The propagation of error in classical geometry constructions (2018) | Hacker News Reader

The propagation of error in classical geometry constructions (2018)

jdh.hamkins.org

88 pointsdownboots27 comments

I bought a copy of TASCHEN's "The Elements of Euclid" with illustrations by Oliver Byrne from his book in 1847. As I was working through a lot of the later problems I could see myself causing these sorts of errors due to a skill issue. For example as I dragged the pencil vertically down a ruler the wooden side of the pencil (above the lead) rested on the ruler. As I dragged the pencil down my arm lowered and my wrist changed orientation so that the lead of the pencil was now resting against the ruler as I completed the line. It was honestly so frustrating to see the corners and arcs of my diagram look like abstract dog shit because I was too much of a dumbass to use a ruler correctly.

Yes, I've learned some techniques after reproducing a lot of Islamic patterns, which can be done entirely with straightedge and compass (although frankly I often 'cheat' by using measures, grid paper and tracing paper, lol).

Indeed the movement of the pencil can matter a lot, as can how good your compasses are. I've found that a bevelled ruler can be better if turned over so that the pencil rests against the bevel. Otherwise you have to take account of the width of the pencil lead and adjust the ruler position to take account of it.

Certainly small errors in line position, length and (especially) angle can have a huge effect over a large diagram that has a lot of symmetries. It's surprising how noticeable even a small error of 0.5 mm between two lines is that should be colinear.

For that kind of things, lead-holders are preferable: you at least get a consistent lead width, despite the wearing, assuming you maintain the orientation of the pencil.

An alternative is to sharpen your pencils with a utility knife & sandpaper: it's more versatile and less wasteful than a regular sharpener. For example, you could sharpen it to an edge (like a slotted screwdriver), and turn the pencil once in a while, so that the wearing helps preserving the edge.

I've also dabbled with Islamic patterns recently. Side-note: I've found some high-quality, cheap second-hand compass sets, that used to be used for wood work a few decades ago (not made-in-China).

It seems to me that the placing of one of the points is “exact.” Unless you are working on graph paper or something (but that sort of goes against the spirit of classic geometry I think), there’s no absolute frame in which the first point can be incorrect, the rest of the construction is relative to it, right?

Also, is the inaccuracy around B actually a circle? I think you could easily end up missing that horizontal line he’s added between the imagined “right A” and “right B.” But, that line shouldn’t exist yet, it connects A and B wherever they are. The position of B is wherever your pencil lands.

The second circle seems to have more room for error. You can miss B with the pointy end, and could flex the compass differently from when you did the first circle, resulting in a different BA distance. Although, it should be possible to visually check this.

Is that right? I somehow never managed to take a geometry class.

I think the circles are for when you create the arc using the compass.

Presumably you already constructed points A and B and joined them into a segment, so there's now a point B to miss.

If you are just trying to show the relationship between the lines, I don’t think there’s any reason to set A and B beforehand.

Of course, it could be the case that there’s a multi-step proof where you do care about the positions of A and B in relation to some other things. But I think that would require doing the analysis with that in mind. Maybe the order in which you construct the more complicated proof can also benefit from picking certain points to be exact.

I think the construction is usually phrased: "given a line, to..", so it assumes a pre-existing line segment, rather then that being part of the construction itself.

A line is defined by a pair of points, isn’t it? It is literally the line AB, wherever A and B are, they define the line.

But you start the construction with the line already drawn. Then you have to place the arms of the compass on the points, so you have error on both ends.

I must be missing something; I don’t see how you can rigorously define the line without A and B, so I don’t see how it can exist before A and B are marked.

I think it is possible that part of what’s missing in your “given a line to..” is that the description of the line in the .. refers to A and B, or possibility the phrasing is just informal.

A, B and the line connecting them exist before you start the construction, and the author assumes there's no error there.

Then to start the construction, you have to put one arm of your compass on the point A, and one on the point B. These points exist and are already marked, but you incur error when you place your compass down on each one. These are the represented by the blue and yellow circles in the diagram.

> I don’t see how you can rigorously define the line without A and B

That line segment isn’t defined, it’s postulated (“there somehow is a line segment with endpoints A and B; you needn’t care how it came about”)

It's not a bug, it's a feature. All students know that any line goes through two big enough points.

I was expecting something related to "exact predicates", like for example:

Efficient Exact Geometric Predicates for Delaunay Triangulations

Which is not "classical geometry", I guess, although it is constructible with ruler and compass (you need to draw circumcircles of triangles and maybe change triangles after that)

> In this sense, the Apollonius perpendicular bisector construction appears to be sensitive to the errors of compass placement.

It looks like errors are reasonably under control. It looks like the input error is a little over 5% of the distance AB and so it's not surprising that after a few steps, the output error in the position of C is around 15-20%. If you started with input error under 1%, the output error would be under 4%. There's no large error blowup here.

Hamkins has some great material, including his book "Lectures on the Philosophy of Mathematics". What's even better is that it is accompanied by a video lecture: https://www.youtube.com/playlist?list=PLg5tKDNI_a86OO6J9HuIn...

I am watching them while doing the dishes :-)

The writer is considering the subject as a physicist might. Euclid’s figures are ideal objects, they don’t have errors. It’s assumed that your construction is an approximation. Euclid might have drawn his figures on sand with lots of “errors”, still the figures are idealized no matter how you draw them. For instance, Euclid represents numbers as lines. There are not units and no errors. His proofs are still valid if I don’t draw his lines exactly as he drew them (I’m talking about Book IX where there are propositins about numbers).

The precision in the physical construction of the proof though does help in assessing it's validity. If the insight was completely divorced from the visual representation then you may as well be doing straight up symbolic algebra.

What an answer to the problem of trisecting the angle!

If you figure your angular accuracy is 0.3 degree or something you can construct a binary fraction like 341/1024.

Also trisecting a line isn’t far off from trisecting an angle, so constructing a pair of angles like 5/16 and 3/8 (=6/16) and then trisecting the chord between them would work. Wolfram MathWorld says that "[a]n approximate trisection is described by Steinhaus" (https://mathworld.wolfram.com/AngleTrisection.html) and this is basically what I have in mind.

You can do a mid-point construction using just a straight edge and some means of drawing parallels. Good example of projective geometry. ;)

Welcome to laying out sheet metal:

https://www.youtube.com/watch?v=9DypIDbAVTc

Interesting topic. I fondly remember, when I was doing geometry at school, I propagated a lot of errors.

Does anyone know where some of these follow-up posts might be?

> Unfortunately, I seem to have lost track of the references I had in mind. If anyone else can post suitable references, I’d be grateful.

(2022)

I thought this is going to be about floating point inaccuracies. For example, here's a Blender Geometry Nodes setup that instead of constructing a circle using a provided node "Circle", or by positioning and connecting points, which coordinates are calculated independently using an incremented angle, sine and cosine, instead draws a circle by walking in a circle: on each step ("extrusion"), the direction of the step is rotated by 360°/total_steps (for 360 steps it's rotated by 1°). The more steps there are, the bigger the accumulated floating point error (the error is guaranteed when positioning a new vertex, which position is stored as a xyz triplet of float32 values, but I imagine in Blender code floats are used as well).

video: https://streamable.com/0xba64

.gif: https://i.imgur.com/4Y33nrf.gif

.blend file: https://e.pcloud.link/publink/show?code=XZvFUnZKrqbyl9h8ab3U...

Going back to the article, I like the illustrations depicting all possible values as areas; it seems you could calculate with a formula the boundaries of those areas and so draw them efficiently and allow to dynamically modify precision, ciechanow.ski style...

2018

Added. Thanks!

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