Prev: Freemasonry in Communism
Next: tried out the prism tonight Chapt10, Fiberglass Experiment #95; ATOM TOTALITY
From: Ken Pledger on 19 May 2010 17:37
In article
<5698e3cb-2d6b-4520-a80d-317225bb5788(a)c13g2000vbr.googlegroups.com>,
Dan Christensen <Dan_Christensen(a)sympatico.ca> wrote:

> .... Triangles (degenerate or otherwise) must be have 3 distinct
> vertices. This is not usually made explicit in standard textbook
> definitions. Shouldn't it be? ....

I'm not sure about the text-books you have in mind. The usual
definition in projective geometry (which makes sense also in Euclidean
and various other geometries) is "three non-collinear points and all
their joins", or dually "three non-concurrent lines and all their
intersections".

Ken Pledger.
From: spudnik on 19 May 2010 17:55
I like all three of those;
note that there is a raw infinity
of trigona, two of whose edges are perpendicular
to the other edge, as far as spherical trig goes,
and I really like that.
From: Dan Christensen on 20 May 2010 13:14
On May 19, 5:37 pm, Ken Pledger <ken.pled...(a)mcs.vuw.ac.nz> wrote:
> In article
> <5698e3cb-2d6b-4520-a80d-317225bb5...(a)c13g2000vbr.googlegroups.com>,
> Dan Christensen <Dan_Christen...(a)sympatico.ca> wrote:
>
> > .... Triangles (degenerate or otherwise) must be have 3 distinct
> > vertices. This is not usually made explicit in standard textbook
> > definitions. Shouldn't it be? ....
>
> I'm not sure about the text-books you have in mind.

The typical introduction to Euclidean geometry.

> definition in projective geometry (which makes sense also in Euclidean
> and various other geometries) is "three non-collinear points and all
> their joins", or dually "three non-concurrent lines and all their
> intersections".
>

In introductory textbooks, a triangle is usually said to be determined
by three non-collinear points. Elsewhere, a so-called degenerate
triangle is usually said to be determined by three collinear points.
(A degenerate triangle is not a triangle??? Not very consistent, I
know.)

In my example, we have three points, A, B and C, where A and B are
distinct and A=C. By the above informal definition, ABC would be a
degenerate triangle since the three points are collinear. As we see
from Henry's reply, however, allowing sides of zero-length presents
problems -- the angles formed are undefined.

If you want to formalize the notion of a triangle, I think you would
have to say that three points determine a triangle iff they are
distinct. This would cover both the degenerate and non-degenerate
cases as loosely defined above.

That student isn't going to like this. :^(

Dan
First  |  Prev  | 
Pages: 1 2
Prev: Freemasonry in Communism
Next: tried out the prism tonight Chapt10, Fiberglass Experiment #95; ATOM TOTALITY