Nima Kalantari

CSCE 441 - Computer Graphics

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Viewing Transformations

Some slides from Ren Ng and Ravi Ramamoorthi

Model Transformation

Model from Pradeep Sen

World Coordinate

Local Coordinate

Z

Y

X

Outline

• View transformation
• Projection transformation

Camera Space

• Camera pose: origin (eye), look at (center), up

Model from Pradeep Sen

Camera

Camera Coordinate

World Coordinate

w

v

u

Z

Y

X

View Transformation

• Construct camera coordinate frame
• Represent vertices in camera coordinate

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Model from Pradeep Sen

Camera

Camera Coordinate

World Coordinate

Z

Y

X

w

v

u

Construct Camera Coordinate Frame

Model from Pradeep Sen

Camera

Camera Coordinate

World Coordinate

Z

Y

X

w

v

u

Coordinate Frames

• Any set of 3 vectors (in 3D) so that

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Construct Camera Coordinate Frame

Model from Pradeep Sen

Camera

Camera Coordinate

World Coordinate

Z

Y

X

w

v

u

up

a

Construct Camera Coordinate Frame

Model from Pradeep Sen

Camera

Camera Coordinate

World Coordinate

Z

Y

X

w

v

u

up

a

Construct Camera Coordinate Frame

Model from Pradeep Sen

Camera

Camera Coordinate

World Coordinate

Z

Y

X

w

v

u

up

a

View Transformation

• Construct camera coordinate frame
• Represent vertices in camera coordinate

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Model from Pradeep Sen

Camera

Camera Coordinate

World Coordinate

Z

Y

X

w

v

u

Coordinate Change: 2D Example

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Coordinate transformation

world

camera

camera to world

world to camera

Camera Control – Assignment 2

• Turntable camera
• Eye is floating around the surface of the sphere
• Camera is looking at the center of the sphere

center

eye

Camera Control – Assignment 2

• 3 operations
• Scroll: zoom in and out
• Right click: translate
• Left click: rotate

center

eye

Camera Control – Assignment 2

• Zoom – change length of a
• Translate – look at and eye
• Y along the local up vector
• X along the right vector
• Rotate – change direction of a & v
• rotation along the right
• first
• rotation along the global up
• second
• glm::lookat(eye, center, v)

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center

eye

local up (v)

a

right (u)

global up (up)

Outline

• View transformation
• Projection transformation

Motivation

• So far transformations in 3D
• Goal: make a 2D picture
• Projection: transformation from 3D to NDC

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x

z

y

+1

-1

+1

-1

-1

+1

Normalized Device

Coordinate (NDC)

x

z

y

Camera coordinate

Projection Transformation

• Orthographic projection
• Perspective projection

Orthographic Projection

• Basic idea: simply project onto xy plane, drop z coordinate

Orthographic Projection

• Parallel lines remain parallel
• Application in technical drawing

Orthographic Projection

• Basic idea: simply project onto xy plane, drop z coordinate

left

right

top

buttom

Orthographic Projections

In general

• We have a cuboid that we want to map to the canonical cube from [-1, +1] in all axes
• First center cuboid by translating
• Then scale into unit cube

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x

z

x

z

Translate

y

x

z

y

Scale

+1

-1

+1

-1

-1

+1

Normalized Device

Coordinates (NDC)

Transformation Matrix

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Scale

Translation (centering)

Caveats

• Looking down –z, f and n are negative (n > f)
• OpenGL convention: positive n, f, negate internally

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x

z

x

z

Translate

y

x

z

y

Scale

+1

-1

+1

-1

-1

+1

Normalized Device

Coordinates (NDC)

Final Result

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Final Result

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• glm::ortho(left, right, bottom, top, near, far)

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Projection Transformation

• Orthographic projection
• Perspective projection

Perspective in Photographs

Pinhole Camera

Perspective view volume to NDC

From Angel and Shreiner, Interactive Computer Graphics

Near

Far

x

z

y

+1

-1

+1

-1

-1

+1

Normalized Device

Coordinate (NDC)

Specifying Perspective Projection

From Angel and Shreiner, Interactive Computer Graphics

Specifying Perspective Projection

Field of view

(fovy)

width

height

Aspect ratio = width / height

Specifying Perspective Projection

From Angel and Shreiner, Interactive Computer Graphics

Near

Far

Summary

• Fovy: vertical angular field of view
• Aspect ratio: width / height
• Near: depth of near clipping plane
• Far: depth of far clipping plane

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Perspective view volume to NDC

• Project all the points to view plane

From Angel and Shreiner, Interactive Computer Graphics

Near

Far

Side View of Our Screen

In Matrices

• Last row affected (no longer 0 0 0 1)
• w coord is no longer = 1
• Must divide at the end by w (perspective division)

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Perspective view volume to NDC

• Project all the points to a normalized plane

From Angel and Shreiner, Interactive Computer Graphics

Near

Far

+1

-1

-1

+1

Side View of Our Screen

fovy

+1

-1

In Matrices

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Taking aspect into account

• If height = 2, width = 2 * aspect

Perspective view volume to NDC

• Project all the points to a normalized plane

In Matrices

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• Aspect ratio taken into account
• Must map z values based on near and far planes

Perspective Projection Transform

Normalized Device Coordinates

(1, 1, 1)

Near Plane

Far Plane

Camera Coordinates

In Matrices

• Homogeneous, simpler to multiply through by d

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• Third row responsible for z
• x and y should not contribute to z

In Matrices

• Homogeneous, simpler to multiply through by d

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• Third row responsible for z
• x and y should not contribute to z
• A and B selected to map n and f to -1, +1 respectively

In Matrices

• Homogeneous, simpler to multiply through by d

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• Third row responsible for z
• x and y should not contribute to z
• A and B selected to map n and f to -1, +1 respectively

Z mapping derivation

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• Two equations to equate z = -n with -1 and z = -f with +1

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Perspective Matrix

Perspective Matrix

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• glm::perspective(fovy, aspect, near, far)