Abstract
The aim of this attempt was to present an efficient algorithm for the evaluation of error bound of triangular subdivision surfaces. The error estimation technique is based on first order difference and this process is independent of parametrization. This technique can be easily generalized to higher arity triangular surfaces. The estimated error bound is expressed in-terms of initial control point sequence and constants. Here, we efficiently estimate error bound between triangular surface and its control polygon after k-fold subdivision and further extended to evaluate subdivision depth of the scheme.
Keywords
Subdivision surfaces; Control polygon; B-spline; Triangular surface; Error bound; Subdivision depth
1. Introduction
Subdivision is a simple and popular method to generate smooth limit curves and surfaces from discrete set of data points. It is an iterative algorithm, which is based on simple refinement rules to generate increasingly dense sequence of points under suitable hypothesis, converging to a continuous and smooth function. Starting from an initial control polygon, a subdivision scheme refers the computed values at the previous step according to the subdivision rules. The scheme is said to be convergent if there exists a limit curve. Efficiency of subdivision schemes is their flexibility and simplicity and they found their way into wide range of applications in computer graphics, medical imaging, industrial design and automotive design, etc. [1], [2] and [3].
Triangular surfaces [4] are one of the fundamental paradigms of Computer Aided Geometric Design (CAGD). These are defined by de Boor nets and have a regular triangular structure. This class of triangular surfaces shares the properties of univariate [5] and tensor product B-splines [6]. The procedure for subdividing triangular surfaces exactly parallels the subdivision for tensor product B-spline surfaces. Actually, these are extension of B-splines surfaces.
For many applications such as rendering, intersection testing or design, it is important to know, how well the control polygon approximate the exact curve or a surface. In the last decade several researchers attempt to answer the question and to improve the rule to estimate error bounds. The techniques presented in [7], [8], [9], [10] and [11] for computation of error bounds are based on parametrization, so they cannot be generalized to subdivision surfaces easily, methods presented in [12], [13] and [14] are based on eigen analysis. Zeng and Chen [15] introduced the concept of neighbor points and by using the first-order difference of control points of Catmull–Clark surfaces, they obtained the rate of convergence of control meshes of Catmull–Clark surface. From the result of convergence, they derived a computational formula of subdivision depth for Catmull–Clark surfaces. Cheng and Yong [16] introduced computational formula for subdivision depth, which is based on second order forward differences for extra-ordinary Catmull–Clark subdivision surface patches. Mustafa et al. [17], [18], [19], [20], [21], [22] and [23] have estimated error bound for binary, ternary, quaternary, non-stationary, n-ary curve, surface and volumetric model in-terms of maximal first order differences of the initial control point sequence and constants that depend on the subdivision mask. Huang et al. [24] derive a bound on the distance between a Catmull Clark subdivision surface patch and its limit face in terms of the maximum norm of the second order differences of the control points and a constant that depends only on the valence of the patch. Later on Mustafa et al. estimate the subdivision depth of Bajaj and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \sqrt{3}}
subdivision schemes for both regular and irregular patches [25] and [26]. Moncayo and Amat [27] presented error bounds for a class of subdivision schemes based on the two-scale refinement equation. In recent years Hashmi et al. [28] estimated the subdivision depth for Li subdivision scheme for regular and irregular patches.
In the present literature survey, it is evident that no such attempt has been made to evaluate subdivision depth for triangular subdivision surfaces. In this paper author successfully articulates the formula for subdivision depth for triangular surfaces based on first order differences by using estimation techniques.
The rest of the paper is arranged in following fashion: Some definition and preliminary notations are given in Section 2. Section 3 is devoted for the proof of main result based on some preliminary results. Future research directions are given in Section 4. To maintain the presentation of paper as simple as possible for readers, notations and typical mathematical proof of basic results are provided in the Appendices.
2. Definition and notations
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k\in R^N} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , denote a sequence of points in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^N} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle N\geqslant 2} , where k is a non-negative integer then binary subdivision process for triangular surfaces [1, pp. 14–19] in our context can be restated as
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(2.1) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha \mbox{,}\beta \in \lbrace 0\mbox{,}1\rbrace }
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \lbrace 1\mbox{,}2\rbrace \mbox{,}m}
is greater than 2, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle a_{\alpha \mbox{,}j\mbox{,}l}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle d_{m\mbox{,}l}}
are defined by
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for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =0\mbox{,}1\mbox{,}2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle j=-m+1\mbox{,}\ldots \mbox{,}m-1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle l=0\mbox{,}\ldots \mbox{,}m} , called subdivision mask. It is cautioned that (2.1) depends on labeling of the control polygon. For example for m = 2, 3, and 4, labeling of old and new points (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A\mbox{,}B\mbox{,}C\mbox{,}D\mbox{,}E\mbox{,}F\mbox{,}G} ) is shown in Figs. 1(a) and (b) and 2 respectively.
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Figure 1. (a) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A=p_{i+\frac{1}{2}\mbox{,}j+\frac{1}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B=p_{i+\frac{1}{2}\mbox{,}j+1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C=p_{i+1\mbox{,}j+\frac{1}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D=p_{i+1\mbox{,}j+1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E=p_{i+1\mbox{,}j+\frac{3}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F=p_{i+\frac{3}{2}\mbox{,}j+1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G=p_{i+\frac{3}{2}\mbox{,}j+\frac{3}{2}}} . (b) Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A=p_{i+1\mbox{,}j+1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B=p_{i+1\mbox{,}j+\frac{3}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C=p_{i+\frac{3}{2}\mbox{,}j+1}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D=p_{i+\frac{3}{2}\mbox{,}j+\frac{3}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E=p_{i+\frac{3}{2}\mbox{,}j+2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F=p_{i+2\mbox{,}j+\frac{3}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G=p_{i+2\mbox{,}j+2}} . |
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Figure 2. Here m = 4, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle A=p_{i+\frac{3}{2}\mbox{,}j+\frac{3}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle B=p_{i+\frac{3}{2}\mbox{,}j+2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle C=p_{i+2\mbox{,}j+\frac{3}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle D=p_{i+2\mbox{,}j+2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle E=p_{i+2\mbox{,}j+\frac{5}{2}}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle F=p_{i+\frac{5}{2}\mbox{,}j+2}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle G=p_{i+\frac{5}{2}\mbox{,}j+\frac{5}{2}}} . |
Given initial values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^0\in R^N} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , then in the limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k\rightarrow \infty } , the process defines an infinite set of points in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle R^N} . A necessary condition for the convergence of the subdivision process (2.1) for arbitrary initial data is that
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(2.2) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha \mbox{,}\beta \in \lbrace 0\mbox{,}1\rbrace }
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left\{1\mbox{,}2\right\}}
.
Let us suppose
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(2.3) |
where
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(2.4) |
Suppose for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =0\mbox{,}1\mbox{,}2} ,
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(2.5) |
where
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Suppose further that
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(2.6) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha \mbox{,}\beta \in \lbrace 0\mbox{,}1\rbrace }
or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left\{1\mbox{,}2\right\}}
.
Also
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(2.7) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle e_{r\mbox{,}l}={\sum }_{p=-m+1}^r(a_{1\mbox{,}p\mbox{,}l}-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a_{2\mbox{,}p\mbox{,}l})
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{r\mbox{,}l}={\sum }_{p=-m+1}^r(a_{0\mbox{,}p\mbox{,}l}-}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a_{1\mbox{,}p\mbox{,}l}) .
Rest of the notations are in Appendix A.
2.1. Subdivision depth
Given control polygon of n -ary subdivision surface and an error tolerance Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \in } , if we subdivide control polygon k times, so that the error between resulting polygon and subdivision surface is smaller than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \in } , then k is called subdivision depth of subdivision surface with respect to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \in } .
3. The error bounds for triangular surfaces
In this Section, the main result for error bounds is presented for triangular surfaces, which is based on some preliminary results. Finally, the section ends on subdivision depth formula.
Lemma 3.1.
Given initial triangular control polygon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^0=p_{i\mbox{,}j}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , let the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k\mbox{,}k\geqslant 0} be defined recursively by subdivision process(2.1)together with(2.2)then
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(3.1) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_t^k} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle k\geqslant 0} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta } are defined by(2.3) and (2.7)respectively.
Proof.
Proof is shown in . □
The proof of following Lemma is shown in .
Lemma 3.2.
Given initial triangular control polygon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^0=p_{i\mbox{,}j}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , let the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k\mbox{,}\quad k\geqslant 0} be defined recursively by subdivision process(2.1)together with(2.2)then
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(3.2) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_t^0\mbox{,}M_{\left(1\mbox{,}1\right)}^k\mbox{,}\delta } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\eta }_1^t} are defined by(2.3), (2.6), (2.7) and (A.1)respectively.
Lemma 3.3.
Given initial triangular control polygon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^0=p_{i\mbox{,}j}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , let the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k\mbox{,}k\geqslant 0} be defined recursively by subdivision process(2.1)together with(2.2)then
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(3.3) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_t^0\mbox{,}M_{\left(2\mbox{,}1\right)}^k\mbox{,}\delta } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\eta }_2^t} are defined by(2.3), (2.6), (2.7) and (A.2)respectively.
Proof.
Proof is given in . □
Similarly, one can prove the following lemma.
Lemma 3.4.
Given initial triangular control polygon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^0=p_{i\mbox{,}j}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , let the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k\mbox{,}\quad k\geqslant 0} be defined recursively by subdivision process(2.1)together with(2.2)then
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(3.4) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_t^0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\chi }^k} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \chi \in \left\{\left(1\mbox{,}2)\mbox{,}(2\mbox{,}2)\mbox{,}(0\mbox{,}1)\mbox{,}(1\mbox{,}0)\mbox{,}(0\mbox{,}0\right)\right\}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\eta }_{\upsilon }^t} are defined by(2.3), (2.6) and (2.7) and (A.3), (A.4), (A.5), (A.6) and (A.7)respectively. There is following correspondence between the values of v and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \chi } : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 3\rightarrow (1\mbox{,}2)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 4\rightarrow (2\mbox{,}2)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 5\rightarrow (0\mbox{,}1)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 6\rightarrow (1\mbox{,}0)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle 7\rightarrow (0\mbox{,}0)} .
Here, we present our main result to estimate error bounds between triangular surface and its control polygon after k-fold subdivision.
Theorem 3.5.
Given initial triangular control polygon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^0=p_{i\mbox{,}j}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , let the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k\mbox{,}k\geqslant 0} be defined recursively by subdivision process(2.1)together with(2.2). Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^k} be the piecewise linear interpolation to the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{\infty }} be the limit triangular surface of the subdivision process(2.1). If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta <1} then error bounds between triangular surface and its control polygon after k-fold subdivision are
|
(3.5) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma } are defined by(2.7) and (A.8)respectively.
Proof.
Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\left\|\cdot \right\|}_{\infty }}
denote the uniform norm. Since the maximum difference between Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{k+1}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^k}
is attained at a point on the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(k+1\right)}
th control polygon (i.e. control polygon after Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \left(k+1\right)} -fold subdivision), then
|
(3.6) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle M_{\left(\alpha \mbox{,}\beta \right)}^k}
is defined by (2.6).
Then from (3.2), (3.3), (3.4) and (3.6) we get
|
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta }
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \gamma }
are defined by (2.7) and (A.8) respectively.
Using triangular inequality we get
|
This completes the proof. □
Remark 3.1.
It is pointed out that, most of the famous binary triangular subdivision schemes satisfies the condition Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta <1} . Our claim is supported by the following corollaries.
Corollary 3.6.
Given initial triangular control polygon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^0=p_{i\mbox{,}j}} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle i\mbox{,}j\in Z} , let the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k\mbox{,}k\geqslant 0} be defined recursively by subdivision process(2.1)for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle m=2\mbox{,}3\mbox{,}4} . Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^k} be the piecewise linear interpolation to the values Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle p_{i\mbox{,}j}^k} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{\infty }} be the limit triangular surface of the subdivision process. Then
|
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\beta }_t^0\mbox{,}t=1\mbox{,}2} are defined by(2.3).
Theorem 3.7.
Let k be subdivision depth and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle d^k} be the error bound between triangular subdivision surface Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^{\infty }} and its k-level control polygon Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle P^k} . For arbitrary Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \in >0} , if
|
(3.7) |
then
|
Proof.
From (3.5), we have
|
This implies, for arbitrary given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \in >0} , when subdivision depth k satisfies the following inequality
|
then
|
This completes the proof. □
4. Future work
The technique presented in this article can be generalized to estimate error bounds between ternary and obviously to n-ary triangular surface and its control polygon after k-fold subdivision. The attempt can also be made to estimate subdivision depth for irregular triangular patches.
Appendix A. Notations
We have the following
|
(A.1) |
|
(A.2) |
|
(A.3) |
|
(A.4) |
|
(A.5) |
|
(A.6) |
|
(A.7) |
and
|
(A.8) |
Appendix B. Proof of Lemma 3.1
Proof.
|
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =1\mbox{,}2} . By using similar approach of Mustafa et al. [17] (Theorem 1, p. 599 and Theorem 7, p. 609), we obtain
|
(B.1) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle e_{r\mbox{,}l}={\sum }_{p=-m+1}^r(a_{1\mbox{,}p\mbox{,}l}-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a_{2\mbox{,}p\mbox{,}l}) .
Similarly from (2.1) and (2.2) we have the following differences, for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =0\mbox{,}1}
|
(B.2) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle f_{r\mbox{,}l}={\sum }_{p=-m+1}^r(a_{0\mbox{,}p\mbox{,}l}-} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): a_{1\mbox{,}p\mbox{,}l}) .
Further for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =1\mbox{,}2} , we get
|
(B.3) |
For Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \alpha =0\mbox{,}1}
|
(B.4) |
Using (B.1) and (B.2) recursively and utilizing (2.3) and (2.7) we get
|
Using (B.3) and (B.4) recursively and utilizing (2.3) and (2.7) we get
|
This completes the proof. □
Appendix C. Proof of Lemma 3.2
Proof.
|
(C.1) |
By expanding innermost summation we get
|
This implies
|
where
|
|
This implies
|
(C.2) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\xi }_{1\mbox{,}l}^1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\xi }_{1\mbox{,}l}^2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{a}}_{1\mbox{,}q\mbox{,}l}}
and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\tilde{\tilde{a}}}_{1\mbox{,}q+1\mbox{,}l}}
are defined by (2.5).
Taking summation on both sides we obtain,
|
(C.3) |
Likewise by expanding summation appear on right hand side of above equation we get
|
(C.4) |
|
(C.5) |
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle {\xi }_{1\mbox{,}l}^3}
is defined by (2.5) and
|
(C.6) |
Substituting (C.4), (C.5) and (C.6) in (C.3) we get
|
where
|
|
Substituting it into (C.1) then by using (2.4) and (2.5) and taking norm we get
|
Using notations (2.3), (2.6) and (A.1) we get
|
Using (3.1) we get,
|
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta }
is defined by (2.7). This completes the proof. □
Appendix D. Proof of Lemma 3.3
Proof.
|
(D.1) |
By expanding two summations in first term as we did in Lemma 3.2 after utilizing (2.5) we get
|
(D.2) |
where
|
|
Similarly by expanding two summations in second term of (D.1) and utilizing (2.5), we get
|
(D.3) |
|
Substituting (D.2) and (D.3) in (D.1) then after simplifying we have
|
where
|
Taking norm and using (2.4) and (2.5) we obtain
|
Using notations (2.3), (2.6) and (A.2) we get
|
Using (3.1), we get
|
where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://test.scipedia.com:8081/localhost/v1/":): {\textstyle \delta }
is defined by (2.7). This completes the proof. □
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