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Thanks for the tip! :)
I don't know anything about Bessel functions. What are they? And btw, no
I'm not sure about anything regarding this :)
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
also tried to integrate simpler [looking] integrals that had something in
common with the actual integral I wanted to integrate. The integral of
cos( sin(x) ) dx was one of the latter.
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
novice...
-Charles
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David Fontaine <dav### [at] faricy net> wrote
> Okay, I sort of follow al this, but integral outputs a function, right?
What's
> the different x-values of the function represent?
'x's in the result of an integration usually mean you didn't
actually sum from a specific x value to another specific x, so
you get a formula instead of a definite value for the integral.
The formula then has variables in it which you can use
when you decide where among the x's the 'actual' summing
is to be done.
(Instead of a definite value which you would get if you started at a
definite place and ended at a definite place.)
>
And what would be the point of
> finding the average of all points in a planet's orbit? it'd just be the
center
> of the ellipse
oh no. the average was just one example of an 'integration'. the planet
was another.
Using the planet example, you might say: start the planet "here"
at coordinates x0, y0 going at velocity vx0, vy0. Say the sun is at 0,0.
Then, knowing the planet's position you can calculate the force the
sun has on the planet. From that you figure (using Newton's laws for
example) how the planet's velocity will change. Add that change to
the planet's velocity. That new velocity takes it to a new position.
There at the new position, it'll feel a different gravity force, and change
its velocity again, and move with that new velocity to a new position,
where it feels a different gravity force, and so on, over and over,
tirelessly adding the effects on the planet's position.
But between 'here' and 'there' the gravity is constantly changing your
velocity so you can't just say it will go with a constant velocity from
'here'
to there'. You jump ahead in small steps so the velocity doesn't change
"much" so where you calculate you will end up at the end of a small
time is pretty close to where you will actually end up.
If you make your steps infinitesimally small you can apply theorems and
rules of 'integral calculus' to get the result. And if the equations for the
forces and motion are 'nice' enough, those rules can lead to a
simple formula which might depend on where you start (x0, y0) and
initial velocity (vx0, vy0).
if there are variables in what you are summing, then the integral could
be a function, but if it is summing numbers, no its not much of a function.
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I appologise for my last post being here double... I re-typed it later
after getting an error message saying it couldn't post... I posted the
second one and now they're both here. ??? I'm still new at ng posting...
-Charles
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On 20 Jan 2000 08:44:04 -0500 ron### [at] povrayorg (Ron Parker) wrote:
>I haven't looked at the whole thing yet, but I can tell you that this
>part at least is incorrect. If the front wheels are parallel, their
>axes will never intersect. In fact when you turn, one or both wheels
>will skid just a little.
Ackerman steering is an attempt to compensate for this. I believe all
modern cars (and race cars) have this ability built into their steering
components.
http://www.auto-ware.com/setup/ack_rac.htm
--
Alan - ako### [at] povrayorg - a k o n g <at> p o v r a y <dot> o r g
http://www.povray.org - Home of the Persistence of Vision Ray Tracer
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On Tue, 20 Jan 2009 04:50:08 -0800, "cc" <coy### [at] fojarcom> wrote:
>Help? Suggestions? Is there a way to approximate these integrals w/o
>iteration?
>
>-Charles
Since I cannot be of great help with your calculus problems
(intergals? Ah yes, but that was two years ago :) ) myay I offer an
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
commonly used technique for doing non-predictable discrete
calculations such as modelling a particle system.
If I haven't understood your problem, please excuse me.
Peter Popov
pet### [at] usanet
ICQ: 15002700
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So is the integral of sin(x) the area between the x-axis and sin(x) between x=0
and x=x?
--
Homepage: http://www.faricy.net/~davidf/
___ ______________________________
| \ |_ <dav### [at] faricynet>
|_/avid |ontaine <ICQ 55354965>
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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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