Enveloid Of Legendre Curves And Its Applications

This thesis mainly studies the differential geometry of curves and surfaces with singular points in the Euclidean space.In the Euclidean plane,the envelope of a one-parameter family of curves is a curve that is "tangent”to each curve of the family at t,he intersections.If the relation of "tangent”is changed into the relation of "fixed angle",we call such a curve the enveloid of the family of curves.Enveloid theory has important applications in differential geometry,differential equations,geometric optics,physics,engineering and many other fields.However,the previous research on enveloids is only focused on the regular condition.For singular curves,on the one hand,due to the uncertainty of the tangent directions of curves at singular points,we can not determine the angular relationships between the curves at singular points,so we can not define enveloids of families of singular curves.On the other hand,as a classical tool to study the differential geometric properties of curves,the "Frenet-Serret type moving frames"cannot be established at singular points.Therefore,the previous results and methods about enveloids in the regular condition cannot be extended to the singular condition.In this thesis,we use the singularity theory of smooth mapping to establish the enveloid theory of one-parameter families of Legendre curves in the unit tangent bundle and unit spherical bundle,which solves the problem that the enveloids of families of singular curves cannot be studied previously.At the same time,we consider the applications of the enveloid theory in differential geometry and geometric optics.Furthermore,we apply the idea of the enveloid theory to the study of singular geometric objects in the Euclidean 3-space,define the normal curves of one-parameter families of framed surfaces and consider the related applications of normal curves.The main results of this thesis are as follows:1.From the perspective of singularity theory,we establish the enveloid theory of oneparameter families of Legendre curves,and use the enveloid theory to reveal the relationships between enveloids and ordinary differential equations.2.We give the relationships between spherical Legendre curves and plane Legendre curves.Furthermore,we give the relationships between the corresponding enveloids.3.As an application of the enveloid theory,we introduce the concept of involutes of spherical Legendre curves,and give the classifications of singular points of involutes.Furthermore,we introduce the concepts of involutoids and evolutoids,and discuss the duality relationships between them.4.As the generalizations of enveloids in high-dimensional space,we introduce the concept of normal curves of one-parameter families of framed surfaces in the Euclidean space.As an application of normal curves,we give the judgment theorem and geometric explanation of parallel curves of framed curves.We also introduce the concept of involutes of framed curves,and discuss the relationships among involutes,evolutes and parallel curves.