摘要
异形超椭圆是一种只有单对称轴的平面闭合曲线，该形状普遍存在于自然界和工程设计中，具有重要学术研究意义。基于宽阔河流平面二维变系数对流扩散物质的等浓度线方程和镜面成像原理，定义了包含半长度（半高度）、半宽度、丰度指数和偏度指数的四参数异形超椭圆标准方程；推导出异形超椭圆的面积公式和绕对称轴旋转体的体积公式；分析发现丰度指数只影响异形超椭圆的丰满程度，偏度指数只影响异形超椭圆最大宽度线偏离原点的相对距离，给出蛋形、飞机截面、鱼雷、飞碟等20种图形/图案相应的异形超椭圆特征参数；分析表明异形超椭圆的连续性、光滑性和整体性好，应用于隧道与地下工程、建筑与桥梁工程、液体运输罐和飞行器与水中航行器等设计中，还需要进行结构与流体力学性能研究。
关键词： 四参数曲线；异形超椭圆；几何性质；丰度指数；偏度指数
Abstract
A heteromorphic superellipse is a planar closed curve with a single symmetric axis, which is frequently encountered in natural environments and human engineering designs, and thus it is of great academic importance. In this study, we define a standard equation for a new heteromorphic superellipse with four parameters comprising the half-length (half-height), half-width, richness index, and skewness index based on the planar two-dimensional variable-coefficient isoconcentration line equation for advection diffusion substances in wide rivers and specular image theory. In addition, we derive equations for computing the area of a heteromorphic superellipse and the volume of a heteromorphic superellipse revolved body. We show that the richness index only affects the degree of fullness and the skewness index only affects the distance of the deviation from the origin to the maximum width line of the heteromorphic superellipse. Furthermore, we present the characteristic parameters for 20 heteromorphic superelliptic shapes, such as an egg, aircraft cross-section, torpedo, and flying saucer. Based on our analysis, we conclude that the continuity, smoothness and integrity of the heteromorphic superellipse are good. The practical applications of the heteromorphic superellipse in areas such as tunnel and underground engineering, civil and bridge engineering, liquid transport containers, aerospace and underwater vehicles designs require further structural and fluid mechanics research.
Key words： Four-Parameter Curve; Heteromorphic Superellipse; Geometric Property; Richness Index; Skewness Index
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