Advanced International Journal for Research

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Volume 7 Issue 5
September-October 2026

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Application of Polar Coordinate Graphs for Industrial Design

Author(s) Chung S Lee, Brendon M Son
Country United States
Abstract Polar coordinate graphs provide a compact mathematical language for products whose geometry is organized around a center, axis, rotation, or angular sequence. This paper examines how equations of the form r = f(θ) can support industrial design from early form exploration through parametric CAD, engineering optimization, and manufacturing. Applications include impellers, turbine and fan components, radial vents and grilles, cams, spiral housings, rotary interfaces, decorative perforations, and other rotationally organized products. The study develops a design methodology based on circles, rose curves, cardioid/limacon families, Archimedean spirals, logarithmic spirals, and Fourier-type radial profiles. It also explains conversion to Cartesian coordinates, geometric constraints, curvature considerations, repeated angular modules, and the extension of two-dimensional polar curves into three-dimensional surfaces. Published CAD and turbomachinery literature demonstrates the practical relevance of coordinate-based parameterization, NURBS/B-spline surfaces, circular-root relations, design automation, reverse engineering, and five-axis manufacturing. The paper concludes that polar graphs are most valuable not as isolated mathematical illustrations but as controllable design generators that connect aesthetics, function, computation, and fabrication.
Keywords polar coordinates; industrial design; parametric design; CAD/CAM; spiral geometry; radial symmetry; impeller; turbine blade; cam; computational design
Field Mathematics
Published In Volume 7, Issue 5, September-October 2026
Published On 2026-09-25
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E-ISSN 3048-7641
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