Geometric Objects and Basic Transformations

Paul Taylor 2010

3D Coordinate systems

http://alt.pluralsight.com/wiki/default.aspx/Craig.DirectX/CoordinateSystemsTutorial.html

DirectX OpenGL

2D Coordinates in Windows0,0

xMax, yMax

0, yMax

xMax, 0

• In both OpenGL and Direct3D Z is the depth of the viewport!!!!hence the Z-buffer (Depth Buffer)

• Z Buffer Depth– Due to the scalar nature of the Pyramid Viewport– Depth precision drops as the distance between

the Near and Far planes are moved apart– This will result in more artefacts and visual errors

Using the Z Buffer

glEnable(GL_DEPTH_TEST);This enables the Z buffer, but is entirely useless

without: glDepthFunc(GL_LEQUAL);You’ll find that the default for the Depth buffer

usually doesn’t work.

Viewport types

• Orthogonal– No depth perception (scaling)

• Projected– This is the typical 3D viewport

• Objects \ Vertices are scaled and rotated based on the ‘virtual’ distance from the screen

Viewport Creation

• glOrtho(left, right, bottom, top, near, far)• gluOrtho2D

– Sets the Near and Far values to 1.0• (Only renders objects with z=1.0)• Same as glOrtho with near & far = 1.0

• GluPerspective(…)– Creates the Viewport Frame based on a FOV

• GlFrustrum(…) – Create the Viewport in a more numerical way

• Create your viewport matrix manually if you like!

Matriceis

• GL_PROJECTION• GL_MODEL_VIEW• GL_TEXTURE_MATRIX

Matrix Math (Joy!)

• Matrix Addition• Matrix Multiplication• Translation and Rotation Matrices

Matrix Addition1 2 3

4 5 6

7 8 9

A B C

D E F

G H I+ =1+A 2+B 3+C

4+D 5+E 6+F

7+G 8+H 9+I

4 0 0

0 6 0

0 0 9

3 0 0

0 1 0

0 0 0+ =7 0 0

0 7 0

0 0 9

Matrix Multiplication Row,Colum1 2 3

4 5 6

7 8 9

A B C

D E F

G H Ix =1A + 2D + 3G

1B + 2E + 3H

1C + 2F + 3I

4A + 5D + 6G

4B + 5E + 6H

4C + 5F + 6I

7A + 8D + 9G

7B + 8E + 9H

7C + 8F + 9I

4 0 0

0 6 0

0 0 9

3 0 0

0 1 0

0 0 0x =12 0 0

0 6 0

0 0 0

That’s great, what do I need them for?

Translation Matrices• Coordinates from Object Coordinates to world

CoordinatesObject Translation World

Typically we use a Matrix Multiplication

100 0 0

0 20 0

0 0 -100+ =110 0 0

0 30 0

0 0 -100

10 0 0

0 10 0

0 0 0

Scalar MatricesObject Scalar (1/10) World

This can be done by glScalef(x,y,z);

100 0 0

0 100 0

0 0 100

0.1 0.1 0.1

0.1 0.1 0.1

0.1 0.1 0.1x =10 0 0

0 10 0

0 0 10

Translation by Multiplication

• This is a computationally longer way to do a simple translation

• glTranslatef(x,y,z);1 0 0 x

0 1 0 y

0 0 1 z

0 0 0 1

x =1 0 0 x

0 1 0 y

0 0 1 z

0 0 0 1

1 0 0 10

0 1 0 5

0 0 1 10

0 0 0 1

Rotational Matrices

http://en.wikipedia.org/wiki/Rotation_matrix#Dimension_threeA Rotation around the unit vector u = (ux,uy,uz)

glRotatef(Theta,x,y,z);

Putting it all together

Typically you will want to:TranslateRotateScale

Matrix Multiplication is the inverse to the logical order!

So:Scale then Rotate, then Translate

Retrieving Matrices

• This is slow!• The Video Matricies are 4x4 hence:GLfloat matrix[16];glGetFloatv(GL_PROJECTION_MATRIX, &matrix[0]);matrix =

Simillarly you can do this for GL_MODEL_VIEW and GL_TEXTURE_MATRIX

Setting your own Matrixvoid glLoadMatrixf( &matrix[0]);

0 4 8 C1 5 9 D2 6 A E3 7 B F

0 1 2 3 4 5 6 7 8 9 A B C D E F

BG115 has

XNA

Silent Hill 5 (Homecoming)

From Jason Allen (Lead Designer)

…” We made a decision during development to support invert aim and not invert camera. Unfortunately none of the team nor any of our test group actually had a preference for playing with an inverted camera, so the issue of its omission never came up.”…

http://72.14.235.132/search?q=cache:ylhQ9HOmPg4J:gamesblog.ugo.com/index.php/gamesblog/more/silent_hill_homecomings_jason_allen_on_inverted_aim/+silent+hill+5+no+camera+invert&cd=3&hl=en&ct=clnk&gl=au&client=firefox-a

Matrix Mathematics Continued

• Matrix Cross Product

• We use this to create Normals

• Matrix Dot Product

• Normalising Matrcies

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