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From: Pier Nardin on 22 Jun 2010 08:52
Does anyone know what the mathematical name for a 2D curve with a certain
thickness is. i.e. a ribbon.

Pier


From: Han de Bruijn on 23 Jun 2010 05:41
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
From: Han de Bruijn on 23 Jun 2010 07:52
On Jun 23, 11:41 am, Han de Bruijn <umum...(a)gmail.com> wrote:
> 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?

Thickened ideal straight lines and circles
==========================================
The equations of an ideal straight line are:

x = a + cos(P).t ; y = b + sin(P).t

Here (x,y) = plane coordinates, P = angle with horizontal (constant),
(a,b) = a point on this line, t = running parameter. All numbers real
valued by default.

Consider the expression:

[x - a - cos(P).t]^2 + [y - b - sin(P).t]^2

= (x-a)^2 + (y-b)^2 + t^2 - 2.[cos(P).(x-a) + sin(P).(y-b)].t

= t^2 - 2.[cos(P).(x-a) + sin(P).(y-b)].t
+ [cos(P).(x-a) + sin(P).(y-b)]^2
- [cos(P).(x-a) + sin(P).(y-b)]^2
+ (x-a)^2 + (y-b)^2

= [t - {cos(P).(x-a) + sin(P).(y-b)}]^2
- cos^2(P).(x-a)^2 - sin^2(P).(y-b)^2 - 2.cos(P).(x-a).sin(P).(y-b)
+ (x-a)^2 + (y-b)^2

= [t - {cos(P).(x-a) + sin(P).(y-b)}]^2
+ sin^2(P).(x-a)^2 + cos^2(P).(y-b)^2 - 2.sin(P).(x-a).cos(P).(y-b)

==> [x - a - cos(P).t]^2 + [y - b - sin(P).t]^2
= [t - {cos(P).(x-a) + sin(P).(y-b)}]^2
+ [sin(P).(x-a) - cos(P).(y-b)]^2 : LEMMA.

The (Gaussian) fuzzyfication of an ideal straight line will be defined
as follows.

L = 1/(sigma.sqrt(2.Pi)) . integral(t = -oo .. +oo) exp(-Q(t)/2) dt
where Q(t) = {[x - a - cos(P).t]^2 + [y - b - sin(P).t]^2}/sigma^2 .

With help of the lemma we find for the integral:

= exp(-{[sin(P).(x-a) - cos(P).(y-b)]/sigma}^2/2).1/(sigma.sqrt(2.Pi))
..int(-oo,+oo) exp(-{[t - {cos(P).(x-a) + sin(P).(y-b)}]/sigma}^2/2) dt

==> L(x,y) = exp(- {[sin(P).(x-a) - cos(P).(y-b)]/sigma}^2 /2)

A function with values between 0 and 1: sort of probability. Hence a
_thickness_ of this fuzzyfied line may be defined as being (sigma).

Same sort of trick may also be done for e.g. a circle, resulting in
C(x,y) , as a fuzzyfication with linewidth (sigma):

Q(x,y) = [(sqrt{(x-a)^2 + (y-b)^2}-R)/sigma]^2 ; C(x,y) = exp(-Q/2)

But a sharply delineated thickness instead of a fuzzy one is feasible
as well. For example a circle within a "block" function C, instead of
a Gaussian (and still other shapes C may be tried as well):

Q(x,y) = [(sqrt{(x-a)^2 + (y-b)^2}-R)/sigma]^2
C(q) = 0 for q > 1 ; C(q) = 1 for q <= 1

Mathematics, what else ..

Han de Bruijn
From: Pier Nardin on 23 Jun 2010 07:54
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.googlegroups.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


From: Brian Chandler on 23 Jun 2010 08:00
Pier Nardin wrote:
> 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.

Well, you said "i.e. a ribbon" -- do you mean "that is, a ribbon"? In
which case you have your answer; you write: "[definition]... In this
paper we will refer to this as a 'ribbon'."

There are no rules about what you can call things in mathematics, as
long as you define your terms, clarify when someone asks, and use the
term consistently.

Brian Chandler
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