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# lec9 - Geometric Constraint Lecture(Mar 21-30 Instructor...

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Unformatted text preview: Geometric Constraint Lecture(Mar 21-30) Instructor: Meera Sitharam, Recorded by Jianhua Fan Mar 31, 2006 1 Problem Categories 2 Five Questions 1. Given graph G , characterize d for which ( G,d ) has a realization. Here d are constraints, for example distance constraints. 2. Given graph G and constraints d , provide a realization. 3. Given grpah G , generically classify it into two categories: It has nite number of realizations. One realization Many realizations It has in nite number of realizations. 4. Given G , generically characterize the realization space. 5. Given nongeneric G , with xed or restricted d , answer question 3 and 4. Give the classi cation and description of its realization space. 3 Working on these Five Questions 3.1 Question 1 Problem: G is a complete distance graph, nd { d : ( G,d ) has a realization in R k space } . 1 Theorem: Cayley-Menger conditions are the necessary and su cient conditions that ( G,d ) has a realization in R k space. 3.2 Question 4 Question 1 and 4 are equivalent in the sense that if we understand one of them, we understand the other. 3.3 Question 3 3.3.1 Laman's theorem: A graph G generically has only nitely many solutions i the following two conditions hold: 1. subgraph S G, 2 | V S | - | E S | 3 2. 2 | V G | - | E G | = 3 3.3.1.1 General Laman theorem: A graph G = ( V,E ) generically has at most nitely many solutions i subgraph G = ( V,E ) with E E such that 1. subgraph S G , 2 | V S | - | E S | 3 2. 2 | V G | - | E G | = 3 3.3.1.2 De nition of Generic Embedding: can be understanded in the following three ways: ( x 1 y 1 x 2 y 2 x n y n ) R 2 n d G R | V G | R 2 n E 2 Given ( G,d ) o/w d G { ( x 1 y 1 x 2 y 2 x n y n ) : ( x v- x w ) 2 + ( y v- y w ) 2 = d 2 vw ( v,w ) E ( G ) E 2 } d G = { ( x a- x b ) 2 + ( y a- y b ) 2 , ( a,b ) / E ( G ) , : ( x v- x w ) 2 + ( y v- y w ) 2 = d 2 vw ( v,w ) E ( G ) } There is one to one map between these two sets.There is one to one map between these two sets....
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lec9 - Geometric Constraint Lecture(Mar 21-30 Instructor...

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