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From: Clifford J. Nelson on 23 Jun 2010 09:43
> Han
>
> I intend defining the thickened curve by saying that
> it is within a certain
> distance from a polygonal line which is my centre
> curve. I just was not sure
> what to call it.
>
> Pier
>
> "Han de Bruijn" <umumenu(a)gmail.com> wrote in message
> news:870d0b86-9391-416c-aa01-f7efb34128fa(a)y4g2000yqy.g
> ooglegroups.com...
> On Jun 22, 2:52 pm, "Pier Nardin" <pi...(a)ramm.co.za>
> wrote:
> > Does anyone know what the mathematical name for a
> 2D curve with a certain
> > thickness is. i.e. a ribbon.
> >
> > Pier
>
> Doubt it. Ideal mathematical curves have NO
> thickness. And IF, how to
> _define_ thickness then, eventually ? What you _can_
> do is, given the
> definition of an _ideal_ curve say [ x = f(t) , y =
> g(t) ] , define a
> thickened curve (known technique); given a thickened
> curve, determine
> its thickness (by contouring & divide area by half
> length of contour)
> There's also a technique named thinning. And more.
> What do you want?
>
> Han de Bruijn
>
>

People think of a curve as a set of points.

And from Synergetics:
527.711 People think of a point as the most primitive thing with which to initiate geometrical conceptioning. A point is a microevent of minutiae too meager, they say, to be dignified with dimensionality: Ergo, they assume a point to be only an "imaginary fix." But speaking in the experiential language of science, whatever is optically point-to-able is a substance, and every substance has insideness and outsideness -- ergo, is systemic: Ergo, all point-to-ables can never be less than the minimum system: the tetrahedron. Points always amplify optically to be identifiable as systemic polyhedra.

Likewise, a point in two dimensions is a polygon and can not be less than a triangle.

You could graph a curve with thickness using Synergetics coordinates described at:
Partial Mathematica Notebook saved as HTML at
http://mysite.verizon.net/cjnelson9/index.htm

SynergeticsAppTen.nb (540.1 KB) - Mathematica Notebook at
http://library.wolfram.com/infocenter/MathSource/600/

Cliff Nelson
From: Han de Bruijn on 24 Jun 2010 02:54
On Jun 23, 7:43 pm, "Clifford J. Nelson" <cjnels...(a)verizon.net>
wrote:

[ .. snip things done .. ]

> People think of a curve as a set of points.

When seen as an IMAGE, a curve is a continuous comb of delta functions
When turning these delta functions into kinda bell shaped curves, with
spread D, the curve becomes visible, and it has a thickness defined as
D. See the above posting of mine.

[ .. snip Synergetics .. ]

Han de Bruijn
From: OwlHoot on 24 Jun 2010 08:44
On 22 June, 13:52, "Pier Nardin" <pi...(a)ramm.co.za> wrote:
>
> Does anyone know what the mathematical name for a 2D curve with a certain
> thickness is. i.e. a ribbon.

I reckon you could reasonably call it a smooth foliation of line
segments.

In other words, think of it as a sequence of line segments,
like a picket fence, whose midpoints lie along the curve.

But, as someone else pointed out, if writing a paper you should
clearly define what you mean by a phrase like that, as it isn't
particularly standard and could be interpreted in various ways.


Cheers

John Ramsden
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