Computer Graphics
Jochen Lang
[email protected]
Faculte de genie | Faculty of Engineering
Jochen Lang, EECS [email protected]
This Lecture
Scenegraph and Viewing Transformations
Tomas Akenine-Moller et al., Chapter 4.2
Marschner and Shirley, Chapters 6.4, 6.5, 12.2 Scenegraph transforms
Viewing and canonical viewing volume
Jochen Lang, EECS [email protected]
Scenegraph Transformations
A Scenegraph is a hierarchical data structure to represent the content of a scene:
Defines spatial relationship between objects relative to each other in a graph
Scenegraphs are extremely useful for animation
OpenGL does not provide a scenegraph; need to implement ourselves
Similar to a kinematic chain in robotics
Scenegraph APIs: Inventor, Java3D, OpenSG, VRML97, X3D, Three.js,
Jochen Lang, EECS [email protected]
Modelling with a 2D Scenegraph: Example
Assemble complex objects from primitives
Instancing
house
door
window
window
Box
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3D Articulation Example:
Animal out of cylinders and spheres:
Sphere radius = , cylinder height = , angle =
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3D Articulation Example:
Transformations:
hip joint
upper leg
lower leg
Jochen Lang, EECS [email protected]
3D Articulation Example:
Knee rotation:
Jochen Lang, EECS [email protected]
3D Articulation Example:
Hip rotation:
Knee rotation unchanged:
Jochen Lang, EECS [email protected]
Example: Java3D Scenegraph
Behavior
Object 1
Table
Chair T BG
Virtual Universe
BG
Locale
BG
T
Viewing
View VP
Plate BG
TT SSS
Appearance
Geometry
Jochen Lang, EECS [email protected]
Scenegraph in Three.js
THREE.Scene is organized a scenegraph
THREE.Object3D is the parent class of most 3D objects
Each object has a parent field, and a
Children field which holds an array of children
Note however, the documentation says to use Object3D as leave nodes and use Group (see below) for building the graph
THREE.Group is a node that serves as a parent for multiple nodes
Jochen Lang, EECS [email protected]
Scenegraph Transforms in Three.js
Each object in the scenegraph holds the following transforms
.matrix Local transform relative to the parent
This is the transform that is typically modified
.modelViewMatrix The model-view matrix for this object in the current image
This is needed to draw the object
.matrixWorld The transform from the object to the root
This is an intermediate step for the above
.normalMatrix The matrix to transform normals by Technically it is the transpose of the inverse of the
upper left 33 sub-matrix of the model-view matrix as discussed.
Jochen Lang, EECS [email protected]
Canonical View Volume
A volume where the output of our program takes place
Square at the origin which a side-length of 2
Chosen as a 3D volume
Need a mapping from the canonical view volume to screen coordinates
Jochen Lang, EECS [email protected]
Mapping to Screen Coordinates
z-coordinate is 1
x and y must map to pixels
pixels are of size 1 with the center at 0.5
corresponds to glViewport( xpixel, ypixel, nx, ny )
Jochen Lang, EECS [email protected]
Mapping in Pictures I
Object Transform
bottom
top
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Transformation Matrices
Changes direction of y-axis
Jochen Lang, EECS [email protected]
Mapping in Pictures II Frame Transform
Jochen Lang, EECS [email protected]
Transformation Matrices
Frame transform is inverse of object transform!
Jochen Lang, EECS [email protected]
Mapping in Pictures III Reverse Frame Transform
Jochen Lang, EECS [email protected]
Transformation Matrices
Frame transform in reverse is the object transform!
Jochen Lang, EECS [email protected]
Next Weeks Lectures
Viewing Transformations
Textbook: Chapter 4.7 Orthographic Projection Perspective Projection Viewing Transformation
Jochen Lang, EECS [email protected]
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