Parametric spiral and its application as transition curve
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Date
2009
Authors
Ahmad, Azhar
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Abstract
The Bezier curve representation is frequently utilized in computer-aided design
(CAD) and computer-aided geometric design (CAGD) applications. The curve is
defined geometrically, which means that the parameters have geometric meaning;
they are just points in three-dimensional space. Since they are also polynomial,
resulting algorithms are convenient for implementation in an interactive computer
graphics environment. However, their polynomial nature causes problems in
obtaining desirable shapes. Low degree (cubic and quartic curve) segments may have
cusps, loops, and inflection points. Since a fair curve should only have curvature
extrema wherever explicitly desired by the designer. But, generally curves do not
allow this kind of behavior. Therefore, it would be required to constrain the proposed
cubics and quartics, so that the spirals are designed in a favorable way.
This thesis investigated the use of cubic spirals and quartic Bezier spirals as the
alternative parametric representations to other spiral functions in the literature. These
new parametric spirals were obtained by algebraic manipulation methods on the
monotone curvature variation of each curve. Results are reported showing that the
additional degree of freedom offers the designer a precise control of total-length and
the ability to fine-tune their curvature distributions. The methods and algorithms to
construct the G1 ,G2, and G3 transition spirals have also been presented. We
explore some common and new cases that may arise in the use of such spiral
segments for practical application of CAD/CAGD.
Description
PhD
Keywords
Mathematical science , Parametric spiral , Transition curve