Re: your sci.math.research article titled "Reflection along the curved line"

[Igor Moiseev]

The question arose in the study of the optimal synthesis of some dynamical
system with controls (if you need some refs just ask). Being an engineer I'm
not able and have no time for doing the precise study of the problem.
What is needed
*
How does the reflection along the curved line work?*

1. Description of the problem
- classification of types of reflection curves: concave/convex/having
inflections (like one is done in optics but in more general case)
- phenomena of self-intersecting and singularities
- multidimensional case and so on
2. Precise study of the class of curves as
smooth/differentiable/continuous ... at least necessary requirements
3. At the end we need a formula which can be evaluated on the computer,
something like you find here in the case of linear algebra
http://en.wikipedia.org/wiki/Reflection_%28linear_algebra%29.
4. Then we'll need to invent and to write the fast algorithm to evaluate
the reflected curve!

This is short expansion of the question.

This week I was trying to do some analysis. The thing I saw was that one can
consider something similar to moving frame along the reflection curve, then
making the change of coordinates the reflection line will be just
straightened, so one may consider now the classical reflection over a line.
(Sorry for quite schematic description)

I cannot ask you to solve this problem!!
What is important for me to get some books-references!

Thank you again for the help!
Igor.

[Igor Moiseev]

Hi Kevin!

Draw a line from the point being reflected to the curve; draw
the tangent at the curve; do the obvious thing.

Yes it's obvious, but it is pointwise. We need something more integral at the end (like convolution or integral formula).
Somthing like trasformation in general!! Yes, maybe i ask for too much, but ...

Thank you!
I.

[Ilya Zakharevich]

On 2009-11-07, Igor Moiseev <moisee...@gmail.com> wrote:
> How does the reflection along the curved line work?*

1) If the distance between eyes is negligible (monocular vision):

a) find a point on the curve where incoming/outgoing angles are equal.

b) make inversion w.r.t. the kissing circle;

2) Otherwise you need to find two points in "a", for left and for
right eye, and take the intersection of corresponding "vision rays".

Hope this helps,
Ilya

[Robert Israel]

This is a repost of Igor Moiseev's article, because the contents of the
original were not visible in some newsreaders.

Igor Moiseev <moisee...@gmail.com> wrote:

--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

[Igor]

> 1) If the distance between eyes is negligible (monocular vision):
>

> ? ?a) find a point on the curve where incoming/outgoing angles are equal.
>
> ? ?b) make inversion w.r.t. the kissing circle;

>
> 2) Otherwise you need to find two points in "a", for left and for

> ? ?right eye, and take the intersection of corresponding "vision rays".
>
Hi Iliya! Thanks for reply, but it seems that you understand the
reflection in the optical sense. I meant the other thing, which occurs
only in math and confusingly named with the same term "reflection"!!

So I meant with reflection this staff:
http://en.wikipedia.org/wiki/Reflection_%28mathematics%29

Igor.

[Peter Spellucci]

In article <hd921e$j4r$1...@dizzy.math.ohio-state.edu>,

compute the point on the curve say xc, which is nearest to your point,
say xp.

then xr=xp+2*(xc-xp)

the computation of xc will be a nonlinear minimization problem in one
variable, the curve parameter s. Provided you have a formula for that
curve this should be manageable by a mimimizer. but you must be aware that
this may be a nonconvex problem with local minimizers. if you have the
curve represented by a list of points, simple do a list search in the
list of distances. this might also give a good starting point for case one.

hth
peter

[Igor]

Thank you Peter!
After you reply I went to search in the other direction "differential
geometry" and got to the idea of evolutes and the related theories
(thanks to the help of Prof. Ernest B. Vinberg).

Here you can see the brief lists of formulas and properties
http://en.wikipedia.org/wiki/Evolute

Thanks again. Igor.