Similarity

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

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 ?
In order to improve the matching accuracy, a traversal similar triangle test method is adopted.
After refining with geometric and similar triangle constraints, the reserved matching points are almost correct except several separate points.
After similar triangle refining, 5 pairs are left in Figure 6.
After consecutive procedure of geometric constraints, similar triangle, and RANSAC matching, the reserved point pairs are all correctly matched.
One of the main novelties is the multimodal matching SURF features refining procedure with geometric, similar triangle, and RANSAC constraints in registration process.
such that [T.sub.n] is derived from [T.sub.n-1] by constructing similar triangles on the sides of [T.sub.n-1], whose opposite angle is 120 degrees and joining the apexes of these triangles.
I now had two measurements and a hunch about similar triangles. I remembered a method for estimating the heights of tall objects by using a small known object, similarity, and shadows, as seen in Figure 4.
I was no longer sure I had any similar triangles with which to work.
Because of all the parallel lines, there are many similar triangles and corresponding angles between the two.
My first solution, while not totally correct, involves a straightforward application of similar triangles and an answer that is not far off my improved solution.
We did not use similar triangles to prove it, rather area, which is surprising!

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