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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