Shoelace Formula: Connecting the Area of a Polygon and the Vector Cross Product

Lee, Younhee; Lim, Woong

doi:10.5951/mathteacher.110.8.0631

Cited by 26 publications

(8 citation statements)

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“…The sound exposure levels (SEL) over the defined noise receptor grid resulting from each modeled FPP and procedural profile are used to determine noise contours at the 90 and 85 SEL dB levels. The contour areas at these SEL dB levels are then computed using the shoelace method [33]. Since the expected contour area at each SEL dB level vary in size, percentage difference is a more useful metric for comparison than the absolute area differences.…”

Section: Comparison Of Performance and Noise Resultsmentioning

confidence: 99%

Takeoff Ground Roll Analysis of Real-World Operations for Improved Noise Modeling

Bhanpato

Behere

Kirby

et al. 2023

AIAA SCITECH 2023 Forum

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The ability to quantify aviation environmental impacts accurately is one of the key enablers for sustainable aviation growth. The Aviation Environmental Design Tool (AEDT) offers the capability to model aircraft operations and quantify the associated environmental impacts in terms of emissions and noise metrics. In a departure, the takeoff ground roll segment has a large contribution to the overall noise footprint from the operation. Discrepancies between modeling assumptions and real-world operations during takeoff ground roll not only lead to inaccuracies in peak noise levels but also affect noise contours throughout the downstream departure operation. This study investigates the modeling assumptions for B737-800 and B737-900 takeoff ground roll at Hartsfield-Jackson Atlanta International Airport (KATL) using high-fidelity real-world flight and surface weather data. Aircraft trajectory, takeoff weight, thrust, and weather conditions are extracted from real-world operations for modeling. This study quantifies the performance and noise differences arising from real-world variability during takeoff ground roll. Additionally, the sensitivity of noise results to inaccuracies during takeoff ground roll is investigated. The outcome of this study is the identification of sources of noise differences arising from modeling assumptions and their relative importance for accurate representation of real-world operations, thereby enabling more accurate noise modeling.

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Section: Comparison Of Performance and Noise Resultsmentioning

confidence: 99%

Takeoff Ground Roll Analysis of Real-World Operations for Improved Noise Modeling

Bhanpato

Behere

Kirby

et al. 2023

AIAA SCITECH 2023 Forum

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“…Where Area(M) is a constant, and it can be calculated either using the shoelace formula [19] or as the sum of its quadrilateral sections. Now we can define properly the objective function of the area f :…”

Section: Lmr Intersection and The Global Objective Functionmentioning

confidence: 99%

The Maximum Cover with Rotating Field of View

Potapov,

Ralph,

Triommatis

2023

Preprint

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Imagine a polygon-shaped platform $P$ and only one static spotlight outside $P$; which direction should the spotlight face to light most of $P$? This problem occurs in maximising the visibility, as well as in limiting the uncertainty in localisation problems. More formally, we define the following maximum cover problem: ''Given a convex polygon $P$ and a Field Of View (FOV) with a given centre and inner angle $\phi$; find the direction (an angle of rotation $\theta$) of the FOV such that the intersection between the FOV and $P$ has the maximum area''. In this paper, we provide the theoretical foundation for the analysis of the maximum cover with a rotating field of view. The main challenge is that the function of the area $A_{\phi}(\theta)$, with the angle of rotation $\theta$ and the fixed inner angle $\phi$, cannot be approximated directly. We found an alternative way to express it by various compositions of a function $A_{\theta}(\phi)$ (with a restricted inner angle $\phi$ and a fixed direction $\theta$). We show that $A_{\theta}(\phi)$ has an analytical solution in the special case of a two-sector intersection and later provides a constrictive solution for the original problem. Since the optimal solution is a real number, we develop an algorithm that approximates the direction of the field of view, with precision $\varepsilon$, and complexity $\mathcal{O}(n(\log{n}+(\log{\varepsilon})/\phi))$.

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“…One of the functions is ar used for computing area (see Fig. 13a)) approximately using the following Shoelace formula [25] such that the area is closed by a polygon created by the points on the profile curve (see Fig. 13b):…”

Section: Areamentioning

confidence: 99%

Rational Generalized Trigonometric Curve: Rationalization of Generalized Trigonometric Curve

Miura¹,

Gobithaasan²,

Misro³

et al. 2022

CADandA

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Designers often constrain their conceptual design as they sketch the idea to make sure the final concept solution has the expected functionality. Proposed system enables userdefined constraints created as mathematical formulas according to a syntax provided using geometrical features, ready-to-use functions, and mathematical operators. In this way, the users will be able to use computers' generative powers for their specific needs without being limited by built-in constraints. Defined constraints can be used in a sampling method as objectives to explore new design samples with better performance or filter out the undesired designs after the samples are generated. Thanks to its flexibility that lets the users define their constraints, the proposed method will also enhance the usability of generative design studies.

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Shoelace Formula: Connecting the Area of a Polygon and the Vector Cross Product

Cited by 26 publications

References 0 publications

Takeoff Ground Roll Analysis of Real-World Operations for Improved Noise Modeling

Takeoff Ground Roll Analysis of Real-World Operations for Improved Noise Modeling

The Maximum Cover with Rotating Field of View

Rational Generalized Trigonometric Curve: Rationalization of Generalized Trigonometric Curve

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