1. Interactive Computer Graphics Viewing
James Gain and Edwin Blake
Department of Computer Science University of Cape Town
July 2002
Collaborative Visual Computing Laboratory

2. Interactive Computer Graphics Contents
Map of the Lecture
Classical Viewing:
Orthographic, Axonometric, Oblique, Perspective
Camera Positioning and Viewing APIs
Simple Projection Matrices
Implementing Projections in OpenGL
Projections and Shadows
Transformations
Viewing
Shading
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3. Viewing Principles
Image formed by intersection of projection plane and lines of projection from object to viewer
Perspective Viewing:
Projectors converge to a finite Centre of Projection (COP)
Objects in the distance are shrunk
Parallel Viewing:
As COP is moved to infinity, projectors become parallel
Projectors oriented along a Direction of Projection (DOP)
Preserves relative lengths and angles
Perspective Projection
Parallel Projection
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4. Classical Viewing: Parallel I
Unlike CG, Classical viewing depends on a specific relationship between object and viewer
Many objects (e.g. buildings) have a box-like shape with 6 principal faces oriented relative to the projection plane
Orthographic:
projection plane is parallel to a single principal face
Preserves both distance and angles
But usually need multiple projections because only on face visible at a time
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5. Classical Viewing: Parallel II
Axonometric:
Projectors are orthogonal to projection plane
Angles are no longer preserved
Isometric: three principal faces symmetric with respect to the plane
Dimetric: two prinicipal faces symmetric with respect to plane
Trimetric: general case
Oblique:
Projectors are parallel to each other but make an arbitrary angle with the projection plane
Not physically realistic
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6. Classical Viewing: Perspective
Characterized by diminution
Distortion is non-linear:
Unable to take measurements
But more realistic
Categorised according to number of visible faces and consequent number of vanishing points for principal directions
All classical views are simply a subset of either general parallel or perspective CG viewing
Three-point Two-Point One-Point
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7. Transformation Pipeline
Modelling Transform
Object in
object
co-ordinates
Object in
world
co-ordinates
Viewing Transform
Object in
2D screen
co-ordinates
Perspective Projection
Object in
viewing
co-ordinates
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8. Positioning of the Camera
The viewpoint (camera) is located at the origin and the projection plane is parallel to the plane at a distance along the negative axis
Camera Transformation:
Must be able to position and orient the camera arbitrarily
Transform objects from modelling coordinates to camera coordinates
Need an appropriate way of specifying the camera position and orientation
OpenGL:
Camera has a left-hand coordinate system
Camera transformation is part of the GL_MODELVIEW state
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9. Interactive Computer Graphics Contents
u-v-n Viewing
Create a camera frame (point and orthonormal vectors):
Positioned at a view-reference point (VRP)
Viewing pointed along the view plane normal ( ). Determines the orientation of the viewplane
Viewing oriented by the view-up vector (VUP). Sets the roll of the viewplane
Construct a camera frame
= Projection of VUP onto plane
= Normalizaion of
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10. Interactive Computer Graphics Contents
Look-At Viewing
Camera located at an eyepoint (eye) specified in world coordinates, pointed towards an atpoint (at) with a view-up vector (up)
gluLookAt(eyex, eyey, eyez, atx, aty, atz, upi, upj, upk);
Alters the modelview matrix to match this camera transform
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11. Euler and Polar Viewing
Other applications require a non-rectilinear coordinate system
Flight Simulation:
Three Euler angles (roll, pitch and yaw) relative to a vehicle’s centre of mass
Stellar Cartography:
Polar coordinates (azimuth, elevation and distance) relative to a ground plane and position
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12. Perspective Projection
From 3D world co-ordinates to 2D screen co-ordinates
Perspective projection provides an illusion of depth in an image by giving distant objects a smaller projected screen size
The screen co-ordinates of a point are calculated using similar triangles
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13. Geometry of Perspective Projection
Triangle is similar to
Similarly:
The Projection Transform is:
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14. Exercise: Binocular Projection
Calculate the separate screen projections and for the left and right eyes of a point The eyes are located at and
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15. Solution: Binocular Projection
Right eye:
Triangle is similar to
Left eye:
By a similar derivation
The co-ordinate projection is unchanged because our eyes are only separated horizontally.
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16. Orthogonal Projection
Projectors are perpendicular to the viewplane
Points retain their and values:
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17. Interactive Computer Graphics Contents
Projections in OpenGL
Need to consider focal length and size of the film plane
View Volume:
Determines the field of view for clipping objects
A truncated pyramid (frustum) with apex at the COP
Specified by front and back clipping planes and a horizontal and vertical angle of view
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18. Perspective Viewing in OpenGL
glFrustum(xmin, xmax, ymin, ymax, near, far);
near and far must both be positive. Note: they are specified in camera coordinates, so
gluPerspective(fovy, aspect, near, far);
fovy is the angle between the top and bottom frustum planes, aspect = width / height
Must make sure to first select the GL_PROJECTION matrix mode:
glMatrixMode(GL_PROJECTION);
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19. Parallel Viewing in OpenGL
glOrtho(xmin, xmax, ymin, ymax, near, far);
Again
But near and far are not restricted to being positive
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20. Hidden Surface Removal
Opaque surfaces cause occlusion
Use Hidden-Surface Removal to delete invisible surfaces
Z-buffer algorithm:
As polygons are rasterized store depth value of projected points. Only render a point with current minimum depth
Worst-case complexity equals number of polygons
Need specialized memory - depth or z buffer
Amenable to hardware implementation
z-buffer Algorithm
OpenGL z-buffer:
glutInitDisplayMode(GLUT_DEPTH); initialize z-buffer in display
glEnable(GL_DEPTH_TEST); enable hidden surface removal
glClear(GL_DEPTH_BUFFER_BIT); clear z-buffer before rendering
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21. Projection Normalization
Projection Requirements:
Handle general view volumes
Retain depth information (required by HSR) as far down the pipeline as possible
Projection Solution:
Decompose projection into a distortion and canonical orthographic projection
Canonical View Volume glOrtho(-1.0, 1.0, -1.0, 1.0, , 1.0);
Perspective Projection = Distortion and Orthographic projection
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22. Orthogonal Projection
glOrtho(xmin, xmax, ymin, ymax, near, far)
 glOrtho(-1.0, 1.0, -1.0, 1.0, -1.0, 1.0)
Translate:
Centre of view volume to origin
Scale:
View volume to cube with sides of length 2
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23. Perspective Normalization
Convert perspective view frustrum to orthographic cube
gluPerspective(90.0, 1.0, zmin, zmax)
 glOrtho(-1.0, 1.0, -1.0, 1.0, -1.0, 1.0)
Near plane at zmin, far plane at zmax, fovy = 90 deg, aspect = square
Mapping is non-linear but preserves the ordering of depth
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24. Projection and Shadows
Shadows are useful shape cues
Fully general shadowing algorithms are expensive
But simple shadows (from a point light source onto a flat plane) can be simulated with projection
Set COP to light source, apply perspective transform to polygon in front of projection plane to create shadow polygon
Example:
Light source
Shadow projection onto
Exercise: Write out
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