Computer Graphics
Jochen Lang
jlang@uottawa.ca
Faculté de génie | Faculty of Engineering
Jochen Lang, EECS jlang@uOttawa.ca
This Lecture
• Drawing and Animation
– Drawing and graphics pipeline – Modelling a Tetrahedron
– Projections from 3D to 2D
– Rasterization
– Animations
Jochen Lang, EECS jlang@uOttawa.ca
Drawing a Simple 3D Shape
• Tetrahedron: 4 triangles, 4 edges or 4 vertices
Jochen Lang, EECS jlang@uOttawa.ca
Modeling
• We can describe the tetrahedron by the positions of the vertices
• We need to use three dimensions
• 4 Vertices:
[-1.0 -1.0 -1.0] [1.0 -1.0 -1.0] [0.0 1.0 0.0] [0.0 -1.0 1.0]
• This means we have a coordinate system
• Redaxisisx,greenisyandblueisz
Jochen Lang, EECS jlang@uOttawa.ca
Image is 2D
• We can map to 2D by simply dropping the z-axis
• 4 Vertices:
[-1.0 -1.0 -1.0] [1.0 -1.0 -1.0] [0.0 1.0 0.0] [0.0 -1.0 1.0]
• Orthographic Projection
• But looks not too nice, why?
Jochen Lang, EECS jlang@uOttawa.ca
Historical Perspective
Source: Wikimedia
Jochen Lang, EECS jlang@eecs.uOttawa.ca
Perspective
Source: Wikimedia
Jochen Lang, EECS jlang@eecs.uOttawa.ca
Perspective Projection is Important
• Distance from viewer determines image size
• Examples:
– Railroad standard gauge is 1,435 mm
– Airport tunnel
Source: Wikimedia
Jochen Lang, EECS jlang@uOttawa.ca
1D Perspective Projection
• Cameras and eyes can be approximated by a camera pinhole model.
• In2D:
Similar triangles
focal length
height
Source: Wikimedia
depth pinhole
projected height
Jochen Lang, EECS jlang@uOttawa.ca
Perspective Viewing
• Size change because of depth
image plane
optical axis
Jochen Lang, EECS jlang@uOttawa.ca
Effect of Focal Length
– Zoom lenses have a large focal length
– Wide angle lenses have a short focal length
Source: Wikimedia
Jochen Lang, EECS jlang@uOttawa.ca
Apply to Tetrahedron
• Perspective view
• Additional information is needed
– Camera placement
– Placement
– Field of view or focal length – FoV
• How to apply the perspective transform?
Jochen Lang, EECS jlang@uOttawa.ca
Relative Vertex Positions
• First step:
– In general: Make the camera focal point the origin of our coordinate system
– Here, we effectively update the z-value
Jochen Lang, EECS jlang@uOttawa.ca
Field of View
• Alternate specification of focal length given image size
– Opening angle of camera
– horizontal, vertical and diagonal are used
Source: Wikimedia
Jochen Lang, EECS jlang@uOttawa.ca
Conversion to Focal Length
image plane
• Assume square image
pinhole
• Let’s assume image plane is from -1…1 in both directions:
Jochen Lang, EECS jlang@uOttawa.ca
2D Positions
• Take the relative coordinate
• Flip the z-axis and hence mirror the image
• Apply
. () . . .
• Result
• Same for other vertices
Jochen Lang, EECS jlang@uOttawa.ca
Another Example
– Cube under perspective projection
– Vanishing points (Parallel line on a tilted plane intersect not at infinity).
vanishing point 2
center of projection
vanishing point 1
image plane
Wikimedia
Source:
Jochen Lang, EECS jlang@uOttawa.ca
Viewing Geometry: Pinhole Camera
• Relate the image plane to the 3D world by projective projection.
• Extrinsic parameters:
– Position of pinhole and orientation of camera
– Rigid body transform
– Orientation specified by optical axis and image orientation
• Intrinsic parameters:
– Focal length possibly different in x and y
– Origin image coordinates – center of projection
Jochen Lang, EECS jlang@uOttawa.ca
Rasterization
• So far only dealt with the vertex positions but need to draw triangles
– Need to know which vertices form a face
– E.g. index the vertices 1,…,4 then faces are: 2 1 3, 3 1 4, 4 1 2 and 3 4 2
– This gives edges, e.g., 2 1 3 implies an edge .
• Display is only pixels in a raster – Need to rasterize the triangles – May add colors
Jochen Lang, EECS jlang@uOttawa.ca
Summary Graphics Pipeline
• Four major tasks: – Modelling
• Produces geometric primitives and description of surface, volume and light properties
– Geometric processing
• Normalization (change to camera, canonical coordinates or normalized device coordinates), clipping, culling, hidden-surface removal and possibly shading
– Rendering: Rasterization or Ray Tracing
• Rasterization: Deciding on mapping of pixels to primitives, e.g., triangle rasterization and line drawing, followed by fragment processing
– Display
• transfer of frame buffer to screen; may require, e.g., anti-aliasing and tone mapping
Jochen Lang, EECS jlang@uOttawa.ca
Animations
• Movement of tetrahedron
– 2D animation or 3D animation – Some techniques:
Source: Wikimedia
• Skeletonal animation or transformation based
– Every ball position is encoded in a transformation matrix and the transformation are interpolated or updated
• Physicis-based
– The position is updated based on some
Jochen Lang, EECS jlang@uOttawa.ca
physical laws, e.g., dynamics
• Key frames plus interpolation
– Every ball position is a keyframe image
– Interpolate the image between keyframes
Next lecture
• Drawing and Animation
– Drawing and graphics pipeline – Modelling a Tetrahedron
– Projections from 3D to 2D
– Rasterization
– Animations
Jochen Lang, EECS jlang@uOttawa.ca