Geometry Question
December 14, 2003 4:26 PM Subscribe
Can anyone settle a 15 year old argument for me? It turned into one of those things "we don't talk about" but it's always bugged me. The question is simple and involves three points and a circle - more inside.
Right... This argument was a simple one but it kept me and my father amused for hours. Okay, it kept me amused for hours and had him swearing. The trigger is simple - I stated that any three points could always be joined togther by a circle that passed though them all. It's a simple theory, and one to which he found many arguments. I think I killed them all appart from one. He loved telling me that three points in a straight line couldn't be linked, but I claimed that if he could have them infinitely straight then I could have a circle of invitite diameter. Ah the wonders of infinity.
Anyway, after all this time I wanted a proper answer, so I though I'd ask you guys. Am I right or was he? Can anyone give me a mathematical proof of it?
Right... This argument was a simple one but it kept me and my father amused for hours. Okay, it kept me amused for hours and had him swearing. The trigger is simple - I stated that any three points could always be joined togther by a circle that passed though them all. It's a simple theory, and one to which he found many arguments. I think I killed them all appart from one. He loved telling me that three points in a straight line couldn't be linked, but I claimed that if he could have them infinitely straight then I could have a circle of invitite diameter. Ah the wonders of infinity.
Anyway, after all this time I wanted a proper answer, so I though I'd ask you guys. Am I right or was he? Can anyone give me a mathematical proof of it?
All circles are curved, all around, even if they are infinitely huge. And these points, whether they are in an 'infintely straight' line, or just a normally straight one, won't all touch the circle, because the circle will curve away from one of them.
posted by armoured-ant at 4:40 PM on December 14, 2003
posted by armoured-ant at 4:40 PM on December 14, 2003
The whole line-is-an-infinite-circle argument is, to use the mathematical term, a bunch of hokum: a circle is defined as the set of all points equidistant from a given point in a 2-dimensional space (R^2, or two axes of real numbers); in an infinite 'circle', that center point (being infinite) isn't part of the R^2, so it can't be used to construct a circle.
posted by boaz at 4:44 PM on December 14, 2003
posted by boaz at 4:44 PM on December 14, 2003
Infinity is a funny thing. It is a limit not a number. You can't always trust your intuition.
An infinite circle has no curve. If it did, you could always make ist curve "less" by expanding the radius a bit more. But of course you cant because the radius is already infinite!
twine42's problem is a special case of the Appollonius problem (use google) and the idea that infinite circles are straight lines is standard practice in something called 'Inversion Geometry'
posted by vacapinta at 4:47 PM on December 14, 2003
An infinite circle has no curve. If it did, you could always make ist curve "less" by expanding the radius a bit more. But of course you cant because the radius is already infinite!
twine42's problem is a special case of the Appollonius problem (use google) and the idea that infinite circles are straight lines is standard practice in something called 'Inversion Geometry'
posted by vacapinta at 4:47 PM on December 14, 2003
depends, is your dad an engineer or a scientist, if the first then the answer is it only has to be relatively close to infinite (take walking down the street as an example, the earth is curved of course), if it's the latter then the answer is a definitive no.
one of my personal favorite bad teaching jokes:
an engineer and a physicist are at opposite ends of a football field and a nice crisp $100 is at the 50 yard line, a bell rings and they walk to half the distance, it rings again and they continue splitting the difference, eventually they're both standing a foot away from the bill. at this point the physicist says "if we keep up this pattern we'll never reach the bill", the engineer leans over and picks up the bill and says "eh, close enough".
posted by NGnerd at 5:27 PM on December 14, 2003
one of my personal favorite bad teaching jokes:
an engineer and a physicist are at opposite ends of a football field and a nice crisp $100 is at the 50 yard line, a bell rings and they walk to half the distance, it rings again and they continue splitting the difference, eventually they're both standing a foot away from the bill. at this point the physicist says "if we keep up this pattern we'll never reach the bill", the engineer leans over and picks up the bill and says "eh, close enough".
posted by NGnerd at 5:27 PM on December 14, 2003
It's quite consistent to think of a line as a circle with an infinite radius of curvature, and mathematicians often do consider them as such.
Example: There are a set of functions on the complex plane known as Mobius functions (also known as linear fractional transformations.) These are functions of the form (az + b)/(cz +d) where a, b, c, d are complex numbers. Ask any mathematician what these things do and they'll say that they map circles to circles, with the implicit understanding that lines are circles. (And they really *are* circles on the Riemann sphere, which is the compactification of the complex plane.)
posted by ptermit at 5:51 PM on December 14, 2003
Example: There are a set of functions on the complex plane known as Mobius functions (also known as linear fractional transformations.) These are functions of the form (az + b)/(cz +d) where a, b, c, d are complex numbers. Ask any mathematician what these things do and they'll say that they map circles to circles, with the implicit understanding that lines are circles. (And they really *are* circles on the Riemann sphere, which is the compactification of the complex plane.)
posted by ptermit at 5:51 PM on December 14, 2003
compactification
Sounds painful.
I for one am completely confused. So the answer depends on whose terms of art you're using?
posted by anathema at 6:41 PM on December 14, 2003
Sounds painful.
I for one am completely confused. So the answer depends on whose terms of art you're using?
posted by anathema at 6:41 PM on December 14, 2003
actually- *explodes*
posted by armoured-ant at 6:56 PM on December 14, 2003
posted by armoured-ant at 6:56 PM on December 14, 2003
Ptermit has it right, yeah, but there's an implicit assumption about the space you're working in. If you consider just the plane, you can't put a point at infinity, because such a thing isn't defined. The solution, as is often the case in math, is to define one. This is usually called R2*, or the extended real plane. As one of my math profs said "A line is just a circle whose center is the point and infinity and the radius is large." Needless to say, the class laughed at that point.
A good way to think about it is to use the Riemann sphere, which ptermit mentioned. This is a way to think of the complex plane (which is basically equivalent to the real plane) with the point at infinity added. Set a sphere of radius one on top of (0,0). Map every point to the sphere by putting a line tangent to the sphere through both the point and the line going through the top and bottom of the sphere. The very top of the sphere, thus, could be reached only by a line starting infinitely far away, but in any direction. This is thus defined to be the point at infinity, and all infinities are the same. It can be shown that both lines and circles in the plane are mapped to circles on this surface.
posted by Schismatic at 7:37 PM on December 14, 2003
A good way to think about it is to use the Riemann sphere, which ptermit mentioned. This is a way to think of the complex plane (which is basically equivalent to the real plane) with the point at infinity added. Set a sphere of radius one on top of (0,0). Map every point to the sphere by putting a line tangent to the sphere through both the point and the line going through the top and bottom of the sphere. The very top of the sphere, thus, could be reached only by a line starting infinitely far away, but in any direction. This is thus defined to be the point at infinity, and all infinities are the same. It can be shown that both lines and circles in the plane are mapped to circles on this surface.
posted by Schismatic at 7:37 PM on December 14, 2003
er, "point at infinity," not "and infinity"
posted by Schismatic at 9:15 PM on December 14, 2003
posted by Schismatic at 9:15 PM on December 14, 2003
A little more down to earth. Maybe you could settle for an ellipse. A circle is a special case of the ellipse and one with an eccentricity = 1 would be a straight line.
posted by golo at 10:40 PM on December 14, 2003
posted by golo at 10:40 PM on December 14, 2003
Since a circle is 2 dimensional, if he gives the points different x, y and z coordinates, then doesn't that kill the idea that you can get a circle through all three?
posted by willnot at 12:52 AM on December 15, 2003
posted by willnot at 12:52 AM on December 15, 2003
incidentally, this is also sometimes useful in numerical programming. since it's tricky to represent infinity in most languages a neat hack is to work with 1/radius rather than radius. straight lines have 1/radius = 0.
my version of the answer you're looking for: in maths infinity isn't a value, but a limit. as you increase the radius of a circle the curvature decreases. as you keep increasing the radius, the curvature keeps decreasing. you can show that if you kept increasing radius "for ever", the curvature would tend to a value of zero - the circle's circumfrence would tend to a straight line. infinity is the word to describe the idea of extrapolating to that ("unreachable") limit, when the circumference *is* a straight line. and the idea that it's "unreachable" comes only from a "physical" model in our heads that things have to be done progressively. in fact, it's perfectly well defined, since maths lets us "go meta" and directly ask the question about the limiting behaviour without spending an infinite amount of time adding ones to the radius to do it.
in other words "a circle of infinite radius" is a kind of mathematical shorthand for something involving limits that is perfectly well defined and does indeed have a circumference that is a straight line.
finally, willnot - you can always select the plane in which you inscribe the circle to intersect the three points. if, for example they were (0,0,0) (0,1,1) (0,1,0) then you'd draw the circle perpendicular to the x axis.
(i'm not a mathematician - studied physics - so there may be a more modern way of describing this; i may also be plain wrong ;o)
posted by andrew cooke at 1:17 AM on December 15, 2003
my version of the answer you're looking for: in maths infinity isn't a value, but a limit. as you increase the radius of a circle the curvature decreases. as you keep increasing the radius, the curvature keeps decreasing. you can show that if you kept increasing radius "for ever", the curvature would tend to a value of zero - the circle's circumfrence would tend to a straight line. infinity is the word to describe the idea of extrapolating to that ("unreachable") limit, when the circumference *is* a straight line. and the idea that it's "unreachable" comes only from a "physical" model in our heads that things have to be done progressively. in fact, it's perfectly well defined, since maths lets us "go meta" and directly ask the question about the limiting behaviour without spending an infinite amount of time adding ones to the radius to do it.
in other words "a circle of infinite radius" is a kind of mathematical shorthand for something involving limits that is perfectly well defined and does indeed have a circumference that is a straight line.
finally, willnot - you can always select the plane in which you inscribe the circle to intersect the three points. if, for example they were (0,0,0) (0,1,1) (0,1,0) then you'd draw the circle perpendicular to the x axis.
(i'm not a mathematician - studied physics - so there may be a more modern way of describing this; i may also be plain wrong ;o)
posted by andrew cooke at 1:17 AM on December 15, 2003
Well, for the common case of the three point not being colinear, there is always a circle that intersects those points. Why prove it when you can derive the formula that will find the circle for you (towards the bottom of the page)? (By the way, if you put the points (0,0), (1,0) and (2,0) into the three-point formula, you end up with the equation y=0.)
As for the case of colinear points, it's purely a matter of definitions, specifically, whether you want to call infinity a number. Well, it's not on the real plain, but then, neither are a lot of numbers (e.g. the complex numbers). A number is just something you can do calculations with, and (as Cantor discovered) you can do calculations with infinity. (Or, rather, the infinities. There are an infinite number or them, you know.) Whether you want to ignore some numbers in a particular context is up to you. It wouldn't make much sense to use infinity in some contexts (Euclidean geometry, engineering), but using infinity also gives rise to things like the Riemann Sphere and Projective and Affine Geometries.
As always, see Mathworld for definitions.
posted by samw at 2:09 AM on December 15, 2003
As for the case of colinear points, it's purely a matter of definitions, specifically, whether you want to call infinity a number. Well, it's not on the real plain, but then, neither are a lot of numbers (e.g. the complex numbers). A number is just something you can do calculations with, and (as Cantor discovered) you can do calculations with infinity. (Or, rather, the infinities. There are an infinite number or them, you know.) Whether you want to ignore some numbers in a particular context is up to you. It wouldn't make much sense to use infinity in some contexts (Euclidean geometry, engineering), but using infinity also gives rise to things like the Riemann Sphere and Projective and Affine Geometries.
As always, see Mathworld for definitions.
posted by samw at 2:09 AM on December 15, 2003
Yeah, it's not on the real plain plane, either.
Dammit.
posted by samw at 2:11 AM on December 15, 2003
Dammit.
posted by samw at 2:11 AM on December 15, 2003
Response by poster: Ladies and Gents, thank you. In my mind I knew that the three points (sensible points) had to form a circle, but couldn't think of a way to prove it.
For the non sensible infinite/inantesimate (?sp) error points I thought it was going to be a case of arguing definitions and ways of seeing it.
Finally, I'm an arse for asking this question because I now want to be able to tell him "I told you so" and the inconsiderate git died last year. ;~)
Thank you one and all. I have some links to read...
posted by twine42 at 4:04 AM on December 15, 2003
For the non sensible infinite/inantesimate (?sp) error points I thought it was going to be a case of arguing definitions and ways of seeing it.
Finally, I'm an arse for asking this question because I now want to be able to tell him "I told you so" and the inconsiderate git died last year. ;~)
Thank you one and all. I have some links to read...
posted by twine42 at 4:04 AM on December 15, 2003
Since a circle is 2 dimensional, if he gives the points different x, y and z coordinates, then doesn't that kill the idea that you can get a circle through all three?
Actually, three points in arbitrary real 3D space should be just as good for forming a circle, because three points define a plane, and yer thus reverting to describing a circle on the 2D plane defined by those points.
posted by cortex at 5:41 AM on December 15, 2003
Actually, three points in arbitrary real 3D space should be just as good for forming a circle, because three points define a plane, and yer thus reverting to describing a circle on the 2D plane defined by those points.
posted by cortex at 5:41 AM on December 15, 2003
I for one am completely confused. So the answer depends on whose terms of art you're using?
Yes. In mathematics, there are lots of sets of different axioms (underlying assumptions) that you can work from. So long as you're consistent about which ones you're using, you're just fine.
Euclidean geometry uses one set of axioms. But there have already been references to some alternative geometries which have different assumptions and different conclusions. If you're willing to say a line is a circle, for example, then it's no longer true that two points define a line/circle, as you've got an infinite family that go through any two points. You need that third point to define a line in this geometry, which corresponds to a point at infinity on the Riemann sphere. This also means that there are an infinite number of parallel circle/lines to a given circle/line through a given point -- violating one of Euclid's axioms (the parallel postulate).
These alternative geometries are sometimes more general than Euclidean geometry and sometimes they're just different, and they all give different insights into the relationships between objects in space, some of which are very, very cool. (At least to math geeks.)
For example, in projective geometry, there's a concept called duality which rests upon a mirror-image relationship between points and lines that exists when there's a point at infinity. Here, two points define a line, but two lines also define a point where they intersect. (Parallel lines intersect at infinity.) This relationship leads to the fact that for every geometric theorem, there's another mirror-image theorem which is also true -- just swap points for lines. This is a really powerful concept in projective geometry which isn't true in Euclidean geometry.
posted by ptermit at 7:15 AM on December 15, 2003
Yes. In mathematics, there are lots of sets of different axioms (underlying assumptions) that you can work from. So long as you're consistent about which ones you're using, you're just fine.
Euclidean geometry uses one set of axioms. But there have already been references to some alternative geometries which have different assumptions and different conclusions. If you're willing to say a line is a circle, for example, then it's no longer true that two points define a line/circle, as you've got an infinite family that go through any two points. You need that third point to define a line in this geometry, which corresponds to a point at infinity on the Riemann sphere. This also means that there are an infinite number of parallel circle/lines to a given circle/line through a given point -- violating one of Euclid's axioms (the parallel postulate).
These alternative geometries are sometimes more general than Euclidean geometry and sometimes they're just different, and they all give different insights into the relationships between objects in space, some of which are very, very cool. (At least to math geeks.)
For example, in projective geometry, there's a concept called duality which rests upon a mirror-image relationship between points and lines that exists when there's a point at infinity. Here, two points define a line, but two lines also define a point where they intersect. (Parallel lines intersect at infinity.) This relationship leads to the fact that for every geometric theorem, there's another mirror-image theorem which is also true -- just swap points for lines. This is a really powerful concept in projective geometry which isn't true in Euclidean geometry.
posted by ptermit at 7:15 AM on December 15, 2003
This thread is closed to new comments.
posted by twine42 at 4:29 PM on December 14, 2003