A new boundary interpolation technique for parameterisation of planar surfaces with four arbitrary boundary curves

Abstract

This article is concerned with construction of planar parametric surfaces enclosed by four arbitrary boundary curves. The available conventional methods for this purpose are algebraic interpolation method (like the Coons method) and partial differential equation method. These methods usually yield some problems. This article proposes a new method namely boundary interpolation method where the geometric properties of the boundary curves have been taken into account. A technique of simultaneous displacement of the interpolated curves from the opposite boundaries has been adopted. The geometric properties considered for displacement include weighted combination of geometric and linear displacement vectors of the two opposite generating curves. The algorithm has two adjustable parameters that control the characteristics of interpolation of one set of curves from their parents. Case studies show that this algorithm gives reasonably smooth boundary-fitted planar parametric surface compared with the other two conventional methods.

A new iso-parametric machining algorithm for free-form surface

“…Because all numerical calculations for the physical problem are performed in the computational plane, equation (2) therefore needs to be transformed to the computational plane and the unknowns (u, v) are determined from the following two elliptic equations; 14,29 au À 2bu þ cu ¼ 0 ð3aÞ (3) produces boundary-fitted parameterised 2D surface. As can be seen from Figure 3, though PDE method resolves the problem of overcrossing produced by the Coons method, it distributes the interpolated curves very unevenly in case of irregular boundaries.…”

“…So re-parameterisation of the truncated surface is necessary to generate a welldefined iso-parametric tool path. For this purpose (re-parameterisation), two different classes of methods have been developed, namely algebraic method (Coons method) [9][10][11][12][13][14][15][16][17][18][19][20][21][22] and partial differential equation (PDE) method. [23][24][25][26][27][28] But application of these methods to highly irregular boundaries yields some shortcomings like overcrossing and uneven distribution of interpolated curves.…”

“…[23][24][25][26][27][28] But application of these methods to highly irregular boundaries yields some shortcomings like overcrossing and uneven distribution of interpolated curves. 12,14 In the present work, first we have discussed the existing parameterization methods such as Coons and Laplace PDE methods and a newly developed parameterization technique called 'Boundary Interpolation' method. 14 Then based on these parameterisations a new tool path generation algorithm has been presented.…”

“…12,14 In the present work, first we have discussed the existing parameterization methods such as Coons and Laplace PDE methods and a newly developed parameterization technique called 'Boundary Interpolation' method. 14 Then based on these parameterisations a new tool path generation algorithm has been presented. Comparative case studies have been presented to show the capability of the new tool path generation algorithm.…”

Velocity control algorithm in glass polishing based on the cubic NURBS curve

“…Du et al 17 and Li et al 18 developed an adaptive nonuniform rational B-splines (NURBS) interpolator with a jerk limitation and feedrate planning method. To solve the axes problem, Bosetti and Bertolazzi, 19 Bharathi and Dong, 20 and Sarkar and Dey 21 proposed optimal control problem formulations considering the dynamics of both axes and the process outcomes. To achieve high machining quality and high speed, Dong et al 22 proposed a real-time smooth feedrate planning algorithm for a short line path.…”