> I don't know what trapezoidal steering mechanisms do...   Do they do what I
> was talking about... about rotating the wheel on the inside of the turn at a
> steeper angle than the wheel on the outside?

Yep!

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Wow!   I'm still new to the idea of getting so many responses so quickly.
Thanks everybody :)

I think there was one thing that I should clarify regarding my original
post.  The system I have now IS able to handle continuous steering-wheel
movement via small incremental changes in steering-angle.    Those
incremental changes add up to a lot of number crunching during parse time.
For movements that leave the steering wheel in one position, we're just
dealing with arks and it's not too difficult.  One or two of my earlier
tests animations involved a steering-angle which went instantly from 45
degrees left to 45 degrees right to instantly straight...... physically
impossible yes, but easy to work with.  :-)

It was after that that I added the sweeping steering-angle movements and the
accelleration and the scripting and all that...    Here's where it's a
choice of working with arcs, and crunching away at a lot of them... or
(sigh) calculus.

-Charles

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> input: a bunch of things to be added up, with a certain weight for each.
> An example is the average. You add up all of them but weight
> them by 1/(number of things).  average of a, b, and c is
> a/3 +b/3+c/3.
> Usually the term 'integration' is used when the things to
> be added for a continuum of values. Thats why integration is
> considered 'calculus'.
> Say you want to know how a planet is going to revolve around its sun.
> The force of gravity on the planet is not usually constant because the
> planet
> can go from one distance from its sun to other distances. To find out where
> it will be tomorrow, you could 'move' it (on paper) in little increments
> calculating the forces on it anew for each second, using its new position
> and new forces. You are actually adding up all these effects, second
> after second. But since it is more accurate to consider time as a
> continuum, you really should add up a continuum of miniscule
> effects. That is, you 'integrate' the effects
>
> output: the sum
>
> oh yeah, 'the area under the curve' explanation comes from finding the
> area under a curve by adding up a bunch of rectangular regions (for
> which you know that area=length X width) areas

Okay, I sort of follow al this, but integral outputs a function, right? What's
the different x-values of the function represent? 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

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Thanks for the tip :)

What exactly is a Bessel function?    And no... I'm not sure about anything
regarding this :)

When I was searching for the integrals (and I'm not that experienced) I was
trying to find ways to simplify the integral I started with into something I
knew what to do with... and I tried to integrate simpler [looking] integrals
which had something in common with the original.   integral of: cos( sin(x)
dx  was one of the latter.

The other thing I wanted to ask you is where I can find out information
about Gradstein/Ryshik and Abramowitz/Stegun.  I'm very much a novice. :)

-Charles

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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] faricynet> 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?

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