Does 'rest' have a precise mathematical definition?

[mathman]

TL;DR Summary

AI question included 'resting'. AI narrowly defined it.

Math qyuestion for AI (Skype) include an expression that x is resting on y (both straight line segments). AI insisted that x coincides with y, while my intent was only placing x on y. Does 'rest' have such a narrow definition?

 

[jbriggs444]

Math qyuestion for AI (Skype) include an expression that x is resting on y (both straight line segments). AI insisted that x coincides with y, while my intent was only placing x on y. Does 'rest' have such a narrow definition?

If $x$ does not coincide with $y$, what is the separation between them? In the standard reals, the only infinitesimal is zero. Two lines with zero separation coincide.

The term "rests on" is more commonly presented as "is tangent to" or "osculates with" and may be further qualified by specifying the point of tangency. i.e. "The parabola given by $y=x^2$ is tangent to the line given by $y=0$ at the point with coordinates $(0,0)$"

Two lines that are tangent at all points along their lengths do coincide. However, we would almost always say that they "coincide" and almost never say that they are "tangent" in this situation.

Similarly, one would never say that one line "rests on" another, even if "rests on" is taken as a synonym for "is tangent to".

 

[Mark44]

Similarly, one would never say that one line "rests on" another, even if "rests on" is taken as a synonym for "is tangent to".

That's my take as well, and I agree with the rest of what @jbriggs444 has said.

If we're talking about line segments in the plane, one line segment "resting on" another would need to coincide at all points they share in common (which would be all points on the shorter segment.

In three-dimensional space, two skew line segments could share just a single point, and then I suppose you could say that one segment rests on another.

 

[jbriggs444]

It occurs to me that the notion of "resting on" would make the most sense in an environment where a line segment or point is free to move within some bounded region. This could occur for a linear optimization problem and is a key way of thinking about the Simplex method.

The idea is that one is trying to optimize a formula which is a linear combination of some finite set of parameters while adhering to a set of constraints. Each constraint is a linear inequality involving some or all of those parameters. The geometric insight is that the constraints amount to oriented hyperplanes cutting through an n-dimensional space. They are the boundaries to a (hopefully non-empty) region. Importantly, the region will be convex. So a hill climbing approach is sure to succeed.

One starts by finding an n-tuple that is a feasible solution (it is the coordinates for a point within the non-empty region). One then walks the solution to a boundary of the region. Once the solution is "resting on" a boundary, one can walk it along the edges until an optimal corner, edge or face is found. The process bears a striking resemblance to Gaussian elimination.

For instance, one can try to optimize a mix of grains for pig feed where each feed has a unit cost and one is trying to satisfy inequalities involving various amino acid requirements that must be met while minimizing feed cost. e.g. https://www.jstor.org/stable/25556356. [It's been over 40 years since I wrote code for such a thing -- I went to school in Iowa, a state where there are about seven times more pigs than people]

 

[mathman]

I used 'placed' instead of 'resting' and it works. My intent was y is a line segment with ends a and b while x is a line sub segment of y with ends c and d and $c\ge a,d\le b$

 

[DaveC426913]

Yeah, I had pointed out elseweb that lines are infinite in length, so two coincident lines have all their infinite points in common. But the OP is asking about two line segments, so they still coincide but they differ in length. I found that's harder to describe rigorously yet concisely.

 

[Mark44]

But the OP is asking about two line segments, so they still coincide but they differ in length. I found that's harder to describe rigorously yet concisely.

I discussed this in post #3.

If we're talking about line segments in the plane, one line segment "resting on" another would need to coincide at all points they share in common (which would be all points on the shorter segment.

A correction to the above is that all points shared by both segments don't necessarily have to be just the shorter segment. For example, if L1 is the segment [1, 5] on the x-axis, and L2 is the segment [3, 6], also on the x-axis, then the points common to both segments are in the segment [3, 5].

 

[DaveC426913]

I discussed this in post #3.

Yes, that's why I circled back to it. It's the crux of the topic.