COORDINATE SYSTEMS
and an introduction to matrices
JEFF CHASTINE
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THE LOCAL COORDINATE SYSTEM
•
Sometimes called “Object Space”
•
It’s the coordinate system the model was made in
JEFF CHASTINE
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THE LOCAL COORDINATE SYSTEM
•
Sometimes called “Object Space”
•
It’s the coordinate system the model was made in
(0, 0, 0)
JEFF CHASTINE
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THE WORLD SPACE
•
The coordinate system of the virtual environment
(619, 10, 628)
JEFF CHASTINE
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(619, 10, 628)
JEFF CHASTINE
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QUESTION
• How did get the monster positioned correctly in the
world?
• Let’s come back to that…
JEFF CHASTINE
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CAMERA SPACE
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It’s all relative to the camera…
JEFF CHASTINE
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CAMERA SPACE
•
It’s all relative to the camera… and the camera never moves!
(0, 0, -10)
JEFF CHASTINE
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THE BIG PICTURE
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How to we get from space to space?
?
JEFF CHASTINE
?
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THE BIG PICTURE
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How to we get from space to space?
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For every model
• Have a (M)odel matrix!
• Transforms from object to world space
M
JEFF CHASTINE
?
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THE BIG PICTURE
•
How to we get from space to space?
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To put in camera space
• Have a (V)iew matrix
• Usually need only one of these
M
JEFF CHASTINE
V
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THE BIG PICTURE
•
How to we get from space to space?
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The ModelView matrix
• Sometimes these are combined into one matrix
• Usually keep them separate for convenience
V
M
MV
JEFF CHASTINE
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MATRIX - WHAT?
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A mathematical structure that can:
• Translate (a.k.a. move)
• Rotate
• Scale
• Usually a 4x4 array of values
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Idea: multiply each point by a matrix to get the new point
•
Your graphics card eats matrices for breakfast
JEFF CHASTINE
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
0.0
0.0
0.0
0.0
1.0
The Identity Matrix
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BACK TO THE BIG PICTURE
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If you multiply a matrix by a matrix, you get a matrix!
•
How might we make the model matrix?
M
JEFF CHASTINE
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BACK TO THE BIG PICTURE
•
If you multiply a matrix by a matrix, you get a matrix!
•
How might we make the model matrix?
Translation matrix T
Rotation matrix R1
Rotation matrix R2
Scale matrix S
M
JEFF CHASTINE
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BACK TO THE BIG PICTURE
•
If you multiply a matrix by a matrix, you get a matrix!
•
How might we make the model matrix?
Translation matrix T
Rotation matrix R1
Rotation matrix R2
Scale matrix S
M
T * R1 * R2 * S = M
JEFF CHASTINE
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MATRIX ORDER
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Multiply left to right
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Results are drastically different
(an angry vertex)
JEFF CHASTINE
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MATRIX ORDER
•
Multiply left to right
•
Results are drastically different
•
Order of operations
• Rotate 45°
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MATRIX ORDER
•
Multiply left to right
•
Results are drastically different
•
Order of operations
• Rotate 45°
• Translate 10 units
JEFF CHASTINE
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MATRIX ORDER
•
Multiply left to right
•
Results are drastically different
•
Order of operations
• Rotate 45°
• Translate 10 units
before
JEFF CHASTINE
after
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MATRIX ORDER
•
Multiply left to right
•
Results are drastically different
•
Order of operations
JEFF CHASTINE
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MATRIX ORDER
•
Multiply left to right
•
Results are drastically different
•
Order of operations
• Translate 10 units
JEFF CHASTINE
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MATRIX ORDER
•
Multiply left to right
•
Results are drastically different
•
Order of operations
• Translate 10 units
• Rotate 45°
JEFF CHASTINE
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MATRIX ORDER
•
Multiply left to right
•
Results are drastically different
•
Order of operations
after
• Translate 10 units
• Rotate 45°
before
JEFF CHASTINE
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BACK TO THE BIG PICTURE
•
If you multiply a matrix by a matrix, you get a matrix!
•
How might we make the model matrix?
Translation matrix T
Rotation matrix R1
Rotation matrix R2
Scale matrix S
M
T * R1 * R2 * S = M
JEFF CHASTINE
Backwards
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BACK TO THE BIG PICTURE
•
If you multiply a matrix by a matrix, you get a matrix!
•
How might we make the model matrix?
Translation matrix T
Rotation matrix R1
Rotation matrix R2
Scale matrix S
M
S * R1 * R2 * T = M
JEFF CHASTINE
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THE (P)ROJECTION MATRIX
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Projects from 3D into 2D
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Two kinds:
• Orthographic: depth doesn’t matter, parallel remains parallel
• Perspective: Used to give depth to the scene (a vanishing point)
•
End result: Normalized Device Coordinates (NDCs between -1.0 and +1.0)
JEFF CHASTINE
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ORTHOGRAPHIC VS. PERSPECTIVE
JEFF CHASTINE
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AN OLD VERTEX SHADER
in vec4 vPosition;
// The vertex in NDC
Originally we passed using NDCs (-1 to +1)
void main () {
gl_Position = vPosition;
}
JEFF CHASTINE
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A BETTER VERTEX SHADER
in vec4 vPosition;
// The vertex in the local coordinate system
uniform mat4 mM;
// The matrix for the pose of the model
uniform mat4 mV;
// The matrix for the pose of the camera
uniform mat4 mP;
// The projection matrix (perspective)
void main () {
gl_Position = mP*mV*mM*vPosition;
}
New position in NDC
JEFF CHASTINE
Original (local) position
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SMILE – IT’S THE END!
JEFF CHASTINE
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