After reviewing the rectangular/polar coordinate conversions we started yesterday, we revisited the original composition of transformations we looked at on Monday, noting that the Dilation x1/2, Rotation 45°, and Translation up 4 corresponds to an iteration of the function Az+B, where
- A = [.5, 45°] = √(2)/4 + √(2)/4i
- B = (0, 4) = 4i
Iterating this function allowed us to find that the coordinates of the attracting fixed point we identified earlier to be roughly positioned at (-2.605, 4.763) (or equivalent to the complex number -2.605+4.763i).
From there, we looked at a few problems from the back of the Complex Transformations Sheet, identifying again the specific geometric transformations that each complex linear function would produce and sketching the new location of a point that underwent that transformation. Your homework is to finish the back of that sheet
With the time we had left, we introduced a new piece of software: Complex Paint, a tool for more easily illustrating the transformations we have been identifying. We will work with this in more detail tomorrow.
We saw yesterday that a sequence of geometric transformations can illustrate the same “fixed point attractor” behavior that we’ve seen when iterating a mathematical expression. We explained this by illustrating that such geometric transformations are analogous to iterating a linear function in the Complex number system. Specifically, adding two complex numbers is analogous to translating, and multiplying two complex numbers is analogous to a combined dilation/rotation.
The conventional rectangular form of a complex number a + bi tells us the horizontal and vertical components of the translation achieved by adding the complex number, but this form is not useful for when the number is being used as a multiplication factor. For that, we need the number’s Polar Form. Once finding it, the value of r tells us the dilation factor and the value of θ tell us the rotation factor.
We’ll dig in to this some more tomorrow, but for now please watch this video recapping polar vs rectangular coordinates (which the video refers to as “Cartesian coordinates”) and how to convert between the two.