Straightedge and Compass Construction - Arclength to Line Length

Q

QuinnJaworski

New member
Joined
Dec 15, 2021
Messages
1
I have recently started working with straightedge and compass geometry for drafting purposes. Unfortunately, it is difficult for me to know if something is possible to be created by these means. The definition of Straightedge and Compass geometry can be found on Wikipedia here:

I would like to know if it is possible to draw a straight line that is the same length of a given arclength. If it is not possible, please direct me to the appropriate proof.

Thank you
 
I wouldn't expect it to be possible. According to the Wikipedia page you reference (more precisely, its chapter on constructible points), you can only construct points with algebraic coordinates, but [imath]\pi[/imath] is not algebraic. But most arcs you can construct will have lengths with [imath]\pi[/imath] in them.
 
Yes it is possible.
I have found it on google search. 1640565990859.png
 

Attachments

  • 1640565826772.png
    1640565826772.png
    53.8 KB · Views: 3
No, this is not possible to do.
Suppose you have a circle of radius 1. Then its circumference is 2pi. Since pi is a transcendental number, it can't be constructed. Similarly, 2pi can't be constructed.
This is basically what blamocur said.
 
blamocur said:
I've found the video and tried to understand it, some important details are not spelled out, and there is no proof whatsoever that the resulting segment has the same length as the arc. The result looks close, but this is no proof.
Click to expand...
Yes. I too agree that results are close. In 1962 when I was studying in Bachelor of Engineering Course , we were using this method in designing problems.
 
P.K.Tandon said:
Yes. I too agree that results are close. In 1962 when I was studying in Bachelor of Engineering Course , we were using this method in designing problems.
Click to expand...
It might be close enough for engineering applications, although I am yet to see a more detailed explanation of the method. But I doubt it is correct in terms of formal geometric constructions.
 
blamocur said:
It might be close enough for engineering applications, although I am yet to see a more detailed explanation of the method. But I doubt it is correct in terms of formal geometric constructions.
Click to expand...
The term "geometric construction" implies "provably exact". That can't be true of this one, because it has been proved that such an irrational length can't be constructed (exactly). (It's equivalent to "squaring the circle", and similar to trisecting an angle, which likewise can easily be approximated, but not done exactly by compass and straightedge.)

But "good approximations" are something different, and are entirely possible. But I would still want an explicit statement about the magnitude of the error, and a proof of that. I'd also want a full description of the construction, which is not at all clear in the video.
 
You must log in or register to reply here.
Top