Creating a circle that goes through three points on your lawn

I get some off-the-wall questions once in a while. Today, I got an off-the-porch question and decided the answer might make a good blog post. I'm probably wrong about that, but here goes.

My friend Dark Laser (I have changed his name to protect his identity) is putting a flower garden in his backyard. He wants the edge of the flower bed to be a circular arc that goes through three points . Those three points have been defined by a higher power. Maybe the higher power is his wife, or maybe it's just cuz of where the house is. I dunno. Below is the drawing that he sent me.  I hope you appreciate his obvious artistic skills.

Before he sent the email, Dark did a little googling and YouTubing. He came up with one answer in a YouTube video, which I show below.

I'm not sure how to interpret an email that asks me a question, and also sends me the answer. I'm not sure what he meant by this juxtaposition of Q & A, but due to my insecurity, I took the email as a challenge. "See if you are as smart as this guy, John!!" I am certainly not going to allow any dufus on YouTube to out-answer me on any dumb old math question!

The MathTuber who answered this question used analytical geometry, which lies between geometry (which is all lemmas and compasses) and algebra (which is all factor this polynomial and take the square root of both sides). But mostly the video is algebra.

Here's my opportunity to show off my brilliance. The abracadabra algebra video is all well and good, but it doesn't completely answer the original Dark question. I mean, how does Dark use this equation? Does he need to go out and buy a huge piece of graph paper to lay on his lawn?!?!?

I chose to forego the algebraic approach and go for a solution that is more along the lines of Euclid and his book Elements. A lot of this ancient text has to do with making constructions of various figures with a pencil, a compass, and a straight edge. But I don't think Dark has these tools, especially not in the size required to lay this out on his lawn. So, I improvised with more appropriate tools: rope and stakes.

(While I am at it, let me take a moment for more self-congratulations. Pure mathematicians are perfectly content with theoretical answers. But I am an applied mathematician, which means that my math isn't happy until it answers a real world problem. You can't get any more practical than building an aesthetically pleasing flower bed!)

Step 1

Put stakes at Points 1 and at Point 2. Make two equal lengths of rope, and tie them together at one end. You don't need the ropes to be red and blue, as in my diagram, but it may help.

Tie the other ends of the ropes to the stakes at Point 1 and to Point 2. Grab the ropes where they join and pull them taut. Place a third stake at that point. This third stake is shown as a blue circle in the drawing below.

To avoid any confusion, I use the word stake in a general sense. If any of the stake positions should happen to be on lawn then a tent stake could meet the purpose. If the point is on a wood porch, then a hefty nail or a bolt could work. If the point is on a cement slab of a porch, then maybe a can of spray paint could be used to mark the point. Or a bathroom plunger? 

Step 2

Shorten the two ropes and repeat the process to locate a position for a fourth stake, as shown below as a second blue circle.

Now the aha! part, which I mention in order to create a sense of making progress. All circles which go through both Point 1 and Point 2 will have a center someplace on the blue dotted line! I hope you are as excited as I am.

A practical comment -- If the lengths of the ropes in Steps 1 and 2 are very close to the same, then the two stakes will be pretty darn close together. This is not such a good thing. This will lead to uncertainty in the angle of the blue dotted line, which will lead to inaccuracy in the position of the final circle.

An even more practical comment -- As I wrote that last comment (the so-called  practical comment), it occured to me that in Step 2, you could have kept the rope the same length, and merely pulled it to a position above Points 1 and 2. Too bad you already went through the process of shortening the ropes.  

Step 3

If you want to get technical on me, Step 3, is really two steps. It's a repeat of Steps 1 and Steps 2, only with different points. Repeat Steps 1 and 2 with Points 2 and 3. You probably will need to stretch the ropes out if you already cut them.

The green dotted line in the drawing below is analogous to the blue dotted line. All circles which go through both Point 2 and Point 3 will have a center someplace on the green dotted line! 

I know some of you may have jumped right ahead to the big climax, but I will state it here for anyone who might not have quote caught the significance: All circles which go through Point 1, Point 2 and Point 3 will have a center at the intersection between the blue dotted line and the green dotted line. Assuming the two dotted lines intersect, and assuming the two dotted lines are not along the same line, we have uniquely defined the center of the circle.

Step 4

A pure mathematician would stop at Step 3, since the point has been theoretically defined. But an applied mathematician, being of a superior breed, would realize that we still need a way to mark that physical intersection point on the porch.

Here is my suggestion. Place a skinny pole at a possible location for the circle center. Move the skinny pole around until it lines up with both of the blue stakes. Then turn your head and line the skinny pole up with the two green stakes. Then go back to the blue stakes to make sure they still line up. This may take several iterations. (My own experience suggests that copious quantities of beer can reduce the number of iterations necessary, not because beer increases your skill level, but because it gives you a more realistic view of just how important the position of the center of the circle is in the grand scheme of things.) Put a nail or a stake or a plunger at the point of intersection. 

If neither nail nor stake nor plunger will work on the porch, then buy another 12 pack and set it next to the intersection point. When a neighbor stops by to see what you're doing, hand him a beer and ask him to sit at the intersection point. The extra beers will keep him from moving.

A comment for those who remembered taking geometry... The task of finding the intersection would have been done with a straight edge. If Dark happens to have a 2 X 4 that is long enough, he could certainly use that to mark the dotted lines. A can of spray paint could serve to make that line indelibly, so the next owners of the house can appreciate the mathematics that went into constructing the flower bed. I haven't looked in Dark's garage lately, but I am guessing that finding a 2 X 4 that is long enough is a tall order. Or a long order. So, we must resort to an iterative procedure which would have been scorned by Euclid. But being scorned by Euclid is not a big deal. We are using non-Euclidean geometry.

Step 5

We're now ready to finish the project. Attach a rope to the nail/stake/plunger/neighbor at the center, and stretch the rope out until it reaches Point 1, Point 2, or Point 3.

Tie a spike to the  rope at that point. Holding the rope taut, move the spike from Point 1 to Point 2 and then on to Point 3, scratching the lawn to indicate the edge of the circle. If you are not skilled in the art of lawn scratching, feel free to tie a can of spray paint to the rope. I suggest a color of paint which is different from that of the grass. Although I show purple in the drawing below, green paint would provide a real good contrast to the color of my lawn.

Step 6

Pull up a lawn chair and finish the beer, content in having accomplished a good day's work.