M. V. Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, J. Vermeulen
{"title":"形状之间,使用豪斯多夫距离","authors":"M. V. Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, J. Vermeulen","doi":"10.4230/LIPIcs.ISAAC.2020.13","DOIUrl":null,"url":null,"abstract":"Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and various related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it.","PeriodicalId":11245,"journal":{"name":"Discret. Comput. Geom.","volume":"9 1","pages":"101817"},"PeriodicalIF":0.0000,"publicationDate":"2020-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"7","resultStr":"{\"title\":\"Between Shapes, Using the Hausdorff Distance\",\"authors\":\"M. V. Kreveld, Tillmann Miltzow, Tim Ophelders, Willem Sonke, J. Vermeulen\",\"doi\":\"10.4230/LIPIcs.ISAAC.2020.13\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and various related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it.\",\"PeriodicalId\":11245,\"journal\":{\"name\":\"Discret. Comput. Geom.\",\"volume\":\"9 1\",\"pages\":\"101817\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-09-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"7\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discret. Comput. Geom.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4230/LIPIcs.ISAAC.2020.13\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discret. Comput. Geom.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.ISAAC.2020.13","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Given two shapes $A$ and $B$ in the plane with Hausdorff distance $1$, is there a shape $S$ with Hausdorff distance $1/2$ to and from $A$ and $B$? The answer is always yes, and depending on convexity of $A$ and/or $B$, $S$ may be convex, connected, or disconnected. We show a generalization of this result on Hausdorff distances and middle shapes, and various related properties. We also show that a generalization of such middle shapes implies a morph with a bounded rate of change. Finally, we explore a generalization of the concept of a Hausdorff middle to more than two sets and show how to approximate or compute it.