Hyperbolicpolynomial Curveswith Shape Parameters
LIU Huayong
(School of Sciences and Physics, AnhuiJianzhuUniversity, Hefei 230022, PR China)
Abstract:Hyperbolic polynomial curves with shape parameters are presented in this paper. The curves with shape parametersarecontinuous.Especially, the hyperbolic polynomial curvesarecontinuous for the shape parameter.With the shape parameters, the hyperbolic polynomial curveswith shape parameters can be close to the polynomial curvesor closer to the given control polygon than the polynomial curves. Thishyperbolic polynomialinterpolation is also discussed.
Keywords: Geometry continuity; hyperbolic polynomial; Shape parameter; Inverse problem
MSC: 65D05; 65D07; TP391
1. Introduction
In several applications it is required to construct smooth functions or parametric curves, interpolating or approximating a given set of data, and reproducing their salient geometric properties. Classical methods based on piecewise polynomials, often do not produce interpolations or approximants satisfying the required constraints. Thus a great deal of research has focused on the study of new function spaces for building constrained curves. In the same time Geometric continuity is widely recognized as the suitable way to fit or interpolate together two curves or surfaces in Computer Aided Geometric Design [1-7]. For curves or surface, the conditions for patches to be of continuity and the constructions of conditions curves or surfaces are important topics in the fields of Computer Aided Geometric Design and Computer Graphics [8-13]. But they cannot exactly represent conics (except parabolas) and some transcendental curves such as the cycloid and the helix, which are often used in engineering.
In recent years, in order to overcome the NURBS (Non-Uniform Rational B-splines) model is inadequate, while maintaining its good geometric properties, some of the mixture based on polynomial and non-polynomial space curves and surfaces on the new model came into being, causing the majority of researchers of great interest. Wang[14-17]givesUniform hyperbolic structure of higher-order polynomial B-spline in the.this hybrid model curves and surfaces[18-25] not only inherits the advantages of a polynomial spline, avoiding the disadvantages generated when using NURBS, and can accurately represent the catenary curve, so they were welcomed by the designer. View of the hybrid model curves and surfaces, the curve has become a hot surface modeling study of the facts, each of them has itsown strong-points, but they have one shortage in common. That isthe shape of themiswelldetermined by their control points.In addition the existinghyperbolic polynomial curves are continuous. Sothe subject of the proposed study of algebraic geometry continuous hyperbolic curve and its nature, the shape adjustment problem of hyperbolic polynomial curve with shape parameters in the theory and application research is verymeaningful.
2. Hyperbolicpolynomial basis function
2.1 The construction of the basis functions
Definition 1.Let, forand, the functions
(1)
are called hyperbolic polynomial basis function of order 4.
Where ,
,.
2.2. The property of the basis functions
Straightforward computation gives the following property. It is used for studying thenonnegativity andthe continuity of the basis functions.
(1) Positivity
(2) Partition of unity
(3) Continuity
In view of the upper equation, we say that the basis functions meet with continuous.
2.3. The effect on the basis functions of altering the value of
Here we discuss the analysis of shape parameters, and. The first four basis functions is on behalf of in thefigure1 to 3, blue line is, greenline is, redline is, yellow line isin the Figs. Now we first fixed = 4, we observe the figures changes by changing the values.
(1) When, then wecan obtain the following figures.
Fig 1: basis function with
(2) When, then we can obtain the following figures.
Fig 2: basis function with
(3) When.Then we can obtain the following figures.
Fig 3: basis function with
From the above we can see, when the fixed, is increased ,basis function begin more and more steep, the maximum of basis function begin greater too, so curve will go outward expansion in the control polygon of the convex hull. When the fixed, is increased, Basis functions appear form smooth to steep, the maximum value firstly begin larger, and then the maximum value begins smaller. In the same time, curve will begin inward contraction, and then curve will go outward expansion in the control polygon of the convex hull.
3. Hyperbolic polynomial curves
3.1The Definitionof the hyperbolic polynomial curves
Definition 2.Given points, or, for,then
(2)
is called the hyperbolic polynomial curve segmentwith shape parameters.
3.2 The effect on the curve shape of altering the value of
Now we take fixed,we observe the figures changes by changing the values.
(a) (b)
Fig 4: hyperbolic polynomial curvewith shape parameters
In the Fig4 (a), if is increased above unity, the velocity of the particle immediately after a knot point increases. Thisserves to pushthe particle further in the direction of the travel before it turns as influenced by the next controlpoint. This is said to biasthe curve to the right. Decreasing below unity, the particle velocity decreasesand thus it biases the paths towards the left. In the Fig4 (b), the parameter controls the tension over the curve. As increases above zero, the knot points of the curve are pulled towards their respective control vertices. Fornonnegative, the knot points are pushed away. In the uniform formulation, these parametersand havethe same value through the entire curve, but in the continuous formulation these parameters are allowedto vary continuously along the curve.
Under normal circumstances, we need to see a value play a leading role ofand. If, then the shape of the curve is controlled byshape parameters.On the contrary, if, then the shape will be controlled byshape parameters.
andare interactive,soandare betterin practice, beyond the upper limit of this range, the basic push the curve into the curve control polygon, smoothing is bad.
If we are to design the curve to go through the first and last control points, knot vectorsare quadruple nodein the first and last vectors.
Fig 5: hyperbolic polynomial curveinterpolating the first and last control points
3.3. Closed hyperbolic curves
The construction of closed curve is the most basic content of curve design; people should know terminal behaviors of the open curve and how to construct a closed curve. Given closed control points, where,if and,then we can construct an closed hyperbolic polynomial curve. In figure 5, the closed curves are generated by altering the value of. Asincreases, the curve is closer to the control polygon.
Fig 5.The closedcurve of different shape parameters
4. Interpolation of the problem
Given data points (requires interpolation points) and two endpoints of tangent vectors , we seek a continuous hyperbolic polynomial curve, so that the curve can interpolate .
Solution:if,, Control points are determined. So that
(3)
4.1 The inverse of hyperbolic polynomial curve
Note that, it has totaln+1 independent equations in the (17).We knowhave n +3 variables, in order to obtain n+3 variables, we must then construct two equations, then we need two boundary conditions to help us.
In order to reverse control points, we construct n +1 equation in (17), so now we just need to construct two boundary conditions, so we can find n +3 control points. In fact, the boundary conditions can have a variety of construction methods. As the above example we can give the boundary conditions
(4)
In general, the curve designs in two ways, namely, open curves and closed curves.For open curves, we can give the boundary conditions
(5)
For closed curves, we give our boundary conditions
(6)
After the given boundary conditions,form (3), (4), (5) and (6), we have the matrix equations
(7)
(8)
(9)
Where
Here we need to discuss the solution of equations; we know that to make the equations solvable. That is, the only solution of interpolation problems, we must prove that the coefficient matrix nonsingular.
Observed matrix of the second and third rowform (7), the molecules of contain only ,the molecular of contain only and.apparently, can not be linearly represented by and, can not be represented by, i.e. the matrix of the second and third row are linear independent.
Similarly, the matrix of all rows is linear independentexcept matrix of the first and last line.
Now we see matrix of the first and second row, the molecules of contain, the molecules of contain onlyby calculating; clearly can not be linearly represented by, can not be represented by,i.e. Matrix of the first and second row are linear independent
Now we see the inverse of the matrix first and second rows,the molecules of containand ,the molecules with and can be calculated only by; can not be linearly represented by, can not be linearly represented by, i.e. The inverse of the matrix first and second rows are linear independent. So (7) is nonsingular matrix.Similarly we can prove that (8) and (9) are nonsingular.
4.2 Example
Now using the above method, we give some examples.
Example 1:for , ,, How demand interpolation curve？
Table 1 Given control points
i / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7vi（x） / 8.125 / 8.4 / 9 / 9.485 / 9.6 / 9.959 / 10.166 / 10.2
vi（y） / 0.0774 / 0.099 / 0.28 / 0.6 / 0.708 / 1.2 / 1.8 / 2.177
Solving: where,,
,,
,.
So we can calculate the control points of the data.
Table 2calculated the control points of the data
i / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9Pi（x） / 7.073 / 8.3334 / 8.2694 / 9.033 / 9.5823 / 9.5104 / 10.005 / 10.196 / 10.193 / 10.205
Pi（y） / 0.058 / 0.0832 / 0.0713 / 0.2355 / 0.6824 / 0.6058 / 1.1782 / 1.8890 / 2.0345 / 3.0854
Fig 6:hyperbolic polynomial interpolation curve
Example 2: the unit circle of the data interpolation.
For,. so we can calculate the control points of the data
Table 3 calculated the control points of the data
i / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7vi（x） / 1.0000 / 0.9135 / 0.6691 / 0.3090 / -0.1045 / -0.5000 / -0.8090 / -0.9781
Vi（y） / 0 / -0.4067 / 0.7431 / 0.9511 / 0.9945 / 0.8660 / 0.5878 / 0.2079
8 / 9 / 10 / 11 / 12 / 13 / 14 / 15
-0.9781 / -0.8090 / -0.5000 / -0.1045 / 0.3090 / 0.6691 / 0.9135 / 1.0000
-0.2079 / -0.5878 / -0.8660 / -0.9945 / -0.9511 / -0.7431 / -0.4067 / 0
Note the control points,, Obtained control points are as follows.
Table 4 Obtained control points
i / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7Pi（x） / 1.0108 / 1.0108 / 0.9451 / 0.6875 / 0.3184 / -0.1077 / -0.5147 / -0.8327
Pi（y） / 0.1485 / -0.1485 / 0.4575 / 0.7547 / 0.9816 / 1.0229 / 0.8916 / 0.6049
8 / 9 / 10 / 11 / 12 / 13 / 14 / 15 / 16 / 17
-1.0068 / -1.0068 / -0.8327 / -0.5146 / -0.1077 / 0.3184 / 0.6875 / 0.9451 / 1.0108 / 1.0108
0.2140 / -0.2140 / -0.6049 / -0.8916 / -1.023 / -0.9816 / -0.7547 / -0.4575 / -0.1485 / 0.1485
We draw the figureby matlab as follows.This error is e=0.0131 by calculation.
Fig 7:hyperbolic polynomial curve interpolation the units circle of the data interpolation
Example 3:for，when all are the same ,Interpolation of dataare obtained,we can seek cubic interpolation spline where meet with Continuous, where.
For,, The following table is data points by calculating.
Table 5 calculated the control points of the data
i / 0 / 1 / 2 / 3 / 4 / 5 / 6vi（x） / 5.0000 / 4.7975 / 4.2063 / 3.2743 / 2.0771 / 0.7116 / -0.7116
vi（y） / 0 / 1.4087 / 2.7032 / 3.7787 / 4.5482 / 4.9491 / 4.9491
vi（z） / 0 / 0.8568 / 1.7136 / 2.5704 / 3.4272 / 4.2840 / 5.1408
7 / 8 / 9 / 10 / 11
-2.0771 / -3.2743 / -4.2063 / -4.7975 / -5.0000
4.5482 / 3.7787 / 2.7032 / 1.4087 / 0
5.9976 / 608544 / 7.7112 / 8.5680 / 9.4248
Obtained control points are as follows.
Table 6 Obtained control points
i / 0 / 1 / 2 / 3 / 4 / 5 / 6Pi（x） / 5.0250 / 5.0250 / 4.8745 / 4.2606 / 3.3198 / 2.1052 / 0.7214
Pi（y） / -0.3009 / -0.3009 / 1.5082 / 2.7187 / 3.8361 / 4.6088 / 5.0170
Pi（z） / -0.1805 / -0.1805 / 0.9050 / 1.7007 / 2.5738 / 3.4263 / 4.2843
7 / 8 / 9 / 10 / 11 / 12 / 13
-0.7214 / -2.1052 / -3.3198 / -4.2606 / -4.8745 / -5.0250 / -5.0250
5.0170 / 4.6088 / 3.8361 / 2.7187 / 1.5082 / -0.3009 / -0.3009
5.1405 / 5.9985 / 6.8509 / 7.7240 / 8.5198 / 9.6053 / 9.6053
Fig 8: hyperbolic polynomial curve interpolation space curve
5.Hyperbolic polynomial surface
Definition 3.If given control pointsand the node vector,.Using the tensor product method,we can define a hyperbolicpolynomial spline surfaces
(10)
Where andis basis function by formula (1).
When,have different values, we can get some shape expressed by hyperbolic polynomial spline curves with different parameters.
(a) when (b)when
Fig 9: hyperbolic polynomial surfacewith parameters
6. Conclusion
The main contents of this paper study hyperbolic polynomial curve and surface, geometry continuous and their application. We raised its inverse problem and prove existence of solutionsin the same time, and finally give some examples.
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