This week the #puzzle is: Can You Trace the Locus? #geometry #area #coding

I have a loop of string whose total length is 10. I place it around a unit disk (i.e., with radius 1) and pull a point on the string away from the disk until the string is taut, as shown below.

I drag this point around the disk in all directions, always keeping the string taut, tracing out a loop. What is the area inside this resulting loop?

And for extra credit:

Now I have a loop of string whose total length is 14, and I place it around two adjacent unit disks. As before, I pull a point on the string away from the disks until the string is taut, as shown below.

I drag this point around the disks in all directions, always keeping the string taut, tracing out a loop. What is the area inside this resulting loop?

Can You Trace the Locus?

Solution, possibly incorrect:

Desmos

Let the center of the circle be at (0, 0). Choose a random point p₂ (x₂, 0) (0 < x₂) and fudge around, until the string has length s = 10. This involves some math first. Let p₁ (x₁, y₁) (0 < x₁) be the point, where the string switches between straight and round. This point is on the tangent to the circle going through p₂. p₁ also represents an angle a. I change a, until s = 10.

(I had some nicer equations planned, but WordPress won’t play along.)

(x₁, y₁) = (cos(a), sin(a))

Round stretch of string:

s₁ = 2 π – 2a = 2 (π – a)

Line through p₁:

m = y₁ / x₁

y = m x

Tangent through p₁:

m = – x₁ / y₁

y – y₁ = m (x – x₁)
y – y₁ = – x₁ / y₁ * x + x₁ / y₁ * x₁
y = – x₁ / y₁ * x + x₁ / y₁ * x₁ + y₁
y = – x₁ / y₁ * x + (x₁² + y₁²) / y₁
y = – x₁ / y₁ * x + 1 / y₁

Point where tangent hits x axis:

0 = – x₁ / y₁ * x₂ + 1 / y₁
0 = -x₁ * x₂ + 1
x₁ * x₂ = 1
x₂ = 1 / x₁

Length of 1 straight bit of string:

s₂ / 2 = √ (y₁² + (x₂ – x₁)²)
= 2 √ (sin²(a) + (1/cos(a) – cos(a))²)

Full length of string:

s = 10 = s₁ + s₂ = 2 (π – a) + 2 √ (sin²(a) + (1/cos(a) – cos(a))²)

As I said, I fudge around in Desmos, until I find a = 1.260529. (Wolfram Alpha confirms this solution.) This means x₂ = 3.27532.

This also means, the loop is a circle with radius 3.27532. This circle has area about 33.70.

And for extra credit:

Desmos Program Formula for tangent lines through point

This puzzle resembles the fiddler. Except the loop isn’t just a nice circle. I have to calculate a new y, every time I change x. (That’s what I ended up doing anyway.) I write a program to do it for me and estimate the area. (Geometry and integrals will be the death of me.) Below I show a quarter of the loop.

When -1 < y < 1, the calculation is the same. This part of the loop will be the same as the circle from the fiddler.

In that situation we have 2 straight bits of length 2, going between the circle, half a circle on the left and then the same situation as the fiddler on the right.

But outside of that band, the 2 straight bits between circles and point go to different circles.

Anyway. Result: area about 52.36.