Similarity

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Similarity

 

a concept in geometry. Geometric figures are said to be similar if they are identical in shape, regardless of whether they are identical in size. The figures F1 and F2 are similar if between their points a one-to-one correspondence can be established such that the ratio of the distances between any two pairs of corresponding points of F1 and F2 a constant k. This constant is called the similarity factor. The angles between corresponding lines of similar figures are equal. Thus, in Figure 1 ∠B1A11C1 = ∠B2A2C2 = ϕ. The ratio of the areas of bounded similar figures is equal to the square of the similarity factor, and the ratio of the volumes of similar solids is equal to the cube of the similarity factor.

Figure 1

A geometric transformation of the plane (or of space) such that all figures in the plane are transformed into similar figures with the same ratio of similitude is called a similarity transformation, which is a special case of an affine transformation. The set of all similarity transformations of the plane or of space forms a group. Any similarity transformation can be represented as the successive application of a homothetic transformation and a proper or improper motion.

The concept of similarity and similarity transformations are used in such engineering applications of geometry as modeling and drawing. For example, the operation of the pantograph is based on the concept of similarity.

References in periodicals archive ?
He shared a screenshot of the then yet-to-be published interview with Payami on Facebook, where the Iranian filmmaker said he was happy for the selection of Newton at the Oscars and had no issue with the thematic similarities between the two films.
88) Erickson held that the usually protectable expression of a melodic sequence had merged with the idea of assigning [pi]'s digits to musical notes, and accordingly found no infringement--all of the similarities between the two works were traceable to this idea.
Research on similarities of fuzzy numbers, relevant in terms of consistency check, resulted with the conclusion that the inconsistency issues appear even with matrices of smaller dimensions.
Furthermore, the present research also showed that sexual prejudice increases as perceived similarities between participants and gay men decreases.
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Along the same lines (but dealing with the specific area of performance ratings), Strauss, Barrick, and Connerley (2001) suggest that people react and make attributions on the basis of perceived similarities, not actual similarities.
PETA's display simply shows the similarities between past abuses to humans and current abuses to other species.
Previous work underestimated genetic similarities in people and chimps, the investigators say, and placed the evolutionary parting of these species about 1 million years too early.
She describes how both Michelangelo and Leonardo were courted by the ruling sultan Beyazid II and points out the similarities between the Hagia Sophia and Bramante's centralised design for St Peter's with hemispherical domes and two towers.