Formal definition of ruler and compass constructions

Question

I would see a formal definition of ruler and compass constructions. I have searched in internet but I haven't found a very formal definiton.

Update:

I founded these lectures about my question:

1 - https://www.isibang.ac.in/~jay/MC/Raghavan%201.pdf

2 - https://conf.math.illinois.edu/~rotman/ruler.pdf

Answer 1

You are given a starting set $C_0$ of points, which are declared to be constructed.

Figures (lines and circles) may be drawn. A point that is not in the starting set is a constructed point must be an intersection point of two drawn figures. A line may only be drawn through 2 constructed points. A drawn circle may only be centered at a constructed point, and a drawn circle's radius must be the distance between 2 constructed points.

A constructed point must be obtained as an intersection point of 2 drawn figures in a finite number of steps:

The set $F_0$ is the empty set. Step $n+1$ ($n\ge 0$) is to draw the set $F_{n+1}$ of all figures that can be drawn using only the set $C_n$ of points and the rules above.

The set $C_{n+1}$ contains (only) the points of intersection of pairs of members of $F_{n+1}$ and the members of $C_0$.

Unless stated otherwise, it is assumed that $C_0$ contains just 2 points.

Answer 2

The answer above fail to define the concept of "eyeballing", which is used normally in solutions to these kind of question. So let me write something different.

You are given at the start a finite set of constructed points, constructed lines and constructed circles.

At any steps, you can construct a circle, construct a line, or construct a point (possibly with eyeballing).

• To construct a circle, choose 2 constructed point: one specified to be the center, and one specified to be on the circumference. Then a new circle become constructed with the specified points.

• To construct a line, choose 2 constructed points. Then a new line become constructed going through the specified point.

• To construct a point, we are allowed to "eyeball" it, as follow. First pick a finite numbers of allowed regions (defined later). Then find a non-empty set obtainable by Boolean operations on these allowed regions, this is the final region to choose from. Then make a new constructed point to be one point in that region. This step is non-deterministic.

The goal of the construction is to make constructed points, lines, circles satisfying some properties.

For the construction to be considered to work, it must be guaranteed to work regardless of the outcome of the non-deterministic step. That means it always terminate in a specific fixed finite number of steps regardless of which points is constructed in the point construction step; and it must always produce the required objects regardless of which points were constructed.

What are the allowed regions?

• Either side of any of the constructed lines.

• On the constructed line.

• Inside or outside of any of the constructed circles.

• On the constructed circle.

• Not on a construct point.

• On a constructed point.

• ("eyeballing" region) an area inside a circle with positive radius (the center and the radius need not be constructed)