The Minkowski Plane Closed Curve Geometric Expansion

The geometric dilation problem, a simple question from computational geome-try begins in the 1970s, but it develops into various fields of computer science and mathematics, like robot motion design, differential and integral geometry, knot the-ory, number theory and convex geometry. The reaserch begins with the Euclidean plane, and lots of results are obtained. The problems include graph dilation, point set dilation, the geometric dilation closed curves and so on.These days, the related study are gradually spread to(normed or)Minkowski planes from Euclidean planes. The thesis studies the geometric dilation of rectifiable simple closed curves in Minkowski planes and its fundamental task is to study the lower bound of geometric dilation of closed curves in Minkowski plane.The thesis summarizes and analyzes systematically the results that we have known and focus on specifying the lower bound of the geometric dilation of rectifiable simple closed curves in Minkowski planes.In this paper, our main work is to give the lower bound for the geometric dila-tion of closed curves in Minkowski planes a quantitative result,δX(C)≥1.5, which is induced by the result that the circumference of the unit circle SX in Minkowski SX of X is equal to 6. By making use of these results and the halving pair trans-formation, we draws the sufficient for the inequality and necessary conditionfor the equality and we specifies that homothet is a transformation preserving geometric dilation.Finally, the thesis contrasts and analyzes the similarity and difference in geo-metric dilation of closed curves of Euclidean planes and Minkowski planes.