similarity in geometry

Two figures $K$ and $K'$ in a Euclidean plane or space are similar iff there exists a bijection $f$ from the set of points of $K$ onto the set of points of $K'$ such that, for any $P,Q \in K$ , the ratio $$\frac{P'Q'}{PQ}$$ of the lengths of the line segments $P'Q'$ and $PQ$ is always the same number $k$ , where $P'=f(P)$ and $Q'=f(Q)$ .

The number $k$ is called the ratio of similarity or the line ratio of the figure $K'$ with respect to the figure $K$ (N.B. the order in which the figures are mentioned!). The similarity of $K$ and $K'$ is often denoted by $$K' \sim K\;\;\;(\mbox{or}\;\;K \sim K').$$

Examples

• All squares are similar.
• All cubes are similar.
• All circles are similar.
• All parabolas are similar.
• All sectors of circle with equal central angle are similar.
• All spheres are similar.
• All equilateral triangles are similar.

Nonexamples

• Not all rectangles are similar.
• Not all rhombi are similar.
• Not all rectangular prisms are similar.
• Not all ellipses are similar.
• Not all ellipsoids are similar.
• Not all triangles are similar.

Properties

• The corresponding angles (consisting of corresponding points) of two similar figures are equal.
• The lengths of any corresponding arcs of two similar figures are proportional in the ratio $k$ .
• The areas of two similar regions are proportional in the ratio $k^2$ when $k$ is the line ratio of the regions.
• The volumes of two similar solids are proportional in the ratio $k^3$ when $k$ is the line ratio of the solids.

Remarks

• In any Euclidean space $E$ , the relation of similarity (denoted $\sim$ ) on the set of figures in $E$ is an equivalence relation.
• If one pair of corresponding line segments in the similar figures $K$ and $K'$ are equal, then all pairs of corresponding line segments are equal, i.e. the figures have also equal sizes: They are congruent ($K' \cong K$ ).