v2004.06.19 - Convex Optimization

278 CHAPTER 8. EDM COMPLETION8.2 Requirements for unique completion8.2.1 Conservative requirements in R 2The well-known geometric technique trilateration requires only three noncollinearabsolute point-positions x 1 ,x 2 ,x 3 in R 2 to uniquely determine absoluteposition of a fourth point x 4 from distance information √ d 14 , √ d 24 ,√d34 . (Figure 8.2(a)) The classical calculation finds the common intersection√of√three circles respectively centered at x 1 , x 2 , and x 3 having radiid14 , d24 , and √ d 34 . This technique is used by GPS receivers (globalpositioning system, §4.11.3.0.1).If only isometric reconstruction is desired, then we need have only relativeposition of the three non-collinear points; and then trilateration is no longerpossible. To determine relative position of x 1 ,x 2 ,x 3 , the triangle inequalityaxiom of the Euclidean metric is necessary and sufficient. (§4.12.1) Combiningwith √ d 14 , √ d 24 , √ d 34 the new distances between unknown x 1 ,x 2 ,x 3represented by the heavy reference triangle in Figure 8.2(b), then we have acomplete EDM for four points;⎡⎢⎣0 d 12 d 13 d 14d 12 0 d 23 d 24d 13 d 23 0 d 34d 14 d 24 d 34 0⎤⎥⎦ (799)(The bold entries represent the reference triangle in a principal 3 × 3 submatrix;all entries here are known.) If a realization exists in R 2 , then theirisometric reconstruction is unique by injectivity of D ; (§4.6) meaning, if theEDM describing the four-point list is unique, then so will be its correspondingisometric realization.Now consider augmentation of our four-point list with a fifth point x 5illustrated in Figure 8.2(c). To unambiguously describe the relative positionof x 5 in R 2 , we need only the reference triangle and √ √ √d 15 , d25 , d35 .If a realization exists in R 2 , then it exists in the plane of the referencetriangle. This means the two isometric realizations thus far ˜x 1 , ˜x 2 , ˜x 3 , ˜x 4and ˜x 1 , ˜x 2 , ˜x 3 , ˜x 5 coexist in R 2 . Since each realization is unique with respectto the reference triangle, then the combined realization is unambiguous inR 2 . Relative position of five points in R 2 is therefore specified uniquely by