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CO1301 - Games ConceptsWeek 20

Matrices continued

CO1301 - Games ConceptsWeek 20

Matrices continued

Gareth BellabyGareth Bellaby

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TopicsTopics

1. Matrices continued

2. Efficiency

3. Numbers

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Topic 1Topic 1

Matrices continued

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MatricesMatrices

●A model has an associated 4 x 4 matrix.

1

0

0

0

zyx

zyx

zyx

zyx

PPP

zzz

yyy

xxxx-axis

y-axis

z-axis

position

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3D Models in graphics3D Models in graphics

●Recorded as a 4x4 matrix.

1

0

0

0

zyx

zyx

zyx

zyx

PPP

zzz

yyy

xxxx-axis

y-axis

z-axis

position

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MatricesMatrices

●A matrix is a rectangular table of numbers.

●A matrix is composed of rows and columns.

● In C++

float matrix[4][4] =

{ { 0.2, 0.4, 0.1, 0 },

{ 1, 1, 1, 0 },

{ 0.3, 0.3, 0.3, 0 },

{ 10, 3, 7, 1 } };

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Matrix maths reminderMatrix maths reminder

●Size is expressed as rrows by columns.

●Matrix addition.

●Scalar multiplication.

●Matrix multiplication.

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Scalar multiplicationScalar multiplication

●Scalar multiplication is simply when each component of a matrix is multiplied by a single value, in effect scaling it.

46

22

23

112

31261423

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Matrix MultiplicationMatrix Multiplication

The location of the result in the new matrix is the position where the row and the column intersect.

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TransposeTranspose

● The transpose of a matrix simply interchanges the rows and the columns of the matrix.

● It is important because vectors in graphics can be ordered by rows (DirectX) or columns (OpenGL and HLSL).

● Represented by a superscript T. The transpose of matrix A is AT.

161

032

10

63

12●So a n x m

matrix becomes an n x m matrix.

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The Identity MatrixThe Identity Matrix● The identity matrix is a special matrix. The identity matrix is a square matrix that has zeros for all elements expect

along the main diagonal.

● Multiplying a matrix by the identity matrix leaves the matrix unchanged.

● For example used as a starting point before additional operations are added to a matrix.

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01

100

010

001

1000

0100

0010

0001

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VectorsVectors●A matrix can have just one column or just one

row.

●A row or column matrix can be used to represent a vector.

ZYX

2.05.01.0 2.0,5.0,1.0

vector is: row matrix is:

●DirectX uses row vectors (OpenGL uses column vectors).

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Transforming a VectorTransforming a Vector

● If you multiply a row matrix by a 3x3 matrix, the result is another row matrix.

●The 3x3 matrix therefore acts to transform the row matrix.

●However, a 3x3 matrix can only produce some of the transformations required in 3D graphics.

●So another component is added at the end to produce a row matrix with a length of 4. The transformation matrix now be a 4x4 matrix.

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Rotation using a matrixRotation using a matrix

●Matrix multiplication is used to perform rotations.

●Well start with Euler rotation. In year 3 you'll look at a faster alternative.

●Euler is pronounced "oiler".

1

1000

0100

00cossin

00sincos

1 zyxzyx

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Rotation using a matrixRotation using a matrix

1000

0100

00cossin

00sincos

1000

0cos0sin

0010

0sin0cos

1000

0cossin0

0sincos0

0001

RotateZ

RotateYRotateX

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Topic 2Topic 2

Efficiency

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EfficiencyEfficiency

●Last week I talked about the need for efficiency. The time available to perform operations within a computer game is small (0.017 to 0.033 seconds per frame) and so efficiency is a dominant concern within games programming.

● If you have 10,000 trees. You need to perform collision detection for all of the trees. That's a lot of processing.

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Tree collisionTree collision

Collision is currently being implemented as:

occured. hascollision athen

tree, theof radius distance If

distance 222

zyx

●What could be done in order to reduce the computational cost?

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Improving efficiencyImproving efficiency

●Firstly, can the collision detection be made more efficient?

●Examine the calculation. Collision detection depends upon a distance check. The most expensive element of the calculation is the square root. But it can be dispensed with:

2222

222

(distance)

distance

zyx

zyx

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Improving efficiencyImproving efficiency

●Rewrite the check so that the square is no longer used.

●Square both sides of the equation.

●Perform the check against the square of the tree radius.

occured. hascollision athen

, tree) theof (radius (distance) If

(distance)

distance

22

2222

222

zyx

zyx

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Improving efficiencyImproving efficiency

●The square of the radius can be pre-calculated so it costs nothing.

●A simple rewriting of the calculation has improved matters.

●The use of squares is common within graphics, e.g. for distance tests.

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EfficiencyEfficiency

●Could anything else be done to improve efficiency?

●Does the program have to check all of the trees?

●Most of the trees are so far away that we can ignore them. It is possible to devise a method so that only likely candidates are checked.

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Partitioning the worldPartitioning the world

●Grid out the world.

●Record the trees according to which grid they are in.

●Only check those trees which are in the same grid as the player.

●10 by 10 grid.

●Each grid has 100 trees within it.

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Partitioning the worldPartitioning the world

●This example will use clear coordinates and scaling, but note that the principle holds true for all contexts. Each cell of the grid willl be 100 units by 100 units.

●Need an efficient way to derive the current grid of the player. Remember that integer devision within C++ produces an integer result and that the result is always rounded down. Divide the player coordinates (x and z) by 100. The result is the grid reference.

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Deriviving the grid cellDeriviving the grid cell

●player location ( 40, 30 )

=> ( 40/100, 30/100 )

=> ( 0, 0 )

●player location ( 140, 130 )

=> ( 140/100, 130/100 )

=> ( 1, 1 )

●player location ( 450, 70 )

=> ( 450/100, 70/100 )

=> ( 4, 0 )

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Spatial PartitioningSpatial Partitioning

● In programming terms you could create a 2D array which corresponds exactly to the grid coordinates. Each cell could then have the locations of its own trees stored.

●This is an simple example of a technique called "spatial partitioning", i.e. partitioning or sub-dividing space into chunks. Spatial partitioning is used extensively within computer games, e.g. in a game level so that only nearby models or visible models are rendered.

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Spatial PartitioningSpatial Partitioning

●For example, it is possible to extend the grid approach to include a check to see whether the trees get rendered.

●There's a nice version of this later on in Frank Luna, Introduction to DirectX, in Chapter 18 when he uses a grid to filter out geometry that is not in view of the camera.

●More sophisticated techniques include Binary Space Partioning (BSP) trees and Oct-trees.