Viewing in 3D - UBC ECE
Viewing in 3D
1. Projections Types:
• perspective
• parallel / orthographic
2. Viewing Parameters:
• position of the viewer’s eye
• location of the viewing plane
• 2-coordinate systems
o the scene (object)
o viewing eye coordinate system
3. Clipping:
• against 3D-view volume
4. Projection and displaying on viewport
Consider Projection Transformation :
[pic]
Projection:
It is a process that transform points in a coordinate system of dimension n into points in a coordinate system of dimension less than n.
Projection from 3D to 2D tools:
• projectors: straight projection rays
• center of projection (COP)
• projection plane ==> (not curved surface).
This type is called Planar Geometric Projection.
Types of Projections:
1. Perspective Projection
• projectors meet at center of projection (eye, camera)
• approximation of human visual system
• distortion close to COP
2. Parallel Projection
• Projectors are parallel
• COP is at infinity
[pic]
[pic]
1. Perspective Projection:
• The size of perspective projection of an object varies inversely with the distance of that object from the center of projection
o not realistic
o distances cannot be taken from the projection
o angles are preserved on only those faces of the objects parallel to the projection plane
• Vanishing point:
o The perspective projection of any set of parallel lines that are not parallel to projection plane converges to a vanishing point. ==> (parallel lines meet at infinity ==> [pic])
• Axis vanishing point:
o When the set of lines is parallel to one of the 3 axes(, x, y and z).
[pic]
What is the maximum number of vanishing points?
[pic]
One-point perspective projection of cube onto plane cutting the z-axis. The projection plane normal is parallel to the z-axis.
[pic]
2. Parallel Projections:
Depending on the relation between the direction of projection and the normal to the projection plane:
a - orthographic :
both directions are the same (or the reverse of each other)
b - oblique:
direction of projection is not normal to the projection plane.
Examples:
(1) Orthographic Projection
[pic]
(2) Oblique Projection
(projectors are not normal to projection plane)
[pic]
(3) Isometric Projection:
It is the kind of projection generated when the projection plane normal makes equal angles with each principle axis.
[pic]
(4) Axonometric Orthographic Projection:
The type of projection that uses planes that are not normal to a principal axis and therefore show several faces of the object at once.
Remarks on Parallel Projection:
• less realistic view
• can be used for exact measurements
• parallel lines remain parallel
• angles are preserved only on faces of the object parallel to the projection plane.
Projection of 3D scene onto 2D screen has two major components:
1 specification of a camera
2 specification of a viewing transformation
The viewing transformation:
specification of the parameters:
- A field-of-view angle, (
- Near and far bounding planes perpendicular to z.
- A 3D view of the camera and its viewing space as in the figure below:
[pic]
Camera Viewing Space
Consider a side view of such a space:
[pic]
Viewing Pyramid
note: u-axis coming out of the paper:
note: ( forms a viewing volume in the shape of a pyramid with the camera at the apex of the pyramid and the negative-w axis of the pyramid
The Viewing Transformation Matrix
Given the specs of parameters [pic], we define the transformation of 3D scene elements to the cube [pic] is:
[pic]
viewing transformation matrix
Development of the matrix:
[pic]
- consider camera at origin
- use similar triangles:
[pic]
[pic]
The transformation that projects
[pic]
This can be expressed in H-D homogeneous coordinate:
[pic]
In a Matrix form:
[pic]
Consider:
Need to transfer the viewing pyramid defined by (, n and f into the cube [pic].
[pic]
To transform the truncated viewing pyramid to the cube, P-matrix can be used:
[pic] (*)
where a and b are chosen constants which will cause the w values of the transformed truncated viewing pyramid to lie in the range [pic].
So we get:
[pic]
and
[pic]
We have:
[pic]
and
[pic]
Projecting back to the 3D we get:
[pic]
and
[pic]
In order that the values on the left map to (0,0,1) and (0,0,-1) respectively we must have:
-dan+b = n
and
-daf+b = -f
subtract these equations and solve for (a)
[pic]
by substitution:
[pic]
Inserting these values in our transformation matrix P indicated by * above
[pic]
[pic]
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