曲线曲面造型中的若干相关问题研究
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摘要
随着产品设计和生产自动化程度的不断提高,许多产品在制作前要进行曲线与曲面的几何造型设计,例如船体的放样与设计,汽车车壳设计,飞机机翼、机身及机舱的设计,甚至刀架、刀片,鞋模和服装的设计都归结于曲线曲面造型的研究,这也是目前计算机辅助几何设计(CAGD)研究的核心内容.曲线曲面造型经历了参数样条方法、Coons曲面、Bezier曲线曲面和B样条方法.另外,利用插值、拟合、逼近、拼接等方法研究满足某种性质的曲线曲面在实际应用中也是非常重要的.本文研究了以Bezier曲线为测地线的可展曲面拼接,插值测地线的极小曲面逼近,过曲率线的参数曲面及可展曲面的构造,还研究了异度隐函数样条曲线曲面并将其应用于参数曲线的分段近似隐式化,给出了一种多层的3次样条拟插值格式,构造了一种简单有效的数值积分公式.主要工作包括：
1.测地线是曲面上一类很重要的曲线.例如,制鞋工业中把皮革看成可展曲面,鞋面上的足背线相当于曲面上的测地线,服装业则以腰线为测地线.而在实际应用中,一片这样的曲面往往是不够用的,所以要考虑曲面的拼接问题.本文研究了以空间Bezier曲线为测地线的可展曲面的拼接问题,并给出了拼接条件.重点研究了以3次Bezier曲线为测地线的可展曲面的拼接,给出了其控制顶点所满足的关系方程.
2.极小曲面具有很多重要的性质,在建筑设计、飞机制造、生物学、分子化学等领域都有重要应用.另外,在产品设计中,我们不仅希望曲面插值某些特征线,比如测地线,还希望消耗更少的材料.基于这一点,本文把测地线和极小曲面相结合,研究了插值测地线的极小曲面逼近问题.
3.曲率线是曲面上另外一类很重要的曲线.曲面上的曲线,如果它在所有点的切方向都是主方向,则称为曲率线.我们利用Frenet标架表示曲面,推导出了曲线为所构造曲面上曲率线的充要条件,并引入了两个控制函数θ(s),λ(s)来控制曲面的形状,根据θ(s)的表达式将充要条件进行分类,给出了曲线既是曲率线又是测地线的条件.
4.可展曲面和曲率线在几何设计和曲面分析中起着很重要的作用.本文给出了插值曲率线的可展曲面的构造方法,并且给出了曲面的具体表达形式.分析了可展曲面分别是柱面、锥面、切线面时的充要条件,并利用控制函数θ(s)控制曲面的形状.
5.在曲线曲面造型中,隐式曲线曲面的应用是非常广泛的.我们通过添加辅助曲线曲面,提出了异度隐函数样条曲线曲面的方法，并对其插值性、凸性以及正则性进行了详细分析.在此基础上提出了一种简单有效的参数曲线分段近似隐式化的方法.
6.在数值逼近的理论和应用研究中,拟插值是非常重要的.径向基拟插值和3次样条拟插值是常用的拟插值格式.本文提出了一种多层的一元3次样条拟插值格式,与LD,LR以及3次样条拟插值进行比较,有很好的逼近效果,并把这种格式应用于数值积分,构造了一种简单且有效的数值积分公式.
With the improvement of product design and production automation, many products need geometrical modeling design of curve and surface before fabricating. For example, loft and design of hull, design of car shell, design of airplane wing, body and cabin, even the design of knife rest, blade, mold, cloth all attribute to the research of curve and surface modeling, this is also the core content of Computer Aided Geometric Design (CAGD). Curve and surface modeling experience parametric spline, Coons surface, Bezeir curve and surface, B-spline, etc. In addition, by using interpolating, fitting, approximation, blending to study curve and surface which satisfy some condition is very importance in practical application. In this paper, we study the blending of developable surface which contains a Bezier curve as a geodesic, designing approximation minimal parametric surfaces with geodesies and the construction of parametric and developable surface through line of curvature. We also discuss functional spline curves and surfaces with different degree of smoothness and apply it to piecewise approximate implicitization of parametric curves, give a multi-level univariate quasi-interpolation scheme and construct an easy and valid integration formula. The main work is as follows:
1. Geodesic is an important curve on a surface. For example, in shoe-making industry the leather is considered as developable surface, and the girth is the geodesic on the shoe surface. In garment-manufacture industry, the waist line is also the geodesic of the cloth. However, only one developable surface is not enough in practical application. So the G1connection of some developable surfaces should be considered. We study the G1connection of the developable surfaces containing G1abutting geodesics, and derive the corresponding conditions. We primarily study the condition of G1connection of two developable surfaces possessing cubic Bezier geodesies, and give the equations that the control points satisfy.
2. Minimal surface has many important properties. It has lots of applications in architectural design, biology, molecular chemistry and so on. Moreover, in product design, we not. only hope the surfaces interpolate some characterizing curves, such as geodesies, but also hope to consume more less material. Based on these, we consider the problem of designing approximation minimal parametric surfaces with geodesies.
3. Line of curvature ia another important characteristic curves on a surface. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. We express the surface pencil by utilizing the Frenet frame and derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface and introduce two control functions θ(s) and λ(s) to control the shape of the surface. Moreover, we classify the condition according to the expression of θ(s), and derive the condition when the given curve satisfy the line of curvature and the geodesic.
4. Developable surface and line of curvature play an important role in geodesic design and surface analysis. We propose a new method to construct a developable surface possessing a given curve as the line of curvature of it, and give the concrete expression of the surface. We analyze the necessary and sufficient conditions when the resulting developable surface is cylinder, cone or tangent surface. The control function can control the shape of the resulting surface.
5. Implicit curves and surfaces are extensively used in curve and surface modeling. By adding auxiliary curves and surfaces, the functional spline curves and surfaces with different degrees of smoothness are presented. We analyze the interpolation, convexity, and regularity in detail. Based on functional spline curves (with different degrees of smoothness), we propose an easy and valid method for implicitization.
6. Quasi-interpolation is very important in the study of the approximation theory and applications. Radial Basis Function (RBF) and cubic spline quasi-interpolation are two common schemes. We propose a multi-level univariate quasi-interpolation scheme with better approximation than LD, LR and cubic spline quasi-interpolation. Moreover, we apply it to numerical integration and construct an easy and valid integration formula.
1.测地线是曲面上一类很重要的曲线.例如,制鞋工业中把皮革看成可展曲面,鞋面上的足背线相当于曲面上的测地线,服装业则以腰线为测地线.而在实际应用中,一片这样的曲面往往是不够用的,所以要考虑曲面的拼接问题.本文研究了以空间Bezier曲线为测地线的可展曲面的拼接问题,并给出了拼接条件.重点研究了以3次Bezier曲线为测地线的可展曲面的拼接,给出了其控制顶点所满足的关系方程.
2.极小曲面具有很多重要的性质,在建筑设计、飞机制造、生物学、分子化学等领域都有重要应用.另外,在产品设计中,我们不仅希望曲面插值某些特征线,比如测地线,还希望消耗更少的材料.基于这一点,本文把测地线和极小曲面相结合,研究了插值测地线的极小曲面逼近问题.
3.曲率线是曲面上另外一类很重要的曲线.曲面上的曲线,如果它在所有点的切方向都是主方向,则称为曲率线.我们利用Frenet标架表示曲面,推导出了曲线为所构造曲面上曲率线的充要条件,并引入了两个控制函数θ(s),λ(s)来控制曲面的形状,根据θ(s)的表达式将充要条件进行分类,给出了曲线既是曲率线又是测地线的条件.
4.可展曲面和曲率线在几何设计和曲面分析中起着很重要的作用.本文给出了插值曲率线的可展曲面的构造方法,并且给出了曲面的具体表达形式.分析了可展曲面分别是柱面、锥面、切线面时的充要条件,并利用控制函数θ(s)控制曲面的形状.
5.在曲线曲面造型中,隐式曲线曲面的应用是非常广泛的.我们通过添加辅助曲线曲面,提出了异度隐函数样条曲线曲面的方法，并对其插值性、凸性以及正则性进行了详细分析.在此基础上提出了一种简单有效的参数曲线分段近似隐式化的方法.
6.在数值逼近的理论和应用研究中,拟插值是非常重要的.径向基拟插值和3次样条拟插值是常用的拟插值格式.本文提出了一种多层的一元3次样条拟插值格式,与LD,LR以及3次样条拟插值进行比较,有很好的逼近效果,并把这种格式应用于数值积分,构造了一种简单且有效的数值积分公式.
With the improvement of product design and production automation, many products need geometrical modeling design of curve and surface before fabricating. For example, loft and design of hull, design of car shell, design of airplane wing, body and cabin, even the design of knife rest, blade, mold, cloth all attribute to the research of curve and surface modeling, this is also the core content of Computer Aided Geometric Design (CAGD). Curve and surface modeling experience parametric spline, Coons surface, Bezeir curve and surface, B-spline, etc. In addition, by using interpolating, fitting, approximation, blending to study curve and surface which satisfy some condition is very importance in practical application. In this paper, we study the blending of developable surface which contains a Bezier curve as a geodesic, designing approximation minimal parametric surfaces with geodesies and the construction of parametric and developable surface through line of curvature. We also discuss functional spline curves and surfaces with different degree of smoothness and apply it to piecewise approximate implicitization of parametric curves, give a multi-level univariate quasi-interpolation scheme and construct an easy and valid integration formula. The main work is as follows:
1. Geodesic is an important curve on a surface. For example, in shoe-making industry the leather is considered as developable surface, and the girth is the geodesic on the shoe surface. In garment-manufacture industry, the waist line is also the geodesic of the cloth. However, only one developable surface is not enough in practical application. So the G1connection of some developable surfaces should be considered. We study the G1connection of the developable surfaces containing G1abutting geodesics, and derive the corresponding conditions. We primarily study the condition of G1connection of two developable surfaces possessing cubic Bezier geodesies, and give the equations that the control points satisfy.
2. Minimal surface has many important properties. It has lots of applications in architectural design, biology, molecular chemistry and so on. Moreover, in product design, we not. only hope the surfaces interpolate some characterizing curves, such as geodesies, but also hope to consume more less material. Based on these, we consider the problem of designing approximation minimal parametric surfaces with geodesies.
3. Line of curvature ia another important characteristic curves on a surface. A curve on a surface is a line of curvature if its tangents are always in the direction of the principal curvature. We express the surface pencil by utilizing the Frenet frame and derive the necessary and sufficient condition for the given curve to be the line of curvature on the surface and introduce two control functions θ(s) and λ(s) to control the shape of the surface. Moreover, we classify the condition according to the expression of θ(s), and derive the condition when the given curve satisfy the line of curvature and the geodesic.
4. Developable surface and line of curvature play an important role in geodesic design and surface analysis. We propose a new method to construct a developable surface possessing a given curve as the line of curvature of it, and give the concrete expression of the surface. We analyze the necessary and sufficient conditions when the resulting developable surface is cylinder, cone or tangent surface. The control function can control the shape of the resulting surface.
5. Implicit curves and surfaces are extensively used in curve and surface modeling. By adding auxiliary curves and surfaces, the functional spline curves and surfaces with different degrees of smoothness are presented. We analyze the interpolation, convexity, and regularity in detail. Based on functional spline curves (with different degrees of smoothness), we propose an easy and valid method for implicitization.
6. Quasi-interpolation is very important in the study of the approximation theory and applications. Radial Basis Function (RBF) and cubic spline quasi-interpolation are two common schemes. We propose a multi-level univariate quasi-interpolation scheme with better approximation than LD, LR and cubic spline quasi-interpolation. Moreover, we apply it to numerical integration and construct an easy and valid integration formula.
引文
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