A modeling method for G2-continuous interpolating blending surface

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

Abstract

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

CLC Number:

TP391

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.

Figures/Tables 14

References 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

A modeling method for G²-continuous interpolating blending surface

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Figures/Tables 14

References 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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