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SE 313 – Computer Graphics

Lecture 8: Transformations and Projections

Lecturer: Gazihan Alankuş

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Plan for Today

• Post-exam talk• Revisit transformations• Projections

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Exam Talk

• Go over exam questions

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Transformations (summary)

• Three types of linear transformations– Translation (point-vector addition)– Rotation (3x3 matrix multiplication)– Scale (vector-scalar multiplication)

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Transformations (summary)

• Three types of linear transformations– Translation (point-vector addition)– Rotation (3x3 matrix multiplication)– Scale (vector-scalar multiplication)

• Cannot combine these operations in one type of operation– Convert them to one type of operation (not

possible unless you use homogeneous coordinates)

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Transformations (summary)

• Homogeneous coordinates enable us to represent translation, rotation and scale using 4x4 matrix multiplications.

• This way we can combine them easily by multiplying matrices together. The resulting matrix is another transformation.

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Transformations (summary)

• 4x4 matrices that are combinations of translation, rotation and scale

0 0 0 1

Rotation and scale Translation

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Transformations (summary)

• You can read the local coordinate frame from 4x4 transformation matrices

0 0 0 1

Rotation and scale Translation

Where in the world the local frame’s origin is

The x, y and z axes of thelocal frame

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Transformations (summary)

• Intuitive understanding of transformations• Local-to-world: insert

new transformations near the wall (world)

• World-to-local: insert new transformations near the object

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Transformations (summary)

• Quaternions: data structure for rotation– Useful for animations

• Ways of representing rotations

Three angles (euler angles)

One axis, one angle 3x3 matrix

Quaternion

Best interpolation (slerp)Great-looking animations

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Plan for Today

• Post-exam talk• Revisit transformations• Projections

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Projection

• Projections from 3D to 2D– Taking pictures of the virtual world

[Images are borrowed from http://db-in.com]

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Projection Types

• Perspective projection– Just like our eyes and

cameras

• Orthographic projection– Architectural drawing

with no distance distortion

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Orthographic vs. Perspective Projection

[Images are borrowed from http://db-in.com]

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Orthographic Projection

• Get the 3D world, compress it on a 2D paper

[engineeringtraining.tpub.com]

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Orthographic Projection

• Great for isometric games (Starcraft, Diablo I-II)

• No depth sensation

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Orthographic Projection in Blender

• Select the camera• The viewport is defined

by the render output size• Camera has – Scale– Start and end clipping distances

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Perspective Projection

• Take the picture of the world from a single point

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Perspective Projection

• What parameters do I need?

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Perspective Projection

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Perspective Projection

• How do you do it mathematically?– Also using a 4x4 matrix

[songho.ca]

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Perspective Projection

• Let’s try to make sense of it very simply

0 0

0 0

0 0

0 0 -1 0

Translating in z

Output’s w depends on input’s zThe further it is in z, the smaller it will get

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Perspective Projection

• What that matrix does

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Perspective Projection in Blender

• Select the camera• The viewport is defined

by the render output size• Camera has – Field of view angle– Start and end clipping distances

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Perspective vs Orthographic Projection

Perspective

Orthographic

Fov=60◦, distance = 1

Fov=30◦, distance = 3

Fov=10◦, distance = 5

Fov=0◦, distance =

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Perspective vs Orthographic Projection

• Orthographic camera is a perspective camera where the camera is at the infinity and the field of view angle is zero

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Perspective vs Orthographic Projection

• In this transition, the size of the arrow in the image stays the same

• This is also "called the “dolly-zoom”, “Hitchcock zoom”, or “vertigo effect”

• Demonstration in Unity and sample scenes from movies

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For next week

• No homework• Study what we learned today, there will be a

quiz• Next week, a part of the lab will be about

projection