v2007.09.13 - Convex Optimization

384 CHAPTER 5. EUCLIDEAN DISTANCE MATRIX5.14.3.3.1 Example. Pyramid.A formula for volume of a pyramid is known; 5.58 it is 1 the product of its3base area with its height. [163] The pyramid in Figure 93 has volume 1 . 3To find its volume using EDMs, we must first decompose the pyramid intosimplicial parts. Slicing it in half along the plane containing the line segmentscorresponding to radius R and height h we find the vertices of one simplex,⎡X = ⎣1/2 1/2 −1/2 01/2 −1/2 −1/2 00 0 0 1⎤⎦∈ R n×N (976)where N = n + 1 for any nonempty simplex in R n .this simplex is half that of the entire pyramid; id est,evaluating (974).√The volume ofc =1found by6With that, we conclude digression of path.5.14.4 Affine dimension reduction in three dimensions(confer5.8.4) The determinant of any M ×M matrix is equal to the productof its M eigenvalues. [247,5.1] When N = 4 and det(Θ T Θ) is 0, thatmeans one or more eigenvalues of Θ T Θ∈ R 3×3 are 0. The determinant willgo to 0 whenever equality is attained on either side of (699), (965a), or (965b),meaning that a tetrahedron has collapsed to a lower affine dimension; id est,r = rank Θ T Θ = rank Θ is reduced below N −1 exactly by the number of0 eigenvalues (5.7.1.1).In solving completion problems of any size N where one or more entriesof an EDM are unknown, therefore, dimension r of the affine hull requiredto contain the unknown points is potentially reduced by selecting distancesto attain equality in (699) or (965a) or (965b).5.58 Pyramid volume is independent of the paramount vertex position as long as its heightremains constant.