SE 313 â Computer Graphics

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# SE 313 â Computer Graphics - PowerPoint PPT Presentation

SE 313 – Computer Graphics. Lecture 8 : Transformations and Projections Lecturer: Gazihan Alankuş. Plan for Today. Post-exam talk Revisit transformations Projections. Exam Talk. Go over exam questions. Transformations (summary). Three types of linear transformations

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### SE 313 – Computer Graphics

Lecture 8: Transformations and Projections

Lecturer: GazihanAlankuş

Plan for Today
• Post-exam talk
• Revisit transformations
• Projections
Exam Talk
• Go over exam questions
Transformations (summary)
• Three types of linear transformations
• Rotation (3x3 matrix multiplication)
• Scale (vector-scalar multiplication)
Transformations (summary)
• Three types of linear transformations
• 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)
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.
Transformations (summary)
• 4x4 matrices that are combinations of translation, rotation and scale

Rotation and scale

Translation

0

0

0

1

Transformations (summary)
• You can read the local coordinate frame from 4x4 transformation matrices

Rotation and scale

Translation

The x, y and z axes of thelocal frame

Where in the world the local frame’s origin is

0

0

0

1

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
Transformations (summary)
• Quaternions: data structure for rotation
• Useful for animations
• Ways of representing rotations

One axis, one angle

3x3 matrix

Quaternion

Three angles (euler angles)

Best interpolation (slerp)

Great-looking animations

Plan for Today
• Post-exam talk
• Revisit transformations
• Projections
Projection
• Projections from 3D to 2D
• Taking pictures of the virtual world

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

Projection Types
• Perspective projection
• Just like our eyes and cameras
• Orthographic projection
• Architectural drawing with no distance distortion
Orthographic vs. Perspective Projection

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

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

[engineeringtraining.tpub.com]

Orthographic Projection
• Great for isometric games (Starcraft, Diablo I-II)
• No depth sensation
Orthographic Projection in Blender
• Select the camera
• The viewport is defined by the render output size
• Camera has
• Scale
• Start and end clipping distances
Perspective Projection
• Take the picture of the world from a single point
Perspective Projection
• What parameters do I need?
Perspective Projection
• How do you do it mathematically?
• Also using a 4x4 matrix

[songho.ca]

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

0

0

0

0

0

0

Translating in z

0

0

-1

0

Output’s w depends on input’s z

The further it is in z, the smaller it will get

Perspective Projection
• What that matrix does
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
Perspective vs Orthographic Projection

Fov=60◦, distance = 1

Fov=30◦, distance = 3

Perspective

Fov=10◦, distance = 5

Fov=0◦, distance =

Orthographic

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
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
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