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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
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