Geometric Properties Of Rational Bézier Curves With SPS Parameter

Rational Bezier curves with SPS (scalar projection scale) parameterization are useful in geometric molding. In order to study their geometric properties, firstly we found that when a rational Bezier curve degenerates into Bezier curve it will have a intuitive geometric character. Secondly, we showed that a rational Bezier curve with SPS parameterization is still SPS parame-terized after degree elevation. Finally, by Descartes'rule of signs, the monotonicity of curvature is discussed for a rational quadratic Bezier curve parameterized by SPS, and the curvature distribution are obtained. Torsion is the geometric properties of a space curve. It contains a derivative, second derivative and three derivative. Therefore, the expression of the torsion has certain complexity. Here we only consider torsion monotony for a special Bezier cubic curve. The paper mainly contains four parts:? In Chapter 1, This paper introduces the SPS parametric rational Bezier curve, its curvature and the cubic Bezier curve of torsion their research background and significance, discuss the research status quo and the trend of future development.? In Chapter 2, This paper introduces the definition of rational Bezier curve and expression, as well as several properties of the curve. Secondly, introducing the definition of SPS parametric curve. A necessary and sufficient condition for SPS parametric curve is also given.? In Chapter 3, Study the curvature monotony of SPS parametric quadratic rational Bezier curve. Introduces the application of curvature monotonicity in the offset, and shows the actual example.? In Chapter 4, Study the cubic Bezier curve of torsion monotonicity, and gives the corre-sponding examples to show.