Construction of lines from points

[Hurkyl]

they have little relation to the problem of constructing space that this thread began talking about.

Sure. But remember that this line of argument spawned because ramsey was effectively declaring the original question as improper -- because it was considering a line as being made out of points. Everything since then was trying to explain how overly narrow-minded such a view is.

 

[wofsy]

Sure. But remember that this line of argument spawned because ramsey was effectively declaring the original question as improper -- because it was considering a line as being made out of points. Everything since then was trying to explain how overly narrow-minded such a view is.

I understand. I was just disappointed that the original problem wasn't pursued.

Perhaps you could talk about how in terms of models one could describe ruler and compass constructions. Can we start form two points, a straight edge and compass, and axioms that govern constructions? All new points would arise as intersections of newly constructed lines and circles. For instance if the original two points (line) are used to construct two new points from two intersecting circles these two new points determine a new line whose intersection with the original we would call the midpoint. And so forth.

 

[ramsey2879]

I understand. I was just disappointed that the original problem wasn't pursued.

Perhaps you could talk about how in terms of models one could describe ruler and compass constructions. Can we start form two points, a straight edge and compass, and axioms that govern constructions? All new points would arise as intersections of newly constructed lines and circles. For instance if the original two points (line) are used to construct two new points from two intersecting circles these two new points determine a new line whose intersection with the original we would call the midpoint. And so forth.

How about calling the intersection of the arc of a compass with the line a point? What axiom of construction would rule that out? For instance is the radius of the compass be required to be in units that are constructed initially from the distance between the two points?

 

[wofsy]

I would think that a compass could only be placed initially at two points that were already constructed. Otherwise the points chosen would be arbitrary and not really constructed.

New points would occur as intersections of already existing lines with the compass arc and of two compass arcs with each other.

Another issue would be how to say how lines intersect. If we go purely by Hurky's example where lines are just pairs of points, I am not sure how we get a midpoint. I was going to ask him this.

 

[ramsey2879]

I would think that a compass could only be placed initially at two points that were already constructed. Otherwise the points chosen would be arbitrary and not really constructed.

New points would occur as intersections of already existing lines with the compass arc and of two compass arcs with each other.

Another issue would be how to say how lines intersect. If we go purely by Hurky's example where lines are just pairs of points, I am not sure how we get a midpoint. I was going to ask him this.

That would be the solved by use of the 1st axiom of construction which is that a line can be drawn between two points, same as to draw a perpendicular bisecter, but you already knew that and was just testing me I think. So constructing a line in effect is to draw non arbitary points on a line by connecting points using a compass and a straight edge all on a single plane. The radius of the compass would always be a distance between the original two points, between constructed points (points of intersection) or a combination thereof. I think you gave a very good rundown of this problem already.

 

[wofsy]

That would be the solved by use of the 1st axiom of construction which is that a line can be drawn between two points, same as to draw a perpendicular bisecter, but you already knew that and was just testing me I think. So constructing a line in effect is to draw non arbitary points on a line by connecting points using a compass and a straight edge all on a single plane. The radius of the compass would always be a distance between the original two points, between constructed points (points of intersection) or a combination thereof. I think you gave a very good rundown of this problem already.

I was not testing you. If I use a compass to create the perpendicular bisector, how do I get the midpoint of the original line if a line is just two points as in Hurky's model? Also I don't think there is an idea of angle yet and the idea that the new line is perpendicular to the first would seem to be replaced by some symmetry idea. Not sure how to so this.

 

[ramsey2879]

I was not testing you. If I use a compass to create the perpendicular bisector, how do I get the midpoint of the original line if a line is just two points as in Hurky's model? Also I don't think there is an idea of angle yet and the idea that the new line is perpendicular to the first would seem to be replaced by some symmetry idea. Not sure how to so this.

Symmetry could be the answer, but how do you "extend" two points?

 

[wofsy]

Symmetry could be the answer, but how do you "extend" two points?

Exactly.

Maybe we need the axiom that two lines intersect in at most one point. It would imply the existence of a new point lying on both lines - a newly constructed point of space. Very cool.

There still is the problem of knowing that the new line determined by the circle arcs is not parallel to the original. Do you see a way out of it?

Maybe the circle arcs must intersect once in each half plane - by symmetry - and so the two points being in opposite half planes must determine a line that intersects the first.

 

[ramsey2879]

Exactly.

Maybe we need the axiom that two lines intersect in at most one point. It would imply the existence of a new point lying on both lines - a newly constructed point of space. Very cool.

There still is the problem of knowing that the new line determined by the circle arcs is not parallel to the original. Do you see a way out of it?

Maybe the circle arcs must intersect once in each half plane - by symmetry - and so the two points being in opposite half planes must determine a line that intersects the first.

Until we know how Hurkyl's model is set forth in detail we would be wiser not to guess. The points could be movable or whatever. The non parallel factor would probably be expressed by mathematical manipulating of the coordinates of the pairs of points. But we are going off course again. I myself care not to brother with Hurkyl's model any more.

 

[Hurkyl]

What exactly do you mean by "get"? The midpoint between two points exists simply by Euclidean geometry -- any choice of model has nothing to do with it.

Some models might give nice non-geometric ways to describe that midpoint. For example, in any of the "usual" models of Euclidean geometry where points are pairs of real numbers, the midpoint of (a,b) and (c,d) is ((a+c)/2, (b+d)/2).


You need to see a model of the type I described explicitly? Okay.
. A point is an ordered pair (x,y) of real numbers
. The points (x,y) and (u,v) are equal iff the ordered pairs (x,y) and (u,v) are equal
. A line is an unordered pair of distinct points
. A point (x,y) lies on the line {(a,b), (c,d)} iff (d-b)x+(a-c)y=ad-bc
. The lines {(a,b),(c,d)} and {(e,f),(g,h)} are equal iff (e,f) and (g,h) lie on {(a,b),(c,d)}
. The point (c,d) lies between (a,b) and (e,f) iff (c-a)²+(d-b)²+(e-c)²+(f-d)²=(e-a)²+(f-b)²
. The line segment between (a,b) and (c,d) and the line segment between (e,f) and (g,h) are congruent iff (c-a)²+(d-b)²=(g-e)²+(h-f)²

This gives semantics to the five elementary undefined terms of (one presentation of) Euclidean geometry: "point", "line", "lies on", "between", and "congruent".

For concreteness, I'll choose a specific meaning for "line segment" too:
. A line segment is an unordered pair of distinct points
. Two line segments are equal iff the unordered pairs are equal
. The "line segment between P and Q" means the unordered pair {P,Q}
. A point Q lies on the line segment {P,R} iff Q lies between P and R, Q is P, or Q is R.

 

[wofsy]

What exactly do you mean by "get"? The midpoint between two points exists simply by Euclidean geometry -- any choice of model has nothing to do with it.

Some models might give nice non-geometric ways to describe that midpoint. For example, in any of the "usual" models of Euclidean geometry where points are pairs of real numbers, the midpoint of (a,b) and (c,d) is ((a+c)/2, (b+d)/2).


You need to see a model of the type I described explicitly? Okay.
. A point is an ordered pair (x,y) of real numbers
. The points (x,y) and (u,v) are equal iff the ordered pairs (x,y) and (u,v) are equal
. A line is an unordered pair of distinct points
. A point (x,y) lies on the line {(a,b), (c,d)} iff (d-b)x+(a-c)y=ad-bc
. The lines {(a,b),(c,d)} and {(e,f),(g,h)} are equal iff (e,f) and (g,h) lie on {(a,b),(c,d)}
. The point (c,d) lies between (a,b) and (e,f) iff (c-a)²+(d-b)²+(e-c)²+(f-d)²=(e-a)²+(f-b)²
. The line segment between (a,b) and (c,d) and the line segment between (e,f) and (g,h) are congruent iff (c-a)²+(d-b)²=(g-e)²+(h-f)²

This gives semantics to the five elementary undefined terms of (one presentation of) Euclidean geometry: "point", "line", "lies on", "between", and "congruent".

For concreteness, I'll choose a specific meaning for "line segment" too:
. A line segment is an unordered pair of distinct points
. Two line segments are equal iff the unordered pairs are equal
. The "line segment between P and Q" means the unordered pair {P,Q}
. A point Q lies on the line segment {P,R} iff Q lies between P and R, Q is P, or Q is R.

Hurky we are trying to derive a new model that allows for the construction of space from simple axioms.

What I was wondering is whether one can preserve the definition of a line as two points. My gut tells me that you can.


While the mid point could be located once a metric is introduced, I was trying to avoid a metric in the spirit of keeping lines as pairs of points. Without a metric the midpoint would have to obey some symmetry - and it seems that symmetry could go a long way in determining the properties of new points.

 

[ramsey2879]

What exactly do you mean by "get"? The midpoint between two points exists simply by Euclidean geometry -- any choice of model has nothing to do with it.

Some models might give nice non-geometric ways to describe that midpoint. For example, in any of the "usual" models of Euclidean geometry where points are pairs of real numbers, the midpoint of (a,b) and (c,d) is ((a+c)/2, (b+d)/2).


You need to see a model of the type I described explicitly? Okay.
. A point is an ordered pair (x,y) of real numbers
. The points (x,y) and (u,v) are equal iff the ordered pairs (x,y) and (u,v) are equal
. A line is an unordered pair of distinct points
. A point (x,y) lies on the line {(a,b), (c,d)} iff (d-b)x+(a-c)y=ad-bc
. The lines {(a,b),(c,d)} and {(e,f),(g,h)} are equal iff (e,f) and (g,h) lie on {(a,b),(c,d)}
. The point (c,d) lies between (a,b) and (e,f) iff (c-a)²+(d-b)²+(e-c)²+(f-d)²=(e-a)²+(f-b)²
. The line segment between (a,b) and (c,d) and the line segment between (e,f) and (g,h) are congruent iff (c-a)²+(d-b)²=(g-e)²+(h-f)²

This gives semantics to the five elementary undefined terms of (one presentation of) Euclidean geometry: "point", "line", "lies on", "between", and "congruent".

For concreteness, I'll choose a specific meaning for "line segment" too:
. A line segment is an unordered pair of distinct points
. Two line segments are equal iff the unordered pairs are equal
. The "line segment between P and Q" means the unordered pair {P,Q}
. A point Q lies on the line segment {P,R} iff Q lies between P and R, Q is P, or Q is R.

Two corrections would give my model

A straight line is defined by an unordered pair of distinct points {P,Q}
The line segment between P and Q means the the portion of the straight line extending from P to Q of the unordered pair of distinct points {P,Q}

Of course both models are otherwise the same, but some one realized that the semantics of the extra language was open to varying interpertations and that one could avoid the ambiguty by omitting the extra language. Now we see that Wofsy is confused by the "more concise language". As to the meaning of "get" Wofsy meant to construct the mid point using a straight edge and compass but did not realized that the line and the line segment between P and Q are in fact given under your model of a line as a pair of unordered points {P,Q} by the statement as to which points lie thereon. As you say the exact semantics are not material once each model is understood.