<div style="text-align:justify;"> The ellipse and the superellipse are both planar closed curves with a double axis of symmetry. Here we show the isoconcentration contour of the simplified two-dimensio...<div style="text-align:justify;"> The ellipse and the superellipse are both planar closed curves with a double axis of symmetry. Here we show the isoconcentration contour of the simplified two-dimensional advection-diffusion equation from a stable line source in the center of a wide river. A new two-parameter heteromorphic elliptic equation with a single axis of symmetry is defined. The values of heights, at the point of the maximum width and that of the centroid of the heteromorphic ellipse, are derived through mathematical analysis. Taking the compression coefficient <em>θ </em>= <em>b/a =</em><em></em><span></span> 1 as the criterion, the shape classification of H-type, Standard-type and W-type for heteromorphic ellipse have been given. The area formula, the perimeter theorem, and the radius of curvature of heteromorphic ellipses, and the geometric properties of the rotating body are subsequently proposed. An illustrative analysis shows that the inner contour curve of a heteromorphic elliptic tunnel has obvious advantages over the multiple- arc splicing cross section. This work demonstrates that the heteromorphic ellipses have extensive prospects of application in all categories of tunnels, liquid transport tanks, aircraft and submarines, bridges, buildings, furniture, and crafts. </div>展开更多
The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane ...The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane significantly influences the value of the geodetic coordinates.This research analyzes this influence and how it can contribute to their solutions.The study investigates the mathematical relation between them and presents an exact formula relating to the geodetic height and the ellipse equation.In addition,a heuristic formula for the relation between the geodetic height and the ellipse equation is proposed,which is independent of the geodetic latitude and has a relative accuracy better than 99.9 %.The calculation is stable,and the cost is low.展开更多
New equations are proposed to describe the shapes and sizes of the contact areas between tire and ground(the contact areas)under the conditions of static load,slip,camber,and rolling.The equations are super ellipse an...New equations are proposed to describe the shapes and sizes of the contact areas between tire and ground(the contact areas)under the conditions of static load,slip,camber,and rolling.The equations are super ellipse and trigonometric function with four parameters.Results indicate that the contact areas under the conditions of static load,slip,camber can be well described by super ellipse,in the most complicated case,fourteen parameters are used.The rolling contact area can be reconstructed by the trigonometric function completely.Symmetrical and asymmetrical shapes can both be described perfectly.The suggested curves will make calculate size of the areas easily.The described shapes and sizes are in good agreement with the measured,the maximum deviation is only 2.88%.This work will provide an instrument for understanding the contact area and be helpful for analyzing tire and road.展开更多
文摘<div style="text-align:justify;"> The ellipse and the superellipse are both planar closed curves with a double axis of symmetry. Here we show the isoconcentration contour of the simplified two-dimensional advection-diffusion equation from a stable line source in the center of a wide river. A new two-parameter heteromorphic elliptic equation with a single axis of symmetry is defined. The values of heights, at the point of the maximum width and that of the centroid of the heteromorphic ellipse, are derived through mathematical analysis. Taking the compression coefficient <em>θ </em>= <em>b/a =</em><em></em><span></span> 1 as the criterion, the shape classification of H-type, Standard-type and W-type for heteromorphic ellipse have been given. The area formula, the perimeter theorem, and the radius of curvature of heteromorphic ellipses, and the geometric properties of the rotating body are subsequently proposed. An illustrative analysis shows that the inner contour curve of a heteromorphic elliptic tunnel has obvious advantages over the multiple- arc splicing cross section. This work demonstrates that the heteromorphic ellipses have extensive prospects of application in all categories of tunnels, liquid transport tanks, aircraft and submarines, bridges, buildings, furniture, and crafts. </div>
文摘The conversion of the cartesian coordinates of a point to its geodetic equivalent coordinates in reference to the geodetic ellipsoid is one of the main challenges in geodesy.The ellipse equation in the meridian plane significantly influences the value of the geodetic coordinates.This research analyzes this influence and how it can contribute to their solutions.The study investigates the mathematical relation between them and presents an exact formula relating to the geodetic height and the ellipse equation.In addition,a heuristic formula for the relation between the geodetic height and the ellipse equation is proposed,which is independent of the geodetic latitude and has a relative accuracy better than 99.9 %.The calculation is stable,and the cost is low.
基金National Natural Science Foundation of China Project(51790502),Open Grant of National Key Laboratory of Science and Technology on Advanced Composites in Special Environments(JCKYS2019603C016)。
文摘New equations are proposed to describe the shapes and sizes of the contact areas between tire and ground(the contact areas)under the conditions of static load,slip,camber,and rolling.The equations are super ellipse and trigonometric function with four parameters.Results indicate that the contact areas under the conditions of static load,slip,camber can be well described by super ellipse,in the most complicated case,fourteen parameters are used.The rolling contact area can be reconstructed by the trigonometric function completely.Symmetrical and asymmetrical shapes can both be described perfectly.The suggested curves will make calculate size of the areas easily.The described shapes and sizes are in good agreement with the measured,the maximum deviation is only 2.88%.This work will provide an instrument for understanding the contact area and be helpful for analyzing tire and road.