Re: Defining a circle
- From: hetware <massless@xxxxxxxxxxxx>
- Date: Sat, 03 Jun 2006 23:20:11 -0400
The Sorcerer wrote:
"hetware" <massless@xxxxxxxxxxxx> wrote in message
news:0-mdnWpqV8vRiRzZRVn-pg@xxxxxxxxxxxxxxxx
| This is one of those things which to me is intuitively obvious, but I
don't
| know how to prove it. If I define a unit circle as the set of all
| points such that x^2+y^2=1, how do I arrive at the definition of
| arclength?
2pi radians is the arc length of a complete circle.
pi radians is the arc length of a half circle.
pi/2 radians is the arc length of a quarter circle.
That is a set of examples, not a definition.
| That gives me some concept of approximate arclength. It is intuitively
| obvious that the more iterations I make in the midpoint selection
| algorithm, the closer I come in my approximation to the real unit
| circle. Taking the limit as the number of divisions goes to infinity
| will lead to the real unit circle. But that doesn't feel like a complete
| proof to me.
Pi is an irrational number like the square root of 2. That means that if
you
suppose there exists two integers p and q such that p/q = sqrt(2) you'll
arrive at a contradiction.
http://www.cut-the-knot.org/proofs/sq_root.shtml
You are attempting to use integers (division by the whole number 2)
to arive at pi, and it cannot be done.
Well, I never claimed it could be done in a finite number of iterations. If
I were to construct a second algorithm to produce an enclosing regular
polygon whose sides share their midpoint with a point in the set defined
for the circle[*], I can show that the circumference of the confining
polygon is always larger than the circumference of the circle (2 pi by
definition), and I can show that the circumference of the circumscribed
polygon is always smaller than that of the circle. Furthermore, I can show
that these values tend toward convergence.
I hesitate to say that I can show that they converge. Even if I can show
that they converge, as you rightly indicated, I cannot define a continuous
angle as an integer multiple of the angle between radial segments drawn
between the center and the point which the circumscribed polygon shares
with the circle.
Interestingly, if you take the calculator value of sqrt(2),
1.4142135623730950488016887242097
and subtract 1.0 to give 0.4142135623730950488016887242097
then take the inverse of that: 2.4142135623730950488016887242097
you can then arrive at
sqrt(2) = 1 + 1/(2 + .4...) as the first approximation,
= 1 + 1/
(2+ 1/
(2+1/
(2+1/
2+....)))
which is an extended fraction and goes on to infinity.
Will you claim that it converges to the actual value sqrt(2)?
| I started thinking
| about this because I was trying to derive the rotation matrix from first
| principles.
Ahhh... you need only cos^2 + sin^2 = 1 for that, which is Pythagoras
again.
That's the problem. It seems to require an assumption beyond the
Pythagorean theorem. With the Pythagorean theorem, I do not need a concept
of arclength.
Euclid gives us these:
http://mathworld.wolfram.com/EuclidsPostulates.html
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight
line.
3. Given any straight line segment, a circle can be drawn having the segment
as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum
of the inner angles on one side is less than two right angles, then the two
lines inevitably must intersect each other on that side if extended far
enough. This postulate is equivalent to what is known as the parallel
postulate.
This may help.
http://www.androcles01.pwp.blueyonder.co.uk/AC/AC.gif
QEQ
[*]I am avoiding the term 'tangent' but that's what I mean) to the
--
http://www.vho.org/GB/c/DC/gcgvcole.html
http://www.vho.org/GB/Books/dth/
http://www.germarrudolf.com/
http://www.ice.gov/pi/news/newsreleases/articles/051115chicago.htm
.
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