a unit vector in the x dimension.
• j = a unit vector in the y dimension.
• k = a unit vector in the z dimension.
The first form is more convenient when working with matrices, whereas the second
form is easier to write in text form.
Relationship to other mathematical quantities
We can extend the concept of vectors (usually by bolting on extra types of
multiplication to add to the built-in addition and scalar multiplication) to form
more complex mathematical structures, alternatively we might think of vectors as
subsets of these structures, for instance:
• As a subset of a matrix or tensor (1 by n, or n by 1 matrix). A matrix is a two
dimensional array with the dot product.
• As a subset of multivectors (Clifford algebra). For example complex
numbers are two element vectors with a certain type of multiplication
added.
What we cannot do is have a vector whose elements are themselves vectors. This is
because the elements of the vector must be a mathematical structure known as a
'field' and a vector is not itself a field because it does not necessarily have
commutative multiplication and other properties required for a field.
Still it would be nice if we could construct a matrix from a vector (drawn as a
column) whose elements are themselves vectors (drawn as a row) :
6 1 7 5
8 4 4 2
2 0 6 9
1 3 0 3
In order to create a matrix by compounding vector like structures we need to do
two things to the 'inner vector':
• We need to take the transpose so that it is a row rather than a column.

6. • We need a multiplication operation which will make it a field.
To do this we create the 'dual' of a vector, this is called a covector as described
on this page.
Vectors can be multiplied by scalars even though they are separate entities, vectors
and scalars can't be added for instance (not until we get to clifford algebra), but we
can define a type of multiplication called scalar multiplication usually denoted by
'*' or the scalar may be written next to the vector with the multiplication implied.
This type of multiplication takes one vector and one scalar. Scalar multiplication
multiplies the magnitude of the vector, but does not change its direction, so:
if we have,
vOut = 2*vIn
where:
• vOut and vIn are vectors
then, vOut will be twice the magnitude of vIn but in the same direction.
Quadratic structure on a linear space
However these linear properties are not enough, on their own, to define the
properties of Euclidean space using algebra alone. To be able to define concepts
like distance and angle we must define aquadratic structure.
For instance pythagoras:
r2
= x2
+ y2
+ z2
in algebraic terms,
if a is a three dimensional vector with bases e1, e2, e3
a = a1 e1 + a2 e2 + a3 e3
so,
a•a = a1
2
+ a2
2
+ a3
2
Other Vector Algebras
In the vector algebra discussed already the square of a vector is always a positive
number:

7. a x a = 0
a • a = positive scalar number
However, we could define an equally valid and consistent vector algebra which
squares to a negative number:
a × a = 0
a • a = negative scalar number
We could also define an algebra where we mix the dimensions, some square to
positive, some square to negative. An example of this is Einsteinean space-time,
space and time dimensions square to different values, if space squares to positive
then time squares to negative and visa-versa.
Applications of Vectors
For 3D programming (the subject matter of this site) we are mainly concerned with
vectors of 2 or 3 numbers.
A vector of dimension 3 can represent a physical quantity which is directional such
as position,velocity , acceleration, force, momentum, etc.
For example if the vector represents a point in space, these 3 numbers represent the
position in the x, y and z coordinates (see coordinate systems). Where x, y and z
are mutually perpendicular axis in some agreed direction and units.
A 3 dimensional vector may also represent a displacement in space, such as a
translation in some direction. In the case of the Java Vecmath library these are two
classes: Point3f and Vector3f both derived from Tuple3f. (Note these use floating
point numbers, there are also classes, ending in d, which contain double values).
The Point3f class is used to represent absolute points and the Vector3f class
represents displacement. In most cases the behavior of these classes is the same, as
far as I know the difference between these classes is when they are transformed by
a matrix Point3f will be translated by the matrix but Vector3f wont.
Here we are developing the following classes to hold a vector and encapsulate the
operations described here,
• sfvec2f for 2D vectors
• sfvec3f for 3D vectors
It would be possible to build a vector class that could hold a vector of any
dimension but a variable dimension class would be less efficient. Since we are
concerned with objects in 3D space it is more important to handle 2D and 3D
vectors efficiently.

8. Other vector quantities
• Normals
• Rays
Alternative interpretation of vectors
Upto now we have thought of the vector as the position on a 2,3 or n dimensional
grid. However for some physical situations there may not be a ready defined
Cartesian coordinate system. An alternative might be to represent the vector as a
linear combination of 3 basis:
σ1
σ2
σ3
These basis don't have to be mutually perpendicular (although in most cases they
probably will be) however they do have to be independent of each other, in other
words they should not be parallel to each other and all 3 should not be in the same
plane.
So a vector in 3 dimensions can be represented by [a,b,c] where a,b and c represent
the scaling of the 3 basis to make the vector as follows:
a σ1 + b σ2 + c σ3
Note that if this vector represents position then it will be a relative position, i.e.
relative to some other point, if we want to define an absolute point we still have to
define an origin.
So the problem remains of how to define the basis, there may be some natural
definition of these in the problem domain. Alternatively we could define the basis
themselves as 3D vectors using a coordinate system. But why bother to do this, if
we have a coordinate system why not just represent the vectors in this coordinate
system? Well we might want to change coordinate systems or translate all the
vectors in some way (see here). For example, we might want to represent points on
a solid object in some local coordinate system, but the solid object may itself be
moving relative to some absolute coordinate system.
Further Reading
Vectors can be manipulated by matrices, for example translated, rotated, scaled,
reflected.

9. There are mathematical objects known a multivectors, these can be used to do
many of the jobs that vectors do, but they don't have some of the limitations (for
example vector cross product is limited to 3 dimensions and does not have an
inverse).
There is also a more general family of algebras (less constrains) than 'vector spaces
over a field' these are 'modules over a ring'.
Field
The term 'field' has two different meanings:
In Geometry and Physics
A 'vector or scalar field' in geometry and physics is a vector or scalar quantity
whose value is a function of its position in a space (described on this page) .
In Algebra
An algebraic structure with two operations, addition and multiplication (described
on this page) a field has the following axioms.
axiom addition multiplication
associativity (a+b)+c=a+(b+c) (a*b)*c=a*(b*c)
commutativity a+b=b+a a*b=b*a
distributivity
a*(b+c)=a*b+a*c
(a+b)*c=a*c+b*c
identity
a+0 = a
0+a = a
a*1 = a
1*a = a
inverses
a+(-a) = 0
(-a)+a = 0
a*a-1
= 1
a-1
*a = 1
if a≠0
Generating Geometric Algebras
Vector multiplication (cross and dot product) can be very useful in physics but it
also has its limitations and Geometric Algebra defines a new, more general, type of
multiplication. This new type of multiplication generates new 'dimensions' so

10. Geometric Algebra takes a vector algebra of dimension 'n' and generates an algebra
of dimension n².
• What are the properties of these new dimensions?
• How much freedom do we have, in in choosing the basis vectors for
example, to modify these properties?
This page discusses these questions .
Vector Cross Product
The vector cross product gives a vector which is perpendicular to both the vectors
being multiplied. The resulting vector A × B is defined by:
x = Ay * Bz - By * Az
y = Az * Bx - Bz * Ax
z = Ax * By - Bx * Ay
where x,y and z are the components of A × B
This page explains this.
Vector Dot Product
The dot product operation multiplies two vectors to give a scalar number (not a
vector).
It is defined as follows:
Ax * Bx + Ay * By + Az * Bz

11. This page explains this.
Transforms
A transform maps every point in a vector space to a possibly different point.
When transforming a computer model we transform all the vertices.
To model this using mathematics we can usematrices, quaternions or other
algebras which can represent multidimensional linear equations.
This page explains this.
Beginning Game Writing
Here are some pages on this site which aim to help start writing games:

12. • What computer language should I use?
• Games tutorials
• Rotation in 3 dimensions
• Writing a car racing game
• Writing a snooker game
Physics In Games
One of the things this site aims to do is to explain the physics required to write
games. For example, this page, explains how to calculate the response of solid
objects to a collision. We can use fairly standard physics concepts such
as impulse to do this.
However, using these concepts can get very complex, especially when we want
complex games to run quickly. Some of the things that can cause problems are:
• The need to calculate linear and rotational quantities such as velocity and
the need to understand how linear and rotational quantities interact with
each other.
• The need to keep changing frames of reference for different tasks.
• The need to use different mathematical notations for collisions and
dynamics.
It would be good to work out how to use different algebras to minimise these
problems. For example dual quaternions are a good way to represent linear and
rotational quantities in one mathematical element. But can dual quaternions be
used when we are working with inertia tensors?