YOUR FRIDAY MATH with KP Hart – Reflections

Does there exist a seamless transition from reflection in a circle (Inversion) to reflection in a line using real mirrors?

So, I am looking for a seamless transition from the formula for Inversion to that of a reflection in a line. Can that be done with hollow mirrors?

A (counter) question could be: what does `seamless transition’ actually mean? Mathematically we can give that a precise meaning by using limits.

The picture above represents what symmetry/inversion with respect to a circle means. The points z₁ and z₂ are said to be symmetric with respect to the circle of radius r with center C if they lie on the same half line emanating from C and if the product of their respective distances to C is equal to r². Geometrically you can construct z₂ from z₁ by drawing a line through z₁ that is tangent to the circle and intersect the perpendicular from the tangent point to the line passing through C and z₁with that line.

We look at a special case: for every positive real number r we take the circle of radius r and with cent-re (-r,0). If you were to draw that circle for a large value of r then the part that fits on the paper would be almost indistinguishable from the y-axis in the plane. (Take r=1000 and calculate what x is when (x,10) lies on the circle. Try r=10000, … as well.) With a little perseverance you will be able to calculate that if (u,v) is the inversion point of (x,y) with respect to the circle then we have the following equalities:

We can take the limits of the right-hand sides for r to ∞, and those limits are equal to -x and y respectively. In this sense the inversions turn seamlessly into the reflection in the y-axis.

This can not be done with real mirrors; not even idealized ones, where we forget about atoms and their effects. The point is that reflecting in a line of plane will preserve distances of points. This would then also have to be true for the points on a line that falls perpendicularly on a (curved) mirror. However a circle inversion takes a line segment from the center to the circle, hence of length r, and transforms it into a line segment of infinite length. Inversion is therefore not the same as (physical) reflection.

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Read all of KP Harts math questions here!

About Dutch Mathematician KP Hart: In the beginning of this year the Dutch government opened a website, The Dutch Science Agenda, where everyone could post questions that they thought were of scientific interest. This was an attempt to involve the whole country in determining what the Dutch science agenda should be in the coming years.

I looked through the questions and searched for terms like `mathematics’, `infinity’ … to see what mathematical questions there were and I noticed various questions that already have answers (and have had for a long time). On a whim I decided to post answers to those questions, in Dutch. For your edification I will translate these posts into English.