一种G2连续插值过渡曲面建模方法

doi:10.11996/JG.j.2095-302X.2026040788

摘要/Abstract

摘要：

过渡曲面的构造是CAD曲面造型技术研究的热点和难点之一。随着实际工程应用对于曲面连续性要求的不断提高，传统滚球法在若干典型场景下已难以满足高阶光顺性的实际需求。现有过渡方法构造的过渡曲面通常为样条拟合形式，难以获取精确的位置与导数信息，且样条表示往往伴随较大的数据存储开销。因此，提出了一种满足G²连续(曲率连续)的过渡曲面插值方法，可实现曲面上点位信息的精确计算。着重分析了过渡曲面的边界约束表达与处理，通过构建合适的边界条件，结合相应的插值策略，保证了过渡曲面在边界处的G²连续性，并给出了过渡曲面的表达形式。该方法基于曲面上曲线进行插值，既能以简洁高效的方式完成过渡曲面建模，又可通过对边界曲线的合理调整以适配不同工程场景的需求。最后给出了曲面上各点导数的精确差分求解方法以及应用该方法生成G²连续过渡曲面的实例，同时通过计算法向量、法曲率以及生成斑马线对所得曲面的连续性进行了验证，结果表明该方法能够有效满足曲率连续性要求，证明了其合理性。

关键词: G²连续, 过渡曲面, 插值, 导数, 曲率

Abstract:

The construction of transition surface is one of the hotspots and difficulties in CAD surface modeling technology. With the continuous improvement of surface-continuity requirements in practical engineering applications, the traditional rolling-ball method can no longer meet the actual needs of high-order smoothness in several typical scenes. Blending surfaces constructed by the existing blending methods are usually represented in the form of spline fitting, making it difficult to obtain the accurate position and derivative information, and the spline representation is often accompanied by large data storage overhead. Therefore, a G² continuity (curvature continuity) interpolation method for blending surfaces was proposed, which can accurately calculate point position information on the surface. The method focused on the analysis of the boundary constraint expression and processing of the blending surface. By constructing appropriate boundary conditions and combining them with the corresponding interpolation strategy, the G² continuity of the blending surface at the boundary was ensured, and the expression form of the blending surface was derived. This method was based on the interpolation of curves on the surface, which not only enabled simple and efficient blending-surface modeling, but also adjusted the boundary curves reasonably to meet the needs of different engineering scenarios. Finally, an accurate difference method for solving the derivatives at each point on the surface and an example of using this method to generate G²-continuous blending surface were provided. At the same time, the continuity of the obtained surface was verified by calculating the normal vector, normal curvature and generating zebra stripes. The results showed that this method can effectively meet the requirements of curvature continuity, thereby confirming its rationality.

Key words: G² continuous, blending surface, interpolation, derivatives, curvature

中图分类号:

TP391

陆相江, 姜豪, 王爱增, 宁涛. 一种G²连续插值过渡曲面建模方法[J]. 图学学报, 2026, 47(4): 788-800.

LU Xiangjiang, JIANG Hao, WANG Aizeng, NING Tao. A modeling method for G²-continuous interpolating blending surface[J]. Journal of Graphics, 2026, 47(4): 788-800.

图/表 14

参考文献 24

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接触曲线参数($\omega $)	原始曲面法向量(N^g)	差分法计算的过渡过程曲面法向量(N^b)	样条拟合的过渡曲面法向量(N ^f)
0	(0.417 56， -0.567 63， 0.709 53)	(0.417 56， -0.567 63， 0.709 53)	(-0.126 94， -0.860 27， -0.493 79)
$\frac{1}{9}$	(0.447 04， -0.567 69， 0.691 29)	(0.447 04， -0.567 69， 0.691 29)	(0.447 13， -0.567 68， 0.691 24)
$\frac{2}{9}$	(0.488 71， -0.541 25， 0.684 26)	(0.488 71， -0.541 25， 0.684 26)	(0.488 79， -0.541 24， 0.684 21)
$\frac{3}{9}$	(0.546 84， -0.473 99， 0.690 15)	(0.546 84， -0.473 99， 0.690 15)	(0.546 92， -0.473 97， 0.690 09)
$\frac{4}{9}$	(0.622 78， -0.340 64， 0.704 35)	(0.622 78， -0.340 64， 0.704 35)	(0.622 86， -0.340 64， 0.704 28)
$\frac{5}{9}$	(0.696 85， -0.114 29， 0.708 06)	(0.696 85， -0.114 29， 0.708 06)	(0.696 90， -0.114 30， 0.708 00)
$\frac{6}{9}$	(0.710 48， 0.190 72， 0.677 38)	(0.710 48， 0.190 72， 0.677 38)	(0.710 54， 0.190 68， 0.677 33)
$\frac{7}{9}$	(0.640 01， -0.418 67， 0.644 28)	(0.640 01， -0.418 67， 0.644 28)	(0.640 07， -0.418 63， 0.644 25)
$\frac{8}{9}$	(0.553 67， -0.522 54， 0.648 41)	(0.553 67， -0.522 54， 0.648 41)	(0.553 67， -0.522 52， 0.648 40)
1	(0.522 16， 0.532 77， 0.665 96)	(0.522 16， 0.532 77， 0.665 96)	(-0.249 32， 0.810 90， -0.529 41)

接触曲线参数($\omega $)	原始曲面法向量(N^g)	差分法计算的过渡过程曲面法向量(N^b)	样条拟合的过渡曲面法向量(N ^f)
0	(0.417 56， -0.567 63， 0.709 53)	(0.417 56， -0.567 63， 0.709 53)	(-0.126 94， -0.860 27， -0.493 79)
$\frac{1}{9}$	(0.447 04， -0.567 69， 0.691 29)	(0.447 04， -0.567 69， 0.691 29)	(0.447 13， -0.567 68， 0.691 24)
$\frac{2}{9}$	(0.488 71， -0.541 25， 0.684 26)	(0.488 71， -0.541 25， 0.684 26)	(0.488 79， -0.541 24， 0.684 21)
$\frac{3}{9}$	(0.546 84， -0.473 99， 0.690 15)	(0.546 84， -0.473 99， 0.690 15)	(0.546 92， -0.473 97， 0.690 09)
$\frac{4}{9}$	(0.622 78， -0.340 64， 0.704 35)	(0.622 78， -0.340 64， 0.704 35)	(0.622 86， -0.340 64， 0.704 28)
$\frac{5}{9}$	(0.696 85， -0.114 29， 0.708 06)	(0.696 85， -0.114 29， 0.708 06)	(0.696 90， -0.114 30， 0.708 00)
$\frac{6}{9}$	(0.710 48， 0.190 72， 0.677 38)	(0.710 48， 0.190 72， 0.677 38)	(0.710 54， 0.190 68， 0.677 33)
$\frac{7}{9}$	(0.640 01， -0.418 67， 0.644 28)	(0.640 01， -0.418 67， 0.644 28)	(0.640 07， -0.418 63， 0.644 25)
$\frac{8}{9}$	(0.553 67， -0.522 54， 0.648 41)	(0.553 67， -0.522 54， 0.648 41)	(0.553 67， -0.522 52， 0.648 40)
1	(0.522 16， 0.532 77， 0.665 96)	(0.522 16， 0.532 77， 0.665 96)	(-0.249 32， 0.810 90， -0.529 41)

接触曲线参数($\omega $)	过渡曲面法向量与原始曲面法向量夹角/°
接触曲线参数($\omega $)	差分法计算	样条拟合
0	0	85.127 00
$\frac{1}{9}$	0	0.005 88
$\frac{2}{9}$	0	0.005 79
$\frac{3}{9}$	0	0.005 97
$\frac{4}{9}$	0	0.005 56
$\frac{5}{9}$	0	0.004 69
$\frac{6}{9}$	0	0.005 08
$\frac{7}{9}$	0	0.004 73
$\frac{8}{9}$	0	0.001 99
1	0	77.899 00

接触曲线参数($\omega $)	过渡曲面法向量与原始曲面法向量夹角/°
接触曲线参数($\omega $)	差分法计算	样条拟合
0	0	85.127 00
$\frac{1}{9}$	0	0.005 88
$\frac{2}{9}$	0	0.005 79
$\frac{3}{9}$	0	0.005 97
$\frac{4}{9}$	0	0.005 56
$\frac{5}{9}$	0	0.004 69
$\frac{6}{9}$	0	0.005 08
$\frac{7}{9}$	0	0.004 73
$\frac{8}{9}$	0	0.001 99
1	0	77.899 00

接触曲线参数($\omega $)	原始曲面法曲率($\kappa_{n}^{g}$)	差分法计算过渡曲面法曲率($\kappa_{n}^{b}$)	样条拟合的过渡曲面法曲率($\kappa_{n}^{f}$)
0	0.287 37	0.287 37	82.753 00
$\frac{1}{9}$	0.227 92	0.227 92	0.217 53
$\frac{2}{9}$	0.178 85	0.178 85	0.168 25
$\frac{3}{9}$	0.135 05	0.135 05	0.124 18
$\frac{4}{9}$	0.072 57	0.072 57	0.062 63
$\frac{5}{9}$	0.012 35	0.012 35	0.020 73
$\frac{6}{9}$	0.059 12	0.059 12	0.049 32
$\frac{7}{9}$	0.088 55	0.088 55	0.078 43
$\frac{8}{9}$	0.062 30	0.062 30	0.057 92
1	0.030 31	0.030 31	0.002 94

一种G²连续插值过渡曲面建模方法

A modeling method for G²-continuous interpolating blending surface

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参考文献 24

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接触曲线参数($\omega $)	差分法计算过渡曲面法曲率与原始曲面法曲率相对偏差	样条拟合的过渡曲面法曲率与原始曲面法曲率相对偏差
0	0	286.970 00
$\frac{1}{9}$	0	0.070 69
$\frac{2}{9}$	0	0.132 18
$\frac{3}{9}$	0	0.227 96
$\frac{4}{9}$	0	0.376 60
$\frac{5}{9}$	0	0.540 22
$\frac{6}{9}$	0	0.658 22
$\frac{7}{9}$	0	0.947 31
$\frac{8}{9}$	0	0.937 58
1	0	34 635.000 00

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