Homothety 1

(This is a series of discussions on Homothety. It is largely derived from the Math Olympiad Classroom Discussion in Cheenta - cheenta.com)

Homothety is a geometric transformation. It has a couple of synonyms: dilation and central similarity.

A geometric transformation is a function. It can be thought of as a machine which takes in a point as input and gives out a point as output.

In algebra or calculus we encounter simple functions in school. One example is $ f(x) = x^2 $ . The rule of processing in this function is "square the input". That is if the input is 3, output will be 9. Geometric Transformations are similar things.  It's set of input (often termed as "domain of the function") are however points instead of real numbers. Algebraically the points are doublets of numbers like (1,2) representing the coordinate of the point. Of course a point can also be represented by triplet of numbers (say (3, 5, -7) ) if we do geometry in three dimension (and similarly by a n-tuple $ (a_1 , a_2 , ... , a_n ) $ of numbers if we work in n-dimension) . Presently we shall confine our discussion in the plane or two dimension.

What kind of function (geometric transformation) is homothety?

To define homothety we need to specify two things: center of homothety and ratio of homothety. Center of homothety is a point on the plane. It can be any point. When we treat homothety algebraically, this point is regarded as the origin (0, 0) of the coordinate axis (assuming we are working in cartesian coordinate system). Ratio of homothety can be any real number positive, negative or zero.

Suppose O is the center of homothety and r is the ratio of homothety. If a point A on the plane is the input we can find the output point A' in the following manner: join OA, extend OA both sides and find a point A' on this line such that $\frac{OA'}{OA} $ = r .

In other words, the segment OA is stretched (or contracted) r times and the direction of this stretching or contraction is decided by the sign of r. Suppose r = 2, then OA will be stretched two times in the direction OA. However if r = -2, A' is on AO (on the other side of A about O) and length of OA' is twice of OA.

In the next installment of this series of articles we will continue with the applications of homothety.

(This is a series of discussions on Homothety. It is largely derived from the Math Olympiad Classroom Discussion in Cheenta - cheenta.com)