Mathematics IV: circles
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  Mathematics IV: circles
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Question: How many possible circles?
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Author Topic: Mathematics IV: circles  (Read 1956 times)

excelsus
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« on: March 28, 2014, 02:34:48 PM »
« edited: March 30, 2014, 04:33:46 PM by excelsus »

In the picture below, the two circles and the straight line are all tangent to each other.
How many circles exist that "touch" (not intersect!!!) each of the three objects?

Caution: The straight line doesn't have any ends.

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True Federalist (진정한 연방 주의자)
Ernest
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« Reply #1 on: March 28, 2014, 06:00:13 PM »

Assuming by "touch" you mean "tangent", then one, assuming you have Euclidean geometry in play.  You'd be able to get a second circle to do that in those non-Euclidean geometries where circles and straight lines are simply two different ways of expressing the same object.
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tik 🪀✨
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« Reply #2 on: March 28, 2014, 06:12:44 PM »

Infinite, because the circles are wheels to a penny-farthing that can run over an infinite series of circles as it moves forwards.

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excelsus
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« Reply #3 on: March 28, 2014, 07:02:05 PM »

Assuming by "touch" you mean "tangent", then one, assuming you have Euclidean geometry in play.  You'd be able to get a second circle to do that in those non-Euclidean geometries where circles and straight lines are simply two different ways of expressing the same object.

Don't make it too complicated. Wink
I'm only referring to the simple Euclidean geometry that every student learns at school.
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angus
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« Reply #4 on: March 28, 2014, 07:03:31 PM »

Assuming by "touch" you mean "tangent", then one


Clearly there's one in the triangle.  I considered that there might be another circumscribing the whole, but I'm having trouble visualizing how that'd happen given how far the line segment sticks out on the sides.  I'm also assuming a Cartesian plane in Euclid space, etc.  If one draws a circle just the right size, it touches the line segment on the right, and the small circle at about 2 o'clock, and the big circle at about 11 o'clock.  Something like this:



That makes two circles, if it is correct.

If you now place two filled circles inside the two unfilled circles, you get this:



If you then draw a few line segments below the original line segment, attaching them at the ends to make vertices and fill in the resulting polygon, you will get this:



Now, if you add a couple of rectangles, a sliced oval, some color, some more line segments, a hypocycloid of four cusps, and some text, you will end up with this remarkable geometric design.



Fascinating.
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excelsus
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« Reply #5 on: March 28, 2014, 07:19:18 PM »

Definitely a freedom post, angus, but ...

I'm sorry to say your answer is wrong. 



Straight lines don't have endings.
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angus
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« Reply #6 on: March 28, 2014, 07:40:05 PM »
« Edited: March 28, 2014, 07:52:23 PM by angus »

Yes, I freely admit that I voted for 1, but I am now convinced that this is incorrect.  I am fairly convinced now that there are at least 2.  Unlike all angus polls however, no excelsus poll allows the user to change the vote.  That is your philosophy and I accept that.  Still, I am increasingly convinced, though I did not draw it well, that at least two circles can be drawn which intersect the three loci at exactly one point.  One in the little triangle (the one everyone immediately notices, presumably), and one about the line segment and the two circles, touching the line segment at its right terminus and the two circles roughly where I indicated in the drawing.

Edit:  I see by the time I post this that you have changed the description of your figure.  If the red text is the case, then my original vote was correct.  More importantly, we can now conclude that two important differences exist between angus polls and excelsus polls.  I can accept not allowing users to change vote, but you changing your question in such a poll after we have voted is highly irregular.  

If you want to set your reputation on this forum as a one-trick pony, or as a niche-poster, that's fine with me.  You can be our next waltermitty, although perhaps a slightly better-informed version, and without the fetish for really old women.  But if you are going to ask serious questions using misleading notation, then that's another issue entirely.  You clearly drew a line segment.  You then, after folks voted--in a poll in which you did not allow users to change their votes!--made it clear that you actually intended to make it a line instead of a line segment.  By this we are left to conclude that you are either (a) too ignorant to use the accepted and customary symbols for denoting lines, rays, and line segments, or (b) you are a troll and a trickster.

I'm inclined to believe that both apply.

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Atlas Has Shrugged
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« Reply #7 on: March 28, 2014, 07:49:58 PM »

I got my SAT scores back. I didn't do well in math....
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excelsus
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« Reply #8 on: March 28, 2014, 08:16:33 PM »

More importantly, we can now conclude that two important differences exist between angus polls and excelsus polls.

Exactly. But I don't use this option out of spite, but rather for reasons of impartiality.

then that's another issue entirely.  You clearly drew a line segment.  You then, after folks voted--in a poll in which you did not allow users to change their votes!--made it clear that you actually intended to make it a line instead of a line segment.  By this we are left to conclude that you are either (a) too ignorant to use the accepted and customary symbols for denoting lines, rays, and line segments, or (b) you are a troll and a trickster.

What terms do you use for lines with (a) no endings, (b) two endings and (c) one beginning and no end?
For the latter I guess "ray". I'd call it half-line.
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True Federalist (진정한 연방 주의자)
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« Reply #9 on: March 28, 2014, 09:54:17 PM »

More importantly, we can now conclude that two important differences exist between angus polls and excelsus polls.

Exactly. But I don't use this option out of spite, but rather for reasons of impartiality.

then that's another issue entirely.  You clearly drew a line segment.  You then, after folks voted--in a poll in which you did not allow users to change their votes!--made it clear that you actually intended to make it a line instead of a line segment.  By this we are left to conclude that you are either (a) too ignorant to use the accepted and customary symbols for denoting lines, rays, and line segments, or (b) you are a troll and a trickster.

What terms do you use for lines with (a) no endings, (b) two endings and (c) one beginning and no end?
For the latter I guess "ray". I'd call it half-line.

No ending is "line"
One ending is "ray"
Two endings is "line segment"
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dead0man
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« Reply #10 on: March 29, 2014, 05:48:35 AM »

Since when does nerd=only doing tricky math problems ?
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SUSAN CRUSHBONE
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« Reply #11 on: March 29, 2014, 07:04:32 AM »

2

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muon2
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« Reply #12 on: March 29, 2014, 08:30:27 AM »

The problem is a specific case of Descartes' theorem that says that four mutually tangent circles have curvatures (ki = 1/ri where r is the radius) given by the following quadratic equation:

(k1 + k2 + k3 + k4)2 = 2(k12 + k22 + k32 + k42)

Since this is a quadratic equation, if three circles are specified then there are at most two solutions for the fourth circle. In this problem one circle is replaced by a straight line which can be considered to be a circle of curvature 0. The two solutions are the small radius inside the "triangle" and the large radius drawn by butafly.

However, the problem said touch without intersecting, but not touch at a single point. If intersecting is taken to mean that one curve has points on either side of another curve (the usual definition), then an object does not intersect itself. In that case the two given circles are also solutions which brings the total to 4.

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True Federalist (진정한 연방 주의자)
Ernest
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« Reply #13 on: March 29, 2014, 12:40:31 PM »

Or we could have been missing the point and its really a grammar problem rather than a math problem.  He asked for circles touching each of the objects, not circles touching all of the object.

For each of the objects there are an infinite number of such circles that touch each individual circle or line without intersecting any of the three.
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excelsus
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« Reply #14 on: March 29, 2014, 01:09:43 PM »

Butafly got it right. Period.
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WalterMitty
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« Reply #15 on: April 18, 2014, 10:03:18 PM »

ha.  just noticed this.  ive always wondered how angus figured out my old woman fetish.  i dont think i was ever that overt about it? 

but the bigger question is why is angus so snarky to the math problems guy?
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Cathcon
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« Reply #16 on: April 19, 2014, 02:01:30 PM »

ha.  just noticed this.  ive always wondered how angus figured out my old woman fetish.  i dont think i was ever that overt about it? 

but the bigger question is why is angus so snarky to the math problems guy?

Because math sucks?
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WalterMitty
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« Reply #17 on: April 19, 2014, 02:40:02 PM »

ha.  just noticed this.  ive always wondered how angus figured out my old woman fetish.  i dont think i was ever that overt about it? 

but the bigger question is why is angus so snarky to the math problems guy?

Because math sucks?

angus was a math major, so i guess he doesnt think it sucks.

so, im confused why he is mean to the math guy.  some people get snarky after a couple of drinks.  i was always more jovial after i got hammered.  Smiley
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Cathcon
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« Reply #18 on: April 19, 2014, 02:42:02 PM »

ha.  just noticed this.  ive always wondered how angus figured out my old woman fetish.  i dont think i was ever that overt about it? 

but the bigger question is why is angus so snarky to the math problems guy?

Because math sucks?

angus was a math major, so i guess he doesnt think it sucks.

so, im confused why he is mean to the math guy.  some people get snarky after a couple of drinks.  i was always more jovial after i got hammered.  Smiley

I'm usually much happier unless I come on here, in which case I start acting like even more of an asshole. This place does bad things to people.
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