Abstract: Geometric modeling plays an important role in the
constructions and manufacturing of curve, surface and solid
modeling. Their algorithms are critically important not only in
the automobile, ship and aircraft manufacturing business, but are
also absolutely necessary in a wide variety of modern applications,
e.g., robotics, optimization, computer vision, data analytics and
visualization. The calculation and display of geometric objects
can be accomplished by these six techniques: Polynomial basis,
Recursive, Iterative, Coefficient matrix, Polar form approach and
Pyramidal algorithms. In this research, the coefficient matrix (simply
called monomial form approach) will be used to model polynomial
rectangular patches, i.e., Said-Ball, Wang-Ball, DP, Dejdumrong and
NB1 surfaces. Some examples of the monomial forms for these
surface modeling are illustrated in many aspects, e.g., construction,
derivatives, model transformation, degree elevation and degress
reduction.
Abstract: Newton-Lagrange Interpolations are widely used in
numerical analysis. However, it requires a quadratic computational
time for their constructions. In computer aided geometric design
(CAGD), there are some polynomial curves: Wang-Ball, DP and
Dejdumrong curves, which have linear time complexity algorithms.
Thus, the computational time for Newton-Lagrange Interpolations
can be reduced by applying the algorithms of Wang-Ball, DP and
Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong
algorithms, first, it is necessary to convert Newton-Lagrange
polynomials into Wang-Ball, DP or Dejdumrong polynomials. In
this work, the algorithms for converting from both uniform and
non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and
Dejdumrong polynomials are investigated. Thus, the computational
time for representing Newton-Lagrange polynomials can be reduced
into linear complexity. In addition, the other utilizations of using
CAGD curves to modify the Newton-Lagrange curves can be taken.
Abstract: In image processing and visualization, comparing two
bitmapped images needs to be compared from their pixels by matching
pixel-by-pixel. Consequently, it takes a lot of computational time
while the comparison of two vector-based images is significantly
faster. Sometimes these raster graphics images can be approximately
converted into the vector-based images by various techniques. After
conversion, the problem of comparing two raster graphics images
can be reduced to the problem of comparing vector graphics images.
Hence, the problem of comparing pixel-by-pixel can be reduced to
the problem of polynomial comparisons. In computer aided geometric
design (CAGD), the vector graphics images are the composition of
curves and surfaces. Curves are defined by a sequence of control
points and their polynomials. In this paper, the control points will be
considerably used to compare curves. The same curves after relocated
or rotated are treated to be equivalent while two curves after different
scaled are considered to be similar curves. This paper proposed an
algorithm for comparing the polynomial curves by using the control
points for equivalence and similarity. In addition, the geometric
object-oriented database used to keep the curve information has also
been defined in XML format for further used in curve comparisons.