CO1301: Games Concepts

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CO1301 Games Concepts
Lecture 8 Vectors

• Dr Nick Mitchell (Room CM 226)
• email npmitchell_at_uclan.ac.uk
• Material originally prepared by Gareth Bellaby

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Essential Reading on Vectors

• Rabin, Introduction to Game Development
• 4.1 "Mathematical Concepts"
• Van Verthe, Essential Mathematics for Games
• Chapter 1 "Vectors and Points"

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Lecture Outline

• Maths Preliminaries
• Coordinates
• Vectors
• Vector arithmetic
• Vector length
• The direction vector and scalars

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• Topic 1 Maths Preliminaries

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Order of precedence

• Mathematical operations have an order of
  precedence.
• Multiplication and division have equal
  precedence.
• Multiplication and division are always carried
  out before addition and subtraction.

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Right-angled triangles

• With a right-angled triangle the length of the
  longest side is equal to the square root of the
  sum of the squares of the other two sides.

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Right-angled triangles
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Topic 2 Coordinates
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Coordinates

• A coordinate represents a point in space.
• A point in 2D is represented using coordinates
  (X, Y)
• In 3D it is represented using coordinates (X, Y,
  Z)
• The convention is
• use round brackets
• a comma separated list, e.g. ( 2, 4, 7 )
• always in the sequence x, y, z.
• Called Cartesian geometry after Rene Descartes.

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Coordinates

• We use a grid to identify points in space.
• Can choose the direction to point Z.
• Use your hand to indicate the geometry. This
  gives the "handedness" of the geometry. We are
  using left-handed axes.

Z
Y

• X is thumb
• Y index finger
• Z middle finger

X
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Coordinates
Y
Z
Z
Y
X
X

• The origin of the grid is at ( 0, 0, 0 )

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Topic 3 Vectors
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Vectors

• We often need to identify a movement or direction
  in 3D - this can also be represented using values
  (X, Y, Z) and is called a vector.

Vectors are more easily shown in 2D using where
we only need (X, Y) values.
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Vector between two points

• Calculated by subtracting one point from the
  other.
• Subtract the x values to produce one new x, the y
  values to produce a new y and then the z values
  to produce a new z.
• The vector V from P1(X1, Y1) to P2(X2, Y2) is
• V( X2 - X1, Y2 - Y1)
• Note that P1 is subtracted from P2

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Vectors

• The arrow of movement from P1 to P2 can be
  represented as the vector V(20,10).

V(X2 - X1, Y2 Y1) (30, 30) - (10, 20)
(20, 10)
The vector V from point P1(X1, Y1) to P2(X2, Y2)
is
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Vector between two points

• Notice how the values in the vector are found by
  subtracting the source point coordinates from the
  destination point coordinates. This means that
  the direction of the vector is important the
  vector in the reverse direction, from P2 to P1,
  is (-20, -10).
• (10, 20) - (30, 30)
• (-20, -10)

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Vectors

• 2D extends simply to 3D. The vector V from point
  P1(X1, Y1, Z1) to P2(X2, Y2, Z2) is
• V(X2 - X1, Y2 Y1, Z2 - Z1).
• Points and vectors are represented in the same
  way. A vector can be directly applied to a point.
• Location and movement can all be considered to be
  the same thing.
• A point is represented by P(X, Y, Z)
• A vector is represented by V(X, Y, Z)
• A vector of force is represented in the same way,
  e.g. the effect of wind.

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Vectors

• The line between two points is the path you need
  to take to travel directly from one point to
  another.
• The vector is the same no matter what its
  coordinates are.
• At the origin this is obviously true.
• But you can also see this will be true wherever
  the points are you can move the vector over
  the grid and it stays the same.

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Vector direction

• Which point is subtracted from which is
  important.
• The order gives us the direction of movement.

P1(1, 1) to P2 (3, 3) (3 - 1, 3 - 1) (2, 2)
P2(3, 3) to P1 (1, 1) (1 - 3, 1 - 3) (-2, -2)
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Vector direction

• If you ever need a reminder about the order when
  calculating the vector between two points just
  think about the direction of the movement.
• Easiest to do this if you imagine one of the
  points being at the origin.

From (0, 0, 0) to (2, 2, 0) ( 2 - 0, 2 - 0, 0 -
0) ( 2, 2, 0 ) Upwards and to the right
From (2, 2, 0) to (0, 0, 0) ( 0 - 2, 0 - 2, 0 -
0) ( -2,- 2, 0 ) Downwards and to the left
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Topic 4 Vector Arithmetic
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Vector Addition

• We visualise vectors by drawing a line with an
  arrowhead at one end.

Two vectors V and W are added by placing the
beginning of W at the end of V.
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Vector Addition

• V(1, 3) W(3, 1) (4, 4)

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Vector Subtraction

• Subtraction works in the same way as addition.
• The direction in which the subtracted vector
  points is effectively reversed.
• V(1, 3) - W(3, 1) (-2,
  2)

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Vector Arithmetic

• The order of operations is irrelevant.
• V W W V
• V - W W - V

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Topic 5 Vector Length
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Length of a vector

• If we have a vector V(X, Y, Z) then we can use
  Pythagoras Theorem to show that

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Length of a vector
Easiest with a 2D example.

• A vector makes a right-angled triangle.
• The shortest sides of the triangle are the x and
  y values.
• 3D is exactly the same - but with the addition of
  tee z axis.

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Maths symbol for length
The length of a vector v(x,y,z) is given by
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Topic 5 The direction vector and scalar
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Direction of movement

• The direction between two points is only partly
  useful.
• The problem is that its size depends on the
  distance between the two points.

Both vectors are pointing in the same direction,
but the black vector is half the size of the red
vector
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Direction of movement

• As you know, the units we use within 3D graphics
  are arbitrary. Sometimes in the TL-Engine a
  distance of 0.1 is large, sometimes it is small.
  It all depends on the scale.
• Direction should be independent of scale.
• The direction vector is a vector which is
  independent of scale.
• The direction vector has a length of 1.
• A vector whose length is 1 is said to be
  normalised.

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The direction vector

• The direction vector is found by
• Calculating the length of the vector
• Dividing each component of the vector by its
  length
• For example, given a vector V( 1, 3, 6 )

direction vector
length
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The direction vector

• Here is a 2D example of the process

length
direction vector
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3D example

• Vector pointing straight forward ( 0, 0, 7.5 )

Vector pointing straight forward ( 0, 0, 3.1 )
length
length
direction vector
direction vector
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"Pointing at" or "Looking at"

• Vectors can be used to represent a direction (in
  the sense of North or North-West).
• To do this we create a vector that points in
  the direction we want, and has a length of 1
  unit.
• We often need to find the direction vector that
  points from one position to another. The
  process we use is
• Get the vector between the two positions V(X, Y,
  Z)
• Find the length of this vector L
• The direction vector is VDir(X/L, Y/L, Z/L)

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Maths for normalised vector

• A normal is any vector whose length is 1
• Represented with a symbol
• A vector is converted to a normal (normalised) by
  dividing the components by the length

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Scalar

• Direction vectors are always length 1.
• They are always the unit length.
• In order to describe an actual movement you
  combine the direction vector with a scalar.
• The scalar is the distance to be moved.
• Imagine that you want to move something by a
  given distance in a particular direction. We
  multiply the direction vector by the distance we
  want to move.
• For example, move the normalised vector ( 2, 4,
  5 ) by 12 units results in ( 24, 48, 60 )

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Scalar

• A normalised vector can be seen as the direction
  of a vector
• A vector can be broken up into normalised vector
  and length, e.g.
• Movement Normalised vector distance
• Velocity Normalised vector speed
• The scalar changes the magnitude of a vector.
• The root word is "scale".
• The operation of the scalar is to scale the
  vector.
• This also reminds us why normalised vectors are
  useful.