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Unread 05-04-2017, 07:45 PM
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Math Quiz 11


There are nine tiny dots, labeled 1 thru 9, on a gym floor.

Jane measures the distances between pairs as follows:

Code:


2	4	47.8017
3	5	69.2026
5	9	148.492
7	8	86.764
5	6	63.0714
2	8	189.528
6	7	65.192
4	9	147.085
2	3	90.5207
6	8	86.0349
3	9	171.956
4	7	88.8144
What is the distance from point 8 to point 9?


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Unread 05-04-2017, 08:14 PM
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Re: Math Quiz 11

I plugged all the points and dimensions into Solidworks, nothing is defined so my answer is:

Unsolvable!

(99% sure I did something wrong)
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Unread 05-04-2017, 08:28 PM
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Re: Math Quiz 11

8.06 units?

Also, no information is given about point number 1. Is that a typo?
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Unread 05-04-2017, 08:38 PM
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Re: Math Quiz 11

157.674
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Unread 05-04-2017, 08:50 PM
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Re: Math Quiz 11

294.325?
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Unread 05-04-2017, 09:01 PM
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Re: Math Quiz 11

I'm getting a whole range of values. I'm using SolidWorks in a similar fashion to Brian. Here's the link to my solution if anyone wants to do a sanity check for me.
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Unread 05-04-2017, 09:08 PM
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Re: Math Quiz 11

Quote:
Originally Posted by DopeCADSwag View Post
I plugged all the points and dimensions into Solidworks, nothing is defined so my answer is:

Unsolvable!

(99% sure I did something wrong)
I think you have two different points for #2.
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Unread 05-04-2017, 09:16 PM
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Re: Math Quiz 11

Quote:
Originally Posted by DopeCADSwag View Post
I plugged all the points and dimensions into Solidworks, nothing is defined so my answer is:

Unsolvable!

(99% sure I did something wrong)
Quote:
Originally Posted by Jon Stratis View Post
I think you have two different points for #2.
Also, your 47.80 dimension is the horizontal distance, not the linear distance between points.

EDIT - Ignore me.
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Last edited by Cothron Theiss : 05-05-2017 at 09:46 AM.
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Unread 05-05-2017, 07:17 AM
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Re: Math Quiz 11

Quote:
Originally Posted by Cothron Theiss View Post
Also, your 47.80 dimension is the horizontal distance, not the linear distance between points.
That's the distance to the top right point, not the one the number is closest to.
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Unread 05-05-2017, 09:46 AM
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Re: Math Quiz 11

Quote:
Originally Posted by Hitchhiker 42 View Post
That's the distance to the top right point, not the one the number is closest to.
Woops! Thanks for catching that. You're right.
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Unread 05-05-2017, 09:48 AM
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Re: Math Quiz 11

174.7256.

Link to my CAD (I found this cool, CAD package online in order to do it... seems to work pretty well, at least for this stuff, and you can have a public free account!)
https://cad.onshape.com/documents/43...9620e77d130f51

Anyone see any mistakes? I even labeled my points

Edit: Out of curiosity Ether, is there a purely mathematical way to arrive at the answer, one that can be done by hand on a single sheet of paper?
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Unread 05-05-2017, 10:26 AM
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Re: Math Quiz 11

Looking at the topology, we have two triangles: 3-5-9 and 8-6-7.

Points 6 and 5 are connected to each other, points 3 and 8 are connected to point 2, 7 and 9 are connected to point 4, and points 2 and 4 are connected. That is:

Code:
          9 ---- 4 ---- 7
          | \    |    / |
          |  3 - 2 - 8  |
          | /         \ |
          5 ----------- 6
Angles within of the two triangles are fixed. But, even given a solution to the mirror ambiguity, there are two unknowns (e.g. angle 9-3-2 and angle 2-8-7) and only one constraint (distance 5-6). Unless this is at some limiting value, there does not appear to be enough information to constrain the distance between 8 and 9.
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Unread 05-05-2017, 11:04 AM
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Re: Math Quiz 11

Quote:
Originally Posted by GeeTwo View Post
Looking at the topology, we have two triangles: 3-5-9 and 8-6-7.

Points 6 and 5 are connected to each other, points 3 and 8 are connected to point 2, 7 and 9 are connected to point 4, and points 2 and 4 are connected. That is:

Code:
          9 ---- 4 ---- 7
          | \    |    / |
          |  3 - 2 - 8  |
          | /         \ |
          5 ----------- 6
Angles within of the two triangles are fixed. But, even given a solution to the mirror ambiguity, there are two unknowns (e.g. angle 9-3-2 and angle 2-8-7) and only one constraint (distance 5-6). Unless this is at some limiting value, there does not appear to be enough information to constrain the distance between 8 and 9.
Gus, you continue to amaze me. You are a true polymath.

I figured you were lurking in the background, waiting to pounce on this.

Here's another way to analyze this:

Jane made 12 distance measurements.

Pick a Cartesian coordinate system with point2 at the origin and point3 on the +X axis.

So the coordinates of point2 are [0,0].

Since Jane measured the distance from 2->3, and point3 lies on the +X axis, the coordinates of point3 are [146,0].

That leaves 6 points (4 thru 9) whose coordinates are unknown. Since there are 2 scalar values per point, you have 12 unknowns.

But you only have 11 measurements left, because you already used the distance from 2->3.

So you have an underdetermined system of nonlinear equations.

There is no unique solution.


Quote:
Originally Posted by Jon Stratis View Post
174.7256.

Link to my CAD (I found this cool, CAD package online in order to do it... seems to work pretty well, at least for this stuff, and you can have a public free account!)
https://cad.onshape.com/documents/43...9620e77d130f51

Anyone see any mistakes? I even labeled my points
Nice work Jon. You found a valid (but not unique) answer.

Quote:
Edit: Out of curiosity Ether, is there a purely mathematical way to arrive at the answer, one that can be done by hand on a single sheet of paper?
What say you, Gus, if Jane had measured the distance from, say, point5 to point8?




Last edited by Ether : 05-05-2017 at 11:13 AM.
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Unread 05-05-2017, 11:26 AM
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Re: Math Quiz 11

I knew I should have posted my musings last night that this wasn't fully constrained! I had gotten nowhere trying to figure it out by hand, and resolved to give CAD a try this morning. When it came back with an answer, I figured why not
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Unread 05-05-2017, 12:46 PM
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Re: Math Quiz 11

Quote:
Originally Posted by Ether View Post
What say you, Gus, if Jane had measured the distance from, say, point5 to point8?
Adding that to the existing measurements should almost do it. You would only need to determine angle 9-5-8 to get distance 9-8. Measuring 5 to 8 would make triangle 5-8-6, fixing those angles and constraining the orientation of segment 8-7. Excepting edge cases, I would expect two solutions, based on alternate mirror-image solutions of the two fixed shapes (9-3-5 and 5-8-7-6).

Edit - or possibly as many as four. I forgot to consider folding the polygon 5-8-7-6 over segment 8-6.

Edit2 - to clarify, by "edge cases" I mean singularity in the system of equations.

Edit3 - Perhaps even more, due to which way the angles at points 2 and 4 bend. Let's go with this: With the length of 5-8 determined, and excluding singularity cases, there would be a (fairly small) finite number of solutions. This arises from the nonlinear nature of the system of equations. For example, the single-unknown equation x5 - 5x3 + 4x = 0 has exactly five real solutions. This set of equations appears to be second order (to simplify, square all of the measured lengths to eliminate those square roots), but the cross terms among the different unknowns increases the number of possible solutions.

Edit4 - I think I have a way to find the solutions when the length of segment 5-8 is given, for all 16 possibilities of the four remaining ambiguities* in a not-too-complex excel spreadsheet, if you don't mind looking for crossing points on graphs. This is stretching Jon's question, but not totally breaking it. I'm going to work this up with a 5-8 distance of 55.17 (to commemorate the date of 5 May 2017), though this will be adjustable if this turns out not to be an interesting value. If I have enough energy left, I will generate all the plots in excel as well**. Ether, if you have a different 5-8 distance in mind for 5-8, please let me know.

* At a geometric level, these ambiguities arise from the law of cosines: c2 = a2 + b2 + 2abcosγ. This equation is ambiguous in that while a and b are always positive, γ may be either positive or negative, and cosγ = cos(-γ).

** As it turns out, the graph of the initial set of distances Ether gave is a "double-down" on the classic Bridges of Konigsberg problem - every one of the eight vertices has three measurements! This will make drawing lines using excel scatter plot more interesting.
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Last edited by GeeTwo : 05-05-2017 at 07:31 PM.
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