Points are actions - Complex Number 2

Spiral Similarity is a combination of rotation and dilation.

It is an interesting geometric transformation for a variety of reasons. In the previous section, we found that any segment can be mapped onto any other segment using spiral similarity.

We wish to treat spiral similarity algebraically.

Let \( A = (r, \theta) \). Suppose we want to rotate it by an angle \( \theta_1 \) and also dilate it (multiply its length by) \( r_1\).

Clearly, after we perform the rotation and dilation, the resultant point is: $$ (r \times r_1, \theta + \theta_1) $$

This action of rotation and dilation can be encoded in an ordered pair of numbers: \( r_1, \theta_1 \).

The first number of this ordered pair says how much we should dilate (by what factor should we multiply the length). The second number gives the angle of rotation.

We will call this ordered pair of numbers an action. Hence \( (r_1, \theta_1) \) acts on \( (r, \theta ) \) to rotate and dilate it to the final position \( (r \times r_1, \theta + \theta_1) \)

A short hand way of writing this is $$ (r_1 \theta_1 ) \searrow (r, \theta) \to (r \times r_1, \theta + \theta_1) $$

Dual nature of points and actions

Notice that the action \( (r_1, \theta_1) \) can be regarded as a polar coordinate of a point. It is the position of a point that is \( r_1 \) distance away from origin and making an angle \( \theta_1 \) with the positive direction of x axis.

On the other hand the point that \( (r_1, \theta_1) \) is acting on, that \( (r, \theta) \) can be itself regarded as an action. It has the power of rotation by \(\theta \) and dilation by r.

We have arrived at the essence of our core idea: points can be regarded at once as objects waiting to be rotated and dilated OR actions that can rotate and dilate.

(This is a supplemental note to live lecture session of Complex Number module of Cheenta I.S.I., C.M.I. Entrance Program and Math Olympiad Program)

Spiral Similarity - Complex Number 1

What is spiral similarity?

Spiral similarity, for our purpose, will be a beautiful motion in the plane. It is a combination of rotation and dilation (expansion or contraction by a constant factor.

Suppose AB is a (finite) segment in the plane. CD be another segment. Assume AB and CD to be disjoint.

It is possible to send AB onto CD using a combination of rotation and dilation.

Spiral Similarity and Complex Number

To accomplish this, we need to find:

How to find center of rotation (and dilation)

Join AC and BD. Suppose they intersect at X. Draw the circles CXD and AXB. Let them intersect at O.

O could be same as X or could be a different point.

Claim: O is the center of rotation and dilation, that sends AB to CD.

Spiral Similarity and Complex Number

Join OA, OX, OB, OC, OD.

Pause and think: Can you find some cyclic quadrilaterals in this picture.

Spiral Similarity and Complex Number

Clearly ABXO is cyclic. This implies \( \angle OBX = \angle OAX \). This is because they are subtended on the circumference by the same segment OX. (Angles subtended by a chord at the circumference are equal, by property of cyclic quadrilaterals).

Similarly CDXO is cyclic implying \( \angle ODX = \angle OCX \)

Pause and think: Can you find a pair of similar triangles?

Hint: Equiangular triangles are similar.

\( \Delta OBD \sim \Delta OAC \)

This is because we have shown pair of corresponding angles to be equal. Hence the two triangles are equiangular.

Pause and imagine:

This implies \( \angle AOC = \angle BOD \). We may imagine this as following:

Since the angles (of rotation) \( \angle AOC, \angle BOD \) are equal, hence this works out nicely.

Moreover, OA grows into OC and OB grows into OD by same constant factor. This is true because \( \frac {OA}{OC} = \frac{OB}{OD} \). This follows from the similarity of triangles.

Hence A goes to C and B goes to D, via rotation and dilation, centered at O.

Rotation and dilation is known spiral similarity.

Complex numbers are algebraic tools to encode this motion in the plane.

(This is a supplemental note to live lecture session of Complex Number module of Cheenta I.S.I., C.M.I. Entrance Program and Math Olympiad Program)