The Theory Of Curves In The Minkowski Plane

In this paper,we mainly study the theory of curves in the Minkowski plane.Under the p-norm and Randers norm,we derive the expressions of Minkowski curvature and normal curvature.By using the variational method,we show that normal curvature of the shortest curves among the variational curves is zero.In addition,we prove that among the variational curves with fixed end points in normed plane with norm ?(x,y)? = |x| + |y|,the shortest ones are the monotonous curves.This paper is organized as follows.In the first chapter,we briefly review the concepts concerned in this work,including Minkowski plane,Birkhoff orthogonal,determinant form,antinorm,etc.Some properties that will be used in this paper are also introduced.In the second chapter,we derive the expressions of Minkowski curvature and normal curvature under the p-norm and Randers norm.Particularly,we find that the two curvatures are the same under the Euclidean norm.In addition,we give some examples.In the third chapter,by using the variational method,we prove that normal curvature of the shortest curves among the variational curves is zero in the smooth Minkowski plane.In the forth chapter,a non-smooth case is discussed.We prove that the shortest ones are the monotonous curves among the variational curves with fixed end points in normed plane with norm ?(x,y)? = |x| + |y|.