Bézier curves as a total approach to measure the upper lid contour: redefining clinical outcomes in palpebral surgery

2021 ◽  
pp. bjophthalmol-2021-319666
Author(s):  
Fernando J Huelin ◽  
Marco Sales-Sanz ◽  
Cristina Ye-Zhu ◽  
Antonio Augusto V Cruz ◽  
Francisco J Muñoz-Negrete ◽  
...  
Keyword(s):  
Normal Population ◽  
Intraclass Correlation ◽  
Normal Subjects ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Upper Lid ◽  
Peak Location ◽  
The Mean

AimsTo define and quantify the upper lid contour by adapting Bézier curves with a newly developed software in normal subjects, assessing their reliability.MethodsFifty eyes of 50 healthy patients with no ocular pathology were included in this study from October 2020 to November 2020. All measurements were performed on Bézier curves adjusted to the upper lid contour. An original software was used to measure the radial and vertical midpupil-to-lid margin distances (MPLD), temporal-to-nasal (T/N) ratios, contour peak location and grade of superposition (GS) and asymmetry (GA) indexes. We calculated differences in the variables measured regarding age, gender or the side of the eye being assessed.ResultsThe mean Bézier curve showed an excellent level of inter-rater reliability (intraclass correlation coefficient of 0.99). The median GS index of each eyelid to the mean Bézier curve was 95.4%, 8.5 IQR, and the median GA index was 3%, 3.4 IQR. The mean contour peak location was −0.35 mm, SD 0.45. Overall, the mean central MPLD was 4.1 mm, SD 0.6. No significant differences were found between male and female patients in variables derived from Bézier curves.ConclusionBézier curves may become a very useful tool for the assessment of upper lid contour, contour peak and symmetry. GS and GA indexes, along with the T/N area ratio are potential outcomes for this purpose. All current variables can be obtained just from one single Bézier curve measurement. Our results offer an in-depth exhaustive description of these variables and their distribution in the normal population.

scholarly journals Geometric Modeling of Novel Generalized Hybrid Trigonometric Bézier-Like Curve with Shape Parameters and Its Applications

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 967 ◽  
Author(s):  
Samia BiBi ◽  
Muhammad Abbas ◽  
Kenjiro T. Miura ◽  
Md Yushalify Misro
Keyword(s):  
Geometric Modeling ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Geometric Continuity ◽  
Shape Parameters ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Curvature Continuity ◽  
Bernstein Basis Functions

The main objective of this paper is to construct the various shapes and font designing of curves and to describe the curvature by using parametric and geometric continuity constraints of generalized hybrid trigonometric Bézier (GHT-Bézier) curves. The GHT-Bernstein basis functions and Bézier curve with shape parameters are presented. The parametric and geometric continuity constraints for GHT-Bézier curves are constructed. The curvature continuity provides a guarantee of smoothness geometrically between curve segments. Furthermore, we present the curvature junction of complex figures and also compare it with the curvature of the classical Bézier curve and some other applications by using the proposed GHT-Bézier curves. This approach is one of the pivotal parts of construction, which is basically due to the existence of continuity conditions and different shape parameters that permit the curve to change easily and be more flexible without altering its control points. Therefore, by adjusting the values of shape parameters, the curve still preserve its characteristics and geometrical configuration. These modeling examples illustrate that our method can be easily performed, and it can also provide us an alternative strong strategy for the modeling of complex figures.

scholarly journals Approximate Multidegree Reduction ofλ-Bézier Curves

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Gang Hu ◽  
Huanxin Cao ◽  
Suxia Zhang
Keyword(s):  
Shape Control ◽  
Control Parameter ◽  
Linear Equations ◽  
Least Square ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Degree Reduction ◽  
Bezier Curves ◽  
Bézier Curves

Besides inheriting the properties of classical Bézier curves of degreen, the correspondingλ-Bézier curves have a good performance in adjusting their shapes by changing shape control parameter. In this paper, we derive an approximation algorithm for multidegree reduction ofλ-Bézier curves in theL2-norm. By analysing the properties ofλ-Bézier curves of degreen, a method which can deal with approximatingλ-Bézier curve of degreen+1byλ-Bézier curve of degreem  (m≤n)is presented. Then, in unrestricted andC0,C1constraint conditions, the new control points of approximatingλ-Bézier curve can be obtained by solving linear equations, which can minimize the least square error between the approximating curves and the original ones. Finally, several numerical examples of degree reduction are given and the errors are computed in three conditions. The results indicate that the proposed method is effective and easy to implement.

scholarly journals Approximate conversion of Bézier curves

1995 ◽  
Vol 51 (1) ◽  
pp. 153-162 ◽  
Author(s):  
Yungeom Park ◽  
U Jin Choi ◽  
Ha-Jine Kimn
Keyword(s):  
Error Analysis ◽  
A Priori ◽  
Approximation Error ◽  
Bézier Curve ◽  
Bezier Curve ◽  
A Priori Bounds ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Geometric Properties

The methods for generating a polynomial Bézier approximation of degree n − 1 to an nth degree Bézier curve, and error analysis, are presented. The methods are based on observations of the geometric properties of Bézier curves. The approximation agrees at the two endpoints up to a preselected smoothness order. The methods allow a detailed error analysis, providing a priori bounds of the point-wise approximation error. The error analysis for other authors’ methods is also presented.

A Bezier Curve Based Key Management Scheme for Hierarchical Wireless Sensor Networks

2012 ◽  
Vol 263-266 ◽  
pp. 2979-2985
Author(s):  
Yong Luo Shen ◽  
Jun Zhang ◽  
Di Wei Yang ◽  
Lin Bo Luo
Keyword(s):  
Key Management ◽  
Wireless Sensor ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Control Points ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Management Scheme ◽  
Secret Keys ◽  
Key Management Scheme

In this paper, we propose a novel key management scheme based on Bezier curves for hierarchical wireless sensor networks (WSNs). The design of our scheme is motivated by the idea that a Bezier curve can be subdivided into arbitrarily continuous pieces of sub Bezier curves. The subdivided sub Bezier curves are easily organized to a hierarchical architecture that is similar to hierarchical WSNs. The subdivided Bezier curves are unique and independent from each other so that it is suitable to assign each node in the WSN with a sub Bezier curve. Since a piece of Bezier curve can be presented by its control points, in the proposed key management scheme, the secret keys for each node are selected from the corresponding Bezier curve’s control points. Comparing with existing key management schemes, the proposed scheme is more suitable for distributing secret keys for hierarchical WSNs and more efficient in terms of computational and storage cost.

Multi-Degree Reduction of DP Curves with Constraints of Endpoints Continuity

2012 ◽  
Vol 215-216 ◽  
pp. 669-673
Author(s):  
Hua Hui Cai ◽  
Yan Cheng ◽  
Yong Hong Zhu
Keyword(s):  
Bézier Curve ◽  
Bezier Curve ◽  
Reduction Algorithm ◽  
Degree Reduction ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Conversion Formula

In this paper, we presented a constrained multi-degree reduction algorithm of DP curves based on the transformation between the DP and Bézier curves. We first correct the conversion formula between Bernstein basis and DP basis. And then, we deal with multi-degree reduction of NP curves by degree reduction of Bézier curve.

scholarly journals A Bézier Curve Based Ship Trajectory Optimization for Close-Range Maritime Operations

2017 ◽  
Author(s):  
Guoyuan Li ◽  
Houxiang Zhang
Keyword(s):  
Fuel Consumption ◽  
Trajectory Optimization ◽  
Bézier Curve ◽  
Ocean Current ◽  
Bezier Curve ◽  
Bezier Curves ◽  
Ship Maneuvering ◽  
Bézier Curves ◽  
Close Range ◽  
Maritime Operations

Ship maneuvering in close-range maritime operations is challenging for pilots, since they have to not only prevent the ship from collisions and compensate environmental impacts, but also steer it close to the target towards a proper heading. This paper presents a path planner to assist the pilots to foresee the optimal trajectory in the scenario. The path planning is formatted as an optimizing problem to minimize the turning variation fluctuation and the fuel consumption of the ship through ocean current while satisfying the constraint of orientations at the start and the end positions. Taking advantages of Bézier curves’ smoothness and adjustability, feasible trajectories are divided into two categories based on the location of the intersection between the start and end directions, and are designed as a set of parameterized Bézier curves. The variables in the Bézier curves become the state space. By searching the space using an evolutionary technique, the candidate of the Bézier curve that has the best turning and the minimized fuel consumption can be obtained. Through two case studies, the feasibility and effectiveness of the proposed planner is verified.

scholarly journals THE SERIES OF FUZZIFIED FUZZY BEZIER CURVE

2015 ◽  
Vol 78 (2-2) ◽  
Author(s):  
Rozaimi Zakaria ◽  
Abd. Fatah Wahab ◽  
R. U. Gobithaasan
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Fuzzy Number ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Bezier Curves ◽  
Number Concepts ◽  
Bézier Curves

This paper discusses about the series of fuzzified Bezier curves. This series of fuzzified Bezier curves constructed based on the fuzzy set theory especially fuzzy number concepts. The series of fuzzified represented by the alpha-cut of fuzzy number with various values of alpha. Then, the results were blended with Bezier curve function to produce the series of fuzzified fuzzy Bezier curves.

scholarly journals Serret-Frenet Frame and Curvatures of Bézier Curves

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 321 ◽  
Author(s):  
Esra Erkan ◽  
Salim Yüce
Keyword(s):  
Euclidean Space ◽  
Euclidean Plane ◽  
Applied Mathematics ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Bezier Curves ◽  
Bézier Curves ◽  
End Points ◽  
Unit Speed

The aim of this study is to view the role of Bézier curves in both the Euclidean plane E 2 and Euclidean space E 3 with the help of the fundamental algorithm which is commonly used in Computer Science and Applied Mathematics and without this algorithm. The Serret-Frenet elements of non-unit speed curves in the Euclidean plane E 2 and Euclidean space E 3 are given by Gray et al. in 2016. We used these formulas to find Serret-Frenet elements of planar Bézier curve at the end points and for every parameter t. Moreover, we reconstruct these elements for a planar Bézier curve, which is defined by the help of the algorithm based on intermediate points. Finally, in the literature, the spatial Bézier curve only mentioned at the end points, so we improve these elements for all parameters t. Additionally, we calculate these elements for all parameters t using algorithm above mentioned for spatial Bézier curve.

scholarly journals An Approximation of Bézier Curves by a Sequence of Circular Arcs

2021 ◽  
Vol 50 (2) ◽  
pp. 213-223
Author(s):  
Taweechai Nuntawisuttiwong ◽  
Natasha Dejdumrong
Keyword(s):  
New Method ◽  
Bézier Curve ◽  
Bezier Curve ◽  
Circular Path ◽  
Arbitrary Degree ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Circular Arc ◽  
Circular Arcs ◽  
The Given

Some researches have investigated that a Bézier curve can be treated as circular arcs. This work is to proposea new scheme for approximating an arbitrary degree Bézier curve by a sequence of circular arcs. The sequenceof circular arcs represents the shape of the given Bézier curve which cannot be expressed using any other algebraicapproximation schemes. The technique used for segmentation is to simply investigate the inner anglesand the tangent vectors along the corresponding circles. It is obvious that a Bézier curve can be subdivided intothe form of subcurves. Hence, a given Bézier curve can be expressed by a sequence of calculated points on thecurve corresponding to a parametric variable t. Although the resulting points can be used in the circular arcconstruction, some duplicate and irrelevant vertices should be removed. Then, the sequence of inner angles arecalculated and clustered from a sequence of consecutive pixels. As a result, the output dots are now appropriateto determine the optimal circular path. Finally, a sequence of circular segments of a Bézier curve can be approximatedwith the pre-defined resolution satisfaction. Furthermore, the result of the circular arc representationis not exceeding a user-specified tolerance. Examples of approximated nth-degree Bézier curves by circular arcsare shown to illustrate efficiency of the new method.

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