Re: metric in which balls are triangles

In article <d31n8r$qd9$1@xxxxxxxxxxxxxxxxxxxxxx>,
Robert Israel <israel@xxxxxxxxxxx> wrote:
>In article <1112821803.685824.241000@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
>lukasz <bbla32@xxxxx> wrote:
>>Can anyone provide a hint on how to define a metric d:R*R->R in which
>>all balls are triangles (any or at least isosceles)? I tried with a
>>function that returned the circumradius for the isosceles triangle with
>>one (bottom) edge parallel to the OX axis, where one coordinate was a
>>point on a triangle and the other was the circumcenter; although these
>>two points unequivocally defined a triangle, such a function obviously
>>did not meet the symmetry condition for a metric.

>>I suppose the function should rather be similar to the metric in which
>>all balls are squares:

>>d( (x,y), (a,b) ) = max{ |x - a|, |y - b| }

>>but I can't think of anything like this for triangles. Any ideas?

>It's not at all obvious that such a metric can exist. Do you
>have any reason to think that it does? Certainly there won't
>be a simple formula: it can't be a norm.

A semi-norm can exist, but it fails to be symmetric; in
fact, any compact convex set with a designated "origin" as
an internal point can produce such a semi-norm. This will
be highly directional, and the semi-norm of x-y and y-x
will not ge equal for a triangle.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.

Relevant Pages

  • Re: Integer tetrahedron
    ... You could make a coordinate system from the whole to the parts, ... and put a smaller triangle of balls on top of the big triangle of balls ... The four planes that defined the ...
    (sci.math)
  • Re: Weekly Video Clips from Greg Kennedy
    ... The problem was that I was connecting ... the balls in the wrong way. ... string triangle. ... My new video Clip, Week #5 - Triad is up at: ...
    (rec.juggling)
  • Re: Simple way of making Golf Ball with Solidworks!
    ... balls. ... sketch to project onto a surface. ... the vertexes or the face centerpoints lie on the sphere. ... split a 2D triangle up into four identical 2D triangles, ...
    (comp.cad.solidworks)
  • Re: Simple way of making Golf Ball with Solidworks!
    ... balls. ... sketch to project onto a surface. ... split a 2D triangle up into four identical 2D triangles, ... the dimples requires a little adjustment. ...
    (comp.cad.solidworks)
  • Re: metric in which balls are triangles
    ... >all balls are triangles? ... >function that returned the circumradius for the isosceles triangle with ... >point on a triangle and the other was the circumcenter; ...
    (sci.math)