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Qiqi Xia

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Proceedings Article | 20 October 2023 Paper

Quadrangular pyramid curve trajectory for communication performance optimization

Qiqi Xia, Chun Luo, Zhengyi Hou

Proceedings Volume 12916, 1291624 (2023) https://doi.org/10.1117/12.3004801

KEYWORDS: Unmanned aerial vehicles, Industry, Monte Carlo methods, Mathematics, Matrices, Mathematical optimization, Analytics

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Based on the definition of ellipse: the sum of the distances from the plane to the fixed point F1 and F2 is equal to the trajectory of the moving point P with a constant ( greater than |F1F2| ). It is extended to the study of the trajectory of the point with the distance from the moving point to the three lines and the fixed value, and the corresponding coefficients are added before the distance from the point to the three lines, so that the sum is fixed, that is k₁ ⋅ d(P,l₁)+k₂ ⋅ d(P,l₂)+k₃ ⋅ d(P,l₃) = M(M ≥ 0). With the help of drawing software to make the trajectory, based on analytic geometry and Auto CAD, a class of quadrilateral pyramid curve trajectory is obtained, which is a geometric projection similar to the conical curve. The trajectory can be intercepted on the quadrilateral pyramid with a rectangular bottom surface, and then the relationship between the variables in the curve trajectory equation is studied. In the variation, the third straight line is changed into a fixed point, and the obtained trajectory is a conic curve with the point as the focus. In addition, in the study of eccentricity, a triangle model is established to determine what kind of curve the trajectory will be. This kind of curve can be applied to the field of UAV communication. Combined with the relevant knowledge of mathematical analysis, the practicability of this kind of curve for signal enhancement and trajectory optimization is further studied from the perspectives of uniform distribution and Gaussian distribution. Finally, Monte-Carlo simulation and covariance verification are used to prove the help and role of pyramid curve in industry and information field.

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