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Excerpt from The Similarity Between Shapes Under Affine Transformation The recognition of similarity between shapes under affine transformation is complicated because we have to consider shapes with rotational, translational, scal ing or stretching degrees of freedom. We simplify this problem by introducing the canonical form of a shape. Any shape can be transformed by an affine transforma tion to a canonical form. The canonical forms of all shapes which are identical under affine transformation can coincide through rotations. A systematical way is provided to find the measure function which can indicate the difference between shapes and keep independent of affine transformation. Then we can reduce the recognition of the similarity between shapes under affine transformation to the computation of the affine-independent difference between their canonical forms, i.e., finding the best matching between a pair of canonical forms by rotation, which is a one dimensional search problem. We have designed a linear algorithm for transforming a shape to its canonical form. The computation of the difference between a pair of canonical forms can be handled by various current techniques and a new method is presented in this paper. This method is robust in the sense that small changes in the shapes will only cause small changes in the final results. Not only similar shapes can be identified but also the quantity of difference between shapes can be computed. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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