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Geometric Objects and Basic Transformations

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Geometric Objects and Basic Transformations. Paul Taylor 2010. 3D Coordinate systems. DirectXOpenGL. http://alt.pluralsight.com/wiki/default.aspx/Craig.DirectX/CoordinateSystemsTutorial.html. xMax , 0. 2D Coordinates in Windows. 0,0. 0, yMax. xMax , yMax. - PowerPoint PPT Presentation
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Geometric Objects and Basic Transformations Paul Taylor 2010
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Page 1: Geometric Objects and  Basic Transformations

Geometric Objects and Basic Transformations

Paul Taylor 2010

Page 2: Geometric Objects and  Basic Transformations

3D Coordinate systems

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

DirectX OpenGL

Page 3: Geometric Objects and  Basic Transformations

2D Coordinates in Windows0,0

xMax, yMax

0, yMax

xMax, 0

Page 4: Geometric Objects and  Basic Transformations

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

Page 5: Geometric Objects and  Basic Transformations

• 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

Page 6: Geometric Objects and  Basic Transformations

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.

Page 7: Geometric Objects and  Basic Transformations

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

Page 8: Geometric Objects and  Basic Transformations

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!

Page 9: Geometric Objects and  Basic Transformations

Matriceis

• GL_PROJECTION• GL_MODEL_VIEW• GL_TEXTURE_MATRIX

Page 10: Geometric Objects and  Basic Transformations

Matrix Math (Joy!)

• Matrix Addition• Matrix Multiplication• Translation and Rotation Matrices

Page 11: Geometric Objects and  Basic Transformations

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

Page 12: Geometric Objects and  Basic Transformations

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

Page 13: Geometric Objects and  Basic Transformations

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

Page 14: Geometric Objects and  Basic Transformations

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

Page 15: Geometric Objects and  Basic Transformations

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

Page 16: Geometric Objects and  Basic Transformations

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);

Page 17: Geometric Objects and  Basic Transformations

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

Page 18: Geometric Objects and  Basic Transformations

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

Page 19: Geometric Objects and  Basic Transformations

BG115 has

XNA

Page 20: Geometric Objects and  Basic Transformations

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

Page 21: Geometric Objects and  Basic Transformations

Matrix Mathematics Continued

• Matrix Cross Product

• We use this to create Normals

Page 22: Geometric Objects and  Basic Transformations

• Matrix Dot Product

Page 23: Geometric Objects and  Basic Transformations

• Normalising Matrcies

Page 24: Geometric Objects and  Basic Transformations

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Page 25: Geometric Objects and  Basic Transformations

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