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International cooperation and technology transfer
Mussio, Luigi

G.Fangi - G. Gagliardini - Ancona University - e.mail:fangi@poscsi.unian.it
KEY WORDS: Virtual reality, representation, computer vision, photogrammetry
ABSTRACT: The virtual reality is a quick and intuitive method to reproduce and understand the reality, by moving it or moving
around it. In the geodetic sciences virtual reality can help in representing the surveys or the objects of the surveys. The first step of
the virtual reality is the multiple transformations. They can be obtained as concatenation of simple transformation called basic
transformation. We examine the basic transformations that are the translation, the rotation, the reflection, the perspective and the
projection. To include in a matrix any basic transformation, and to represent a point at infinity, homogeneous co-ordinates are used.
After the scene transformations, the clipping takes place, say removing hidden lines. Finally to make realistic the object, the
rendering is the final step. We present some examples of architectural surveys and their animation.
1. The scene transformations
A quick and intuitive method to reproduce the reality, by
moving around it.
The virtual reality reproduces these movements.
As scene we intend the set of objects used to represent the
The scene transformations are the transformations applied to
the scene to obtain these movements.
To depict and represent three-dimensional objects on planes
supports, has always created many problems to the craftsman
and designers, because of the need to translate their ideas in
understandable ‘formats’ by those who will be involved in
the realisation of the project. The introduction of the
computer is modifying the things. Actually it is possible to
reproduce virtual models of the real objects and use them to
represent the realty. A man placed in front of an unknown
object, examines it by moving it. The scene transformation
reproduces, virtually, this phenomenon. Actually a three-
dimensional virtual reality doesn't exist, so that the three-
dimensional model must be projected on a plane support, that
generally is that of the screen. In this manner the information
relative to the depth of the object, are lost. The scene
transformations, permitting the motion of the virtual object,
cope with this problem.
In this paper we will examine these transformations, that in
combination reproduce the motion of the object or the motion
of the operator around the object.
The representation of a point in the 3D space is got with a
four dimensions vector, the homogeneous co-ordinates. The
use of the homogeneous co-ordinates, is needed to represent
points to infinity and to include a translation in a
transformation matrix. It is sufficient that the fourth
component of the vector is equal to zero to represent a point
to infinity. The representation in homogeneous co-ordinates
of a point of the 3D space is not unique.
Note how the representation changes with the parameter h.
For sake of representation, since the fourth component has
been added to solve the inconvenient of the infinity, we pose
h equal to the unity when the point is to the finite. Anyhow,
independently from that, a 4x4 matrix must represent any
transformation applied to a point in homogeneous co
ordinates. In this manner:
[hx, hy, hz, h] = [x,y,z, 1 ] [ T] (1)
~a b c
d e f
g 1 J
~1 m n~
Factor of scale
c. omog.
where [T\ is the matrix of the transformation. Such a matrix
in general will have the form:
In a general forma eqns. (1) represent the 3D transformations.
Let’s analyse the transformations one by one.
2. The basic transformations
All the transformations can be arranged by matrix
concatenation of small amount of them: the basic
• translation
• scale variation
• shear deformations
• rotation through one co-ordinate axis
• reflection about one co-ordinate plane
• Transformations of translation
It is the simplest transformation. It moves the object, without
modifying the form or the orientation respect to the reference
system axes. We get a matrix having the general form (3):
10 0 0
0 10 0
0 0 10
l m n 1
The translated homogeneous co-ordinates are obtained by
writing (3):
By expansion of the products we have:
[x' ,y',z',h]= [(* + /), (y + m), (z + H),l] (4)
The transformed co-ordinates are (5):
The present research has been financed by Cofm97