G10: Transformation and Scene Graphs Flashcards
Why are transformations important in computer graphics?
They allow us to position, rotate, scale, and project objects within a 3D scene.
How is projection modeled in rendering?
As a transformation, just like rotation, translation, and scaling.
What are the benefits of representing transformations as matrices?
Easy to store
Efficient to apply
Can be composed (multiplied)
Enables uniform and fast algorithms and hardware support
What does the product of transformation matrices represent?
A sequence of transformations combined into a single transformation.
How is rotation represented in 2D and 3D?
2D: Using a 2×2 rotation matrix
3D: Rotation about a fixed axis using a 3×3 matrix, or as part of a 4×4 transformation matrix in homogeneous coordinates
What properties do rotation matrices have?
Orthonormal
Their transpose is equal to their inverse
Preserves lengths and angles
Map standard basis axes x, y, z to an orthonormal basis e1, e2, e3
How is translation represented in 3D?
Using a 4×4 matrix and homogeneous coordinates.
Why do we use homogeneous coordinates for transformations?
To uniformly represent all types of transformations (rotation, translation, scale) as 4×4 matrices.
What’s the difference between points and vectors in homogeneous coordinates?
Points: last coordinate is 1
Vectors: last coordinate is 0 (they’re not affected by translation)
What is a scene graph?
A hierarchical data structure representing objects and their transformations in a scene.
What is object instancing in a scene graph?
Reusing the same geometry data across multiple objects by pointing to one copy and applying different transforms.
Why are scene graphs memory-efficient?
They avoid duplicating geometry by sharing instances and applying different transformations per node.
How are transformations composed in a scene graph?
By multiplying transformation matrices as you traverse the graph from root to leaf.
What’s the result of composing multiple 4×4 transformation matrices?
A single matrix representing the cumulative transformation.
