الفهرس 
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Abstract
In this work, the problem of construct ing mathemat ical
models of a 3-D surfaces of arbitrary shapes is considered. A
survey is presented for previous work and a comparison between
existing models imply that the field is dominated by models
based on parametric funct ion. However, the parametr ic
functions used is a subclass of the implicit algebraic
functions, the parametric functions are not closed under many
geometric operations under which the implicit algebraic
functions are closed, the algebraic degree of a degree
n-parametric surface is n2 and a bidegree n-parametric surface
can be of algebraic degree up to 2n2, whereas for a surface
defined by an implicit algebraic equation its algebraic degree
is the same as the degree of the defining equation, more~ver
the point classification problem is greatly simplified when
using implicit algebraic representation. For these reasons and
others, attention is focused on modeling with implicit
function, especially that of low degree.
A mathemat ical model is developed to control
singularities of quadrics and cubics, also a mathematical
model for detrmining the equations of the projections of the
intersection of two algebraic surfaces is established.
The above entities are utilized in constructing three
models for 3-D surfaces of arbi trary shapes based on low
degree implicit functions. In the first model the building
unit is a quadratic patch with natural intersection boundaries
and a near et overall continuity. In the second model the
building unit is a cubic implicit patch with quadratic
parametric boundary curves. A CO overall continuity is
gauranteed wi th at least two degrees of freedom used to
control singularities and to enhance the overall continuity to
be eO+. In the third model a side vertex interpolation scheme
is used to construct three cubic patches for every input
triangular face, these patches are then combined to produce a
surface with an overall et continuity.
The necessary transformations, hidden-surfaces elimination,
shades, and shadows are examined for completeness of the work.