Distance geometry

Distance geometry studies geometric configurations using the distances between pairs of points. It asks which distance data are realizable in a given space and whether they determine a configuration uniquely.

Euclidean distance matrix: A matrix whose entries are pairwise distances between points in Euclidean space. It packages the measurements that distance geometry tests for realizability.
Euclidean distance matrix completion: The problem of filling in missing entries of a partial matrix so it represents Euclidean distances. Many distance-geometry problems begin with incomplete pairwise measurements.
Molecular conformation: A molecule's three-dimensional arrangement of atoms that can change through rotations around bonds. Interatomic distances constrain candidate molecular shapes in reconstruction problems.
Coordinate geometry: The study of geometry using coordinates and algebraic equations. It represents points directly, while distance geometry starts from pairwise separations.
Gram matrix: A matrix of inner products among vectors, encoding their lengths and mutual angles. Centering a squared distance matrix can recover the point configuration's Gram matrix.
Embedding dimension: The smallest dimension of a space into which a given metric configuration can be embedded isometrically. It states the least Euclidean dimension needed to realize specified distances.
Nuclear magnetic resonance spectroscopy: A spectroscopy method that uses nuclear magnetic resonance to probe molecular environments and structure. NMR-derived distance restraints help infer three-dimensional molecular configurations.
Affine geometry: The study of properties preserved by affine transformations, including collinearity and ratios along lines. Affine transformations can change distances, so affine structure is not the focus here.
Cayley–Menger determinant: A determinant built from pairwise squared distances that gives simplex volume information. It expresses dimension and volume constraints using distances alone.
Multidimensional scaling: Methods that construct low-dimensional point configurations from pairwise dissimilarities. It turns distance data into coordinates, often approximately when exact realization fails.

Linked from 13 pages