cc wrote:

> I don't know anything about Bessel functions.  What are they?  And btw, no
> I'm not sure about anything regarding this :)

They are a sort of special functions. One obtains them either as
solutions of the wave equation in polar coordinates or as integrals
(hard ones, which cannot expressed by using only those functions
which are taught at school).

> When I was searching for the integrals, (and I'm not very exprienced) I
> tried simplifying the integral into something I knew how to deal with.   I

This can't work -- if it were possible, there were no need to
invent special functions.

> The other thing I wanted to ask you is where can I get information about
> Gradstein/Ryshik and Abramowitz/Stegun and Runge Kutta?  I'm very much a

The first is probably the most famous table of integrals and related
stuff, the other a collection of mainly definitions with an emphasis
on correctness (the less-used special functions are sometimes defined
by different specialists with different factors in front of them,
A/S tries to straighten this out). Both are probably useless unless
one is a mathematician, engineer or scientist.

I still believe that differentiation is better suited to your
problem than integration for the following reasons:

1. Even if you manage the integrals, you would have to compute
   the Bessel functions (or whatever) numerically. This either
   requires hacking the whole Netlib into POV (which would be a
   nice idea, btw. -- it would give a cool gnuplot replacement :-)
   ) or doing them slowly using #macro.
2. If you do it by integration, you input a strategy (steering
   and speed) and get some curve of the car. Since you probably
   want the curve to be where the street is, you have to use
   trial and error. The other way around, you just describe the
   street by a mathematical expression, and get the steering
   by some derivatives.

Ralf

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On Fri, 21 Jan 2000 15:53:04 -0600, David Fontaine <dav### [at] faricynet>
wrote:

>So is the integral of sin(x) the area between the x-axis and sin(x) between x=0
>and x=x?

The integral of sin(x) (with respect to x :) ) is -cos(x). This is the
general form. In order to evaluate it for a particular range of x, or
in your case to find the area enclosed by the sine function and some
segment of the +x axis, you need to calculate the integral for that
range. Say the range is (a,b) then the integral evaluates to
-cos(b)-(-cos(a)) or cos(a)-cos(b). Of course this may give 0 as a
result and even though it represents an area this should not surprise
you because the part of the curve below the x axis is weighted
negatively.


Peter Popov
pet### [at] usanet
ICQ: 15002700

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An easy way to look at the problem might be to use a simple spiral with the
radius as a direct function of the turned angle. Dead simple to calculate.

Ralf Muschall <rmu### [at] t-onlinede> wrote in message
news:388### [at] t-onlinede...
> cc wrote:
>
> > I don't know anything about Bessel functions.  What are they?  And btw,
no
> > I'm not sure about anything regarding this :)
>
> They are a sort of special functions. One obtains them either as
> solutions of the wave equation in polar coordinates or as integrals
> (hard ones, which cannot expressed by using only those functions
> which are taught at school).
>
> > When I was searching for the integrals, (and I'm not very exprienced) I
> > tried simplifying the integral into something I knew how to deal with.
I
>
> This can't work -- if it were possible, there were no need to
> invent special functions.
>
> > The other thing I wanted to ask you is where can I get information about
> > Gradstein/Ryshik and Abramowitz/Stegun and Runge Kutta?  I'm very much a
>
> The first is probably the most famous table of integrals and related
> stuff, the other a collection of mainly definitions with an emphasis
> on correctness (the less-used special functions are sometimes defined
> by different specialists with different factors in front of them,
> A/S tries to straighten this out). Both are probably useless unless
> one is a mathematician, engineer or scientist.
>
> I still believe that differentiation is better suited to your
> problem than integration for the following reasons:
>
> 1. Even if you manage the integrals, you would have to compute
>    the Bessel functions (or whatever) numerically. This either
>    requires hacking the whole Netlib into POV (which would be a
>    nice idea, btw. -- it would give a cool gnuplot replacement :-)
>    ) or doing them slowly using #macro.
> 2. If you do it by integration, you input a strategy (steering
>    and speed) and get some curve of the car. Since you probably
>    want the curve to be where the street is, you have to use
>    trial and error. The other way around, you just describe the
>    street by a mathematical expression, and get the steering
>    by some derivatives.
>
> Ralf

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Peter Popov wrote in message ...
>alternative solution? As I understand the problem, you don't want to
>have to calculate all previous steps when doing an animation. Why
>don't you use the file I/O directives to write the current state
>(position, linear velocity etc.) of the car and then, in the next
>frame, read it and continue your calculations from there? It is a


Actually I already do have a system which works pretty well for the final
animation, although it doesn't work on a frame-by-frame basis.  I run a pov
file which reads the motion script and calculates through the motions of the
car from start to finish.   For every motion segment in the motion script,
the starting position-data of the car is recorded into a file.   For final
rendering, only one motion segment (or partial segment) needs to be
calculated per frame.   This is more calculation per frame than it'd be to
save data per frame, true.

Ultimately both systems have trouble when doing trial and error test frame
renders (and Ralf Muschall correctly pointed out that this is something I
currently have to do).  I usually add a little bit to the end of the motion
script I'm editing and then render the final frame.  One of the next things
I plan on doing is making a modification of my current system that doesn't
need to precalculate the entire sequence of motions, and which doesn't need
them in any particular order...  I.e. the new system would record relative
positon data for any given initial-speed, initial-steering-angle,
speed-delta and steering-angle-delta, but Wouldn't record initial car
orientation or position for each segment.   I'm thinking this should work
well for testing... the other would work better for final rendering... er um
parsing speed.

-Charles

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This is true...   Yes I have made animations where the road follows the car
instead of the usual car-follows-the-road situation. :)   And yes I have
been having to do trial and error if I want the car to follow the road.
Trial and error is what I'm trying to make easier... faster.   I like your
idea of differentiating from a pre-defined path...  Really turns the problem
around. :)   I'd still like to figure out how to control the car directly in
a clean, sybolic way if possible (as oppposed to numeric approximations,
Riemann (sp?) sums etc.)  (I'm ignoring floating point roundoff)   When you
say I "would have to compute the Bessel functions (or whatever)
numerically," do you mean something like for example taking a sum of some
formula as n goes from 1 to something aribitrarily large?  Anyway it's
starting to sound doubtful that it can reasonably be done symbolically... or
maybe I should say, not without inventing some new symbols... something to
go along with the trig functions etc?...

-Charles

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cc wrote:

> say I "would have to compute the Bessel functions (or whatever)
> numerically," do you mean something like for example taking a sum of some
> formula as n goes from 1 to something aribitrarily large?  Anyway it's

Yes, someting like that. The zeroth Bessel function is like cos,
just the factorials in the denominator are squared, and the
powers are taken from x^2/4 instead of x. But usually one uses
a more complicated, but faster and preciser formula (which is
different for different values of the argument).

A pointer to a numerics site (which is considered doubtful
by professionals, but good enough for simple stuff) can be found
near the bottom of http://math.jpl.nasa.gov/nr/nr.html .

Ralf

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Hi,

i've found a cool homepage with many links about (physics) simulation.

http://members.nbci.com/Kourdakov/Links/mathematics_links.htm

There are also some links to homepages about car physics.

Bye
 Paul

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I dont know about anyone else, but you're writing from the distant
future, 2009 to be precise... How come?

--
Tim Nikias
Homepage: http://www.digitaltwilight.de/no_lights/index.html
Email: Tim### [at] gmxde

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"cc" <coy### [at] fojarcom> wrote in news:388702d1@news.povray.org

[...]

I guess that this post will stay on top of list until 2009 :) ?

-- 
#macro g(U,V)(.4*abs(sin(9*sqrt(pow(x-U,2)+pow(y-V,2))))*pow(1-min(1,(sqrt(
pow(x-U,2)+pow(y-V,2))*.3)),2)+.9)#end#macro p(c)#if(c>1)#local l=mod(c,100
);g(2*div(l,10)-8,2*mod(l,10)-8)*p(div(c,100))#else 1#end#end light_source{
y 2}sphere{z*20 9pigment{function{p(26252423)*p(36455644)*p(66656463)}}}//M

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In article <Xns### [at] 204213191226>, "Rafal 'Raf256' Maj"
<raf### [at] raf256com> wrote:

> I guess that this post will stay on top of list until 2009 :) ?

No, just in your misconfigured or broken newsreader.  Turn off receiving of
_all_ messages, the unsubscribe to the group and subscribe again such that
your newsreader does not get a list of _all_ messages.

    Thorsten

____________________________________________________
Thorsten Froehlich, Duisburg, Germany
e-mail: tho### [at] trfde

Visit POV-Ray on the web: http://mac.povray.org

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