Viewing Transformations

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Viewing Transformations

CS425: Computer Graphics I

Fabio Miranda

https://fmiranda.me

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• Coordinate systems

• Camera

• Viewing transformations

• Orthographic and perspective projections

• Hidden surface removal

Overview

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• Viewing transformation is the mapping of coordinates of points and lines from world coordinates into screen space pixels.

Viewing transformations

Modeling Camera Projection Viewport

World coordinates

Screen

Space

(pixels)

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• Previously, we saw how transformations can manipulate primitives (points, vectors) in space.

• Today, we will see how transformations can manipulate primitives to 2D screen coordinates (that will be later rasterized).

Viewing transformations

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Perspective projection

Distant objects appear smaller

Parallel lines converge at the

horizon

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Early paintings

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Perspective in art

Giotto di Bondone (14

^th

century) Giotto di Bondone (14

^th

century)

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Birth of perspective in art:

One-point perspective

Giovanni di Paolo (15

^th

century) Piero della Francesca (15

^th

century)

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Birth of perspective in art: One-point perspective

Perugino (15

^th

century)

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Birth of perspective in art: One-point perspective

The Ideal City (15

^th

century)

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Birth of perspective in art: One-point perspective

Rafael (16

^th

century)

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Birth of perspective in art:

Two-point perspective

Jean Pélérin (16

^th

century) Gustave Caillebotte (19

^th

century)

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Birth of perspective in art:

Three-point perspective

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Multi perspective

The Frozen City by Matthias A. K.

Zimmermann (2006)

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Perspective in computer graphics

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Rejection of perspective in CG

SimCity 2000

“Automated Generation of Interactive 3D

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Coordinate spaces

From: Mark Pauly

Object coordinates World coordinates Camera coordinates Screen coordinates

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Viewing transformation

Model

Camera Projection

Viewport

Object space Camera space

Canonical view volume

Screen space

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• Construct the camera reference system:

• The eye position 𝐞𝐞.

• The forward direction 𝐝𝐝.

• The view-up vector 𝐮𝐮.

• A view matrix transform all coordinates into view coordinates.

Camera transformation

𝐞𝐞 𝐮𝐮 𝐠𝐠

𝐮𝐮 × 𝐠𝐠

𝐌𝐌 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 = 𝐮𝐮 × 𝐠𝐠 𝐮𝐮 𝐝𝐝 𝐞𝐞 0 0 0 1

−1

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Orthographic and perspective projections

https://en.wikipedia.org/wiki/File:Various_projections_of_cube_above_plane.svg

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View frustum

Camera space

(𝑥𝑥

_{𝑐𝑐𝑐𝑐𝑙𝑙𝑙𝑙}

, 𝑦𝑦

𝑏𝑏𝑐𝑐𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏

, 𝑧𝑧

_{𝑏𝑏𝑐𝑐𝑐𝑐𝑐𝑐}

) × (𝑥𝑥

_{𝑐𝑐𝑟𝑟𝑟𝑟𝑟𝑙𝑙}

, 𝑦𝑦

_{𝑙𝑙𝑐𝑐𝑡𝑡}

, 𝑧𝑧

_{𝑙𝑙𝑐𝑐𝑐𝑐}

)

Far plane

Near plane

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View frustum

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Orthographic transformation

Camera space

Projection

𝑥𝑥 𝑦𝑦 1 𝑧𝑧

=

2 (𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟 − 𝑙𝑙𝑙𝑙𝑙𝑙𝑟𝑟) 0 0 −𝑥𝑥

_{𝑐𝑐𝑟𝑟𝑐𝑐}

0 2

(𝑟𝑟𝑡𝑡𝑡𝑡 − 𝑏𝑏𝑡𝑡𝑟𝑟𝑟𝑟𝑡𝑡𝑏𝑏) 0 −𝑦𝑦

_{𝑐𝑐𝑟𝑟𝑐𝑐}

0 0 −2

𝑥𝑥 𝑦𝑦 1 𝑧𝑧

Canonical view volume

(−1, −1, −1) × (1,1,1)

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Basic orthographic projection

𝐱𝐱 ^′ = 𝐱𝐱, 𝐲𝐲′ = 𝐲𝐲

y’ y

z

𝑥𝑥

^′

𝑦𝑦′ z′

1 =

1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1

𝑥𝑥 𝑦𝑦 1 𝑧𝑧

Projection plane

(z = 0)

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Basic perspective projection

-f y’

y

z

𝑥𝑥 ^′ = 𝑙𝑙 𝑥𝑥

𝑧𝑧 , ^{𝑦𝑦′ = 𝑙𝑙}

𝑦𝑦 𝑧𝑧

f

y’ y

z

𝑟𝑟𝑙𝑙 𝑙𝑙 = 1:

𝑥𝑥 ^′ = 𝑥𝑥

𝑧𝑧 , 𝑦𝑦 ^′ = 𝑦𝑦

𝑧𝑧 , 𝑙𝑙 = 1

𝑥𝑥

^′

𝑦𝑦

^′

𝑧𝑧′ 1

→ 𝑥𝑥 𝑦𝑦 𝑧𝑧 𝑧𝑧

=

1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0

𝑥𝑥 𝑦𝑦

1 𝑧𝑧

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Perspective transformation

Camera space

Projection

𝑥𝑥 𝑦𝑦 1 𝑧𝑧

=

𝑙𝑙

(𝑓𝑓𝑎𝑎𝑡𝑡𝑙𝑙𝑎𝑎𝑟𝑟) 0 0 0

0 𝑙𝑙 0 0

0 0 𝑛𝑛𝑙𝑙𝑓𝑓𝑟𝑟 + 𝑙𝑙𝑓𝑓𝑟𝑟 (𝑛𝑛𝑙𝑙𝑓𝑓𝑟𝑟 − 𝑙𝑙𝑓𝑓𝑟𝑟)

2 ∗ 𝑙𝑙𝑓𝑓𝑟𝑟 ∗ 𝑛𝑛𝑙𝑙𝑓𝑓𝑟𝑟 (𝑛𝑛𝑙𝑙𝑓𝑓𝑟𝑟 − 𝑙𝑙𝑓𝑓𝑟𝑟)

𝑥𝑥 𝑦𝑦 𝑧𝑧 1

Canonical view volume (−1, −1, −1) × (1,1,1)

𝑙𝑙 = 1 tan(𝑙𝑙𝑡𝑡𝑓𝑓 2 )

, 𝑓𝑓𝑎𝑎𝑡𝑡𝑙𝑙𝑎𝑎𝑟𝑟 = 𝑤𝑤𝑟𝑟𝑤𝑤𝑟𝑟𝑟

𝑟𝑙𝑙𝑟𝑟𝑟𝑟𝑟𝑟𝑟

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• Why?

• Makes clipping much easier! GL can quickly discard geometry outside -1,1.

Canonical view volume

From: https://paroj.github.io/gltut/

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