Re: Triangle with more than 180 degrees-

In article <1202ae20-dec1-40fb-bf8a-789d9d107012@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> Hero <Hero.van.Jindelt@xxxxxx> writes:
*** wrote:
Hero wrote:
*** wrote:
> That i tried to explain in my last letter.
> Just practical, given two points their distance is taken between the
> tips of a circle. Now let a line connect these two points. If the
> middle between these two points does not lie on this line, the line
> is not straight.
.
It is less straight-forward then you think. To get the middle between
two points, you need a straight line, so you can not define straight
lines using the middle between two points. Or tell me how to get the
middle of a point without using the concept of straight lines?
.
This is a nice problem.
Just for the definition of straight this would do already:

No, it will not.

Given two points, draw a circle around one of them, which is cutting
through the other point. Without changing the circle, draw around the
other point intersecting the first circle, from this intersection
again without changing the circle draw a circle intersecting the first
circle, and one more time. Now this point and the given two points are
three points and one of them is in the middle between two of them.
That is doubling the distance, You say. Okay, but proceed a few steps
further on and one can find the middle between the first two points as
well, just using a circle.

Perhaps it can be done, but I still do not see the definition of a
straight line. If I want to distill a definition for that, I get:
a line between two points is straight if its midpoint is on the
line, and the two halves of the line are also straight.
A recursive definition.

But this all is a bit moot. Consider the following:
take a function f: [0,1) -> [0,oo) that is monotonously increasing
(x > y -> f(x) > f(y)), has f(0) = 0 and when x goes to 1, f(x)
grows without bound.
Now consider the inside of the unit circle in Euclidean 2D space.
Define on it a distance as follows (using polar coordinates):
given x = (r_x, phi_x) and y = (r_y, phi_i),
d(x, y) = sqrt(f(r_x)^2 + f(r_y)^2 - 2.f(r_x).f(r_y).cos(phi_x - phi_y)).
Define the straight lines as:
a.f(r).cos(phi) + b.f(r).sin(phi) = c
with any real a, b and c.

You may check that this defines an Euclidean geometry on the inside of
that circle.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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