This paper describes the use of trigonometric spline to visualize the given planar data. The goal of this work is to determine the smoothest possible curve that passes through its data points while simultaneously satisfying the shape preserving features of the data. Positive, monotone, and constrained curve interpolating schemes, by using a
Data visualization, the technique of using images to represent information, has its history in the days back to the second century AD. But most of the developments are made in the last couple of centuries, predominantly during the last 30 years. It has extensively been used in industrial design, image processing, computer vision, computer aided geometric design, computer graphics, and many more. Shape preserving interpolation is a powerful tool to visualize the data in the form of curves and surfaces. The problem of curve interpolation to the given data has been studied with various requirements. One may be concerned with the smoothness of the interpolating curves, the preservation of the underlying shape features of the data, the computational complexity, or the fulfillment of certain constraints. Shape preserving signifies preserving the three basic and crucial geometrical features such as positivity, monotonicity, and convexity of the data. These shape characteristics can be easily observed when data arises from a physical experiment. In this case, it becomes vital that the interpolant produces curves more smooth and represent physical reality as close as possible. For this purpose, designers and engineers want such approximation methods that represent such physical situations accurately.
At present, spline methods have become the main tools for solving the majority of problems involving the approximation of functions, which also includes interpolation problems. Many spline functions exist that generate smooth and visually pleasant curves. Sarfraz et al. [
In recent years, polynomial splines and NURBS are replaced by trigonometric splines in order to prevail over the difficulties faced in using the former. Polynomial splines are not able to represent circular arcs and conics which are the most basic geometrical entity in almost every modeling system [
In this paper, we present a
This paper is arranged as follows. In Section
In this section, we develop a
Suppose that for knot spacing
Using conditions (
In this section, we utilize
A
Consider a data set
These conditions on the shape parameters can also be expressed as
A 2D positive dataset.
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
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0 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.12 | 0.13 |
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0.82 | 1.2 | 0.978 | 0.6 | 0.3 | 0.1 | 0.15 | 0.48 |
A 2D positive dataset.
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
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0 | 3.25 | 15 | 26.5 | 30 | 32 | 37 | 40 | 42.5 | 44 |
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8.8 | 3 | 0.025 | 3.1 | 6.2 | 9.6 | 20 | 22.5 | 21.519 | 20 |
Nonpositivity preserving rational cubic trigonometric curve.
A
Nonpositive rational cubic trigonometric curve.
Positive curve by rational cubic trigonometric spline with different values of free parameters.
A 2D data set
The
Let
For monotonicity, the necessary conditions on derivatives are
Also
To produce a monotone curve using a monotone data, the restrictions on the shape parameters can be rearranged as
A 2D monotone dataset.
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
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1 | 4 | 6.5 | 7 | 11 | 15 | 20 | 25 | 40 | 44 | 45 |
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1 | 1 | 2 | 3.5 | 5.5 | 5.5 | 10 | 10 | 12.5 | 18 | 20 |
A 2D monotone dataset.
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 |
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0 | 2 | 3 | 5 | 6 | 8 | 9 | 11 | 12 | 14 | 15 |
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10 | 10 | 10 | 10 | 10 | 10 | 10.5 | 15 | 50 | 60 | 80 |
Nonmonotonicity preserving curve.
Monotonicity preserving curve with different values of shape parameters.
Nonmonotonicity preserving curve.
Monotone data visualization with specified values of free parameters.
In this section, we generalize the curve scheme for positive data developed in Section
The
Let
As
2D data set lying above the line
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1 | 2 | 3 | 4 | 5 |
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0 | 1.1 | 2 | 3 | 4.5 |
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1.5 | 0.4 | 4 | 6.2 | 6 |
2D data set lying above the line
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1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
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0 | 0.3 | 0.5 | 1 | 1.5 | 2 | 2.5 | 3.05 | 4 |
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2 | 0.6 | 0.33 | 0.35 | 1 | 0.5 | 1.1 | 0.45 | 0.6 |
Rational cubic trigonometric curve lying below the given line.
Rational cubic trigonometric curve lying below the given line.
A
The authors are grateful to the anonymous referees for their valuable comments which improved this paper significantly. This work is supported by School of Mathematical Sciences, Universiti Sains Malaysia.