JEFF CHASTINE 1

and an introduction to matrices

COORDINATE SYSTEMS

JEFF CHASTINE 2

THE LOCAL COORDINATE SYSTEM• Sometimes called “Object Space”

• It’s the coordinate system the model was made in

JEFF CHASTINE 3

THE LOCAL COORDINATE SYSTEM• Sometimes called “Object Space”

• It’s the coordinate system the model was made in

(0, 0, 0)

JEFF CHASTINE 4

THE WORLD SPACE• The coordinate system of the virtual environment

(619, 10, 628)

JEFF CHASTINE 5

(619, 10, 628)

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QUESTION

• How did get the monster positioned correctly in the world?

• Let’s come back to that…

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CAMERA SPACE• It’s all relative to the camera…

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CAMERA SPACE• It’s all relative to the camera… and the camera never moves!

(0, 0, -10)

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THE BIG PICTURE• How to we get from space to space?

? ?

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THE BIG PICTURE• How to we get from space to space?

• For every model

• Have a (M)odel matrix!

• Transforms from object to world space

M ?

JEFF CHASTINE 11

THE BIG PICTURE• How to we get from space to space?

• To put in camera space

• Have a (V)iew matrix

• Usually need only one of these

M V

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THE BIG PICTURE• How to we get from space to space?

• The ModelView matrix

• Sometimes these are combined into one matrix

• Usually keep them separate for convenience

M V

MV

JEFF CHASTINE 13

MATRIX - WHAT?• A mathematical structure that can:

• Translate (a.k.a. move)

• Rotate

• Scale

• Usually a 4x4 array of values

• Idea: multiply each point by a matrix to get the new point

• Your graphics card eats matrices for breakfast

[1.0 0.0 0.0 0.00.0 1.0 0.0 0.00.0 0.0 1.0 0.00.0 0.0 0.0 1.0

]The Identity Matrix

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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?

M

JEFF CHASTINE 15

BACK TO THE BIG PICTURE• If you multiply a matrix by a matrix, you get a matrix!

• How might we make the model matrix?

M

Translation matrix TRotation matrix R1

Rotation matrix R2

Scale matrix S

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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?

M

Translation matrix TRotation matrix R1

Rotation matrix R2

Scale matrix S

T * R1 * R2 * S = M

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MATRIX ORDER• Multiply left to right

• Results are drastically different

(an angry vertex)

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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 20

MATRIX ORDER• Multiply left to right

• Results are drastically different

• Order of operations

• Rotate 45°

• Translate 10 units

before after

JEFF CHASTINE 21

MATRIX ORDER• Multiply left to right

• Results are drastically different

• Order of operations

JEFF CHASTINE 22

MATRIX ORDER• Multiply left to right

• Results are drastically different

• Order of operations

• Translate 10 units

JEFF CHASTINE 23

MATRIX ORDER• Multiply left to right

• Results are drastically different

• Order of operations

• Translate 10 units

• Rotate 45°

JEFF CHASTINE 24

MATRIX ORDER• Multiply left to right

• Results are drastically different

• Order of operations

• Translate 10 units

• Rotate 45°

before

after

JEFF CHASTINE 25

BACK TO THE BIG PICTURE• If you multiply a matrix by a matrix, you get a matrix!

• How might we make the model matrix?

M

Translation matrix TRotation matrix R1

Rotation matrix R2

Scale matrix S

T * R1 * R2 * S = M 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?

M

Translation matrix TRotation matrix R1

Rotation matrix R2

Scale matrix S

S * R1 * R2 * T = M

JEFF CHASTINE 27

THE (P)ROJECTION MATRIX• Projects from 3D into 2D

• 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)

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ORTHOGRAPHIC VS. PERSPECTIVE

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AN OLD VERTEX SHADERin vec4 vPosition; // The vertex in NDC

void main () {

gl_Position = vPosition;

}

Originally we passed using NDCs (-1 to +1)

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A BETTER VERTEX SHADERin 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;

}

Original (local) positionNew position in NDC

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SMILE – IT’S THE END!

HOW ABOUT MORE THAN ONE OBJECT?

• Hierarchical Transformations

• Composing transformations

• Coordinate systems/frames

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COMPOSING TRANSFORMATIONS: ROTATION ABOUT A FIXED POINT

Basic idea:1) Move fixed point to origin2) Rotate3) Move the fixed point backRemember, postmultiplication applies transforms in reverse

Result: M = T RT –1

What does this look like graphically?

ROTATE AROUND A FIXED POINTT-1

ROTATE AROUND A FIXED POINTR

Ө

ROTATE AROUND A FIXED POINTR

Ө

ROTATE AROUND A FIXED POINTT

Ө

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OPENGL/GLM EXAMPLE

• Rotation about z axis by 30 degrees with a fixed point of (1.0, 2.0, 3.0)

• Remember that last transform specified in the program is the first applied

model *=glm::translate(1.0, 2.0, 3.0)*glm::rotate(30.0, 0.0, 0.0, 1.0)*glm::translate(-1.0, -2.0, -3.0);cube.render(view*model, &shader);...

TRANSFORMATION HIERARCHIES

• For example, a robot arm

Transformation Hierarchies

• Let’s examine:

Transformation Hierarchies

• What is a better way?

Transformation Hierarchies

• What is a better way?

Transformation Hierarchies• We can have transformations be in relation to each other• How do we do this in openGL and glm?

World Coordinates

Upper Arm Coordinates

Lower Arm Coordinates

Hand Coordinates

Transformation: Upper Arm -> World

Transformation: Lower -> Upper

Transformation: Hand-> Lower

Transformation Hierarchies• Activity: how you would have an object B orbiting object A, and

object A is constantly translating.

World Coordinates

Upper Arm Coordinates

Lower Arm Coordinates

Hand Coordinates

Transformation: Upper Arm -> World

Transformation: Lower -> Upper

Transformation: Hand-> Lower