We presented a general formula to generate the family of even-point
ternary approximating subdivision schemes with a shape parameter for describing curves. Some sufficient conditions for
There are numbers of binary subdivision schemes in the literature. The interest in investigating arities higher than two has been started by Hassan et al. [
The 2-, 3-,…, 6-point binary and ternary schemes are very common in the literature. The schemes involving convex combination of more or less than six points at coarse refinement level to insert a new point at next refinement level is introduced by Ko et al. [
This motivates us to present the family of even-point ternary schemes with high smoothness and more degree of freedom for curve design. Proposed schemes not only provide the mask of even-point schemes but also generalize and unify several well-known schemes. Moreover, we measured curvature and torsion that can be used to describe the quality of curve. Also we compared plot of curvature and torsion, obtained by proposed schemes with the other existing schemes.
A general compact form of univariate ternary subdivision scheme
Let
A scheme
The paper is organized as follows: the family of even-point ternary approximating scheme and analysis of two even-point ternary schemes are presented in Section
Here we offer a general formula for even-point ternary approximating subdivision schemes with one parameter in the form of Laurent polynomial
Despite the fact that we can generate
From (
Since by (
For
From (
From above discussion, we reach the conclusion shown in Table
The order of continuity of proposed 4-point and 6-point ternary schemes for certain ranges of parameter.
Scheme | Parameter | Continuity | Scheme | Parameter | Continuity |
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4-point |
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6-point |
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For If we set In case
In this section, we discuss approximation order and support of basic limit function of even-point ternary approximating schemes.
Here we only find the approximation order of proposed 4-point ternary scheme. The approximation order of other even-point ternary schemes can be computed in a similar fashion.
A 4-point ternary approximating scheme has approximating order 4 for
A 6-point ternary approximating scheme has approximating order 7 for
We carry out this result by taking our origin the middle of an original span with ordinate
If
If
If
If
If
The basic limit functions,
The basic limit functions of proposed schemes at
4-point scheme
6-point scheme
The basic limit functions
The support width “
In order to show the performance of the proposed schemes, we compare continuity, support, approximation order, and shape of limit curves. We also discuss curvature and torsion.
Comparison of various 4- and 6-point ternary schemes are given in Tables
Comparison of 4-point ternary schemes.
Scheme | Type | Support | Order |
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Ternary 4-point [ |
Interpolating | 5 | 3 | 2 |
Ternary 4-point [ |
Interpolating | 5 | 3 | 2 |
Ternary 4-point [ |
Interpolating | 5 | 4 | 1 |
Ternary 4-point [ |
Interpolating | 5 | 3 | 1 |
Ternary 4-point [ |
Approximating | 5.5 | 4 | 2 |
Ternary 4-point [ |
Approximating | 5 | 4 | 3 |
Ternary 4-point [ |
Approximating | 5.5 | 4 | 2 |
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Ternary 4-point proposed | Approximating | 5.5 | 5 | 4 |
Comparison of 6-point ternary schemes.
Scheme | Type | Support | Order |
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Ternary 6-point [ |
Interpolating | 8 | 5 | 2 |
Ternary 6-point [ |
Interpolating | 8 | 4 | 2 |
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Ternary 6-point proposed | Approximating | 8.5 | 8 | 7 |
Comparison: bold solid continuous curves are generated by proposed 4-point and 6-point ternary approximating schemes (a) Ko et al. [
4-point
4-point
4-point
6-point
6-point
The quality of subdivided curves can be assessed quantitatively by measuring curvature and torsion, as functions of cumulative chord length. Curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, while in the elementary geometry of space curves, torsion measures the rate at which a twisted curve tends to depart from its osculating plane. When the torsion is zero, osculating plane never changes, and we have a plane curve. We used the method described in [
Comparison of existing and proposed 4-point schemes sampled from control polygon by using the 5th iteration. The results are shown on the left together with their corresponding curvature on the right.
Left: control data sampled from a space curve and four iteration of 4-point subdivision scheme. The corresponding curvature and torsion are shown in the center and right column, respectively.
The family of even-point approximating schemes for curve design has been established. The 4- and 6-point ternary schemes introduced by Zheng et al. [
This work is supported by the Indigenous Ph.D Scholarship Scheme of Higher Education Commission (HEC) Pakistan.