The Study Of Geodesics On Several Special Curved Surfaces

Geodesic plays an important role in the study of differential geometry.On the one hand,geodesics can be used to study the properties of curved surfaces,and on the other hand,it has a wide range of applications in the channel design of aircraft and ships.Although geodesics can be understood mathematically as the promotion of straight lines on a plane on a surface,it is still difficult to figure out the specific shapes of geodesics on some surfaces.It is understood that geodesics on most surfaces with orthogonal coordinate networks can only be expressed in the form of integral equations,which makes it difficult to sense intuitively.This thesis takes geodesics as the research subject and from the isometric invariance and geometric significance of geodesic curvature,it studies the geodesics on the surface under the nonorthogonal coordinate network;meanwhile,the geodesic images are drawn by the mathematical software Mathematica.The main work includes the following three aspects:1.The geodesic family of rotational hyperbolic surface,positive spiral surface and plane under the orthogonal coordinate network are studied with the Liouville formula,and then the geodesic families on the spiral surface,catenoid and parabolic cylinder are obtained from the isometric correspondence between surfaces.The images of geodesics and the parallel vector fields on the geodesics are drawn.2.From the perspective of the geometric significance of geodesic curvature,the geodesics of the type of plane curves passing through the origin on elliptic paraboloid and hyperbolic paraboloid are explored,and images of geodesics on paraboloids are drawn.3.The semi-geodesic coordinate nets on the two special rotating surfaces of catenoid and rotational hyperboloid surface are taken into consideration;the family of constant geodesic curvature curves of the semi-geodesic network on these surfaces under the isometric mapping is derived;the images of the semi-geodesic coordinate network and the family of constant geodesic curvature curves under the isometric mapping are drawn.