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Distribution of slope in a terrain sample
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DGoncz@aol.com
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 122

PostPosted: Thu Jun 08, 2006 8:07 am    Post subject: Distribution of slope in a terrain sample Reply with quote

Hello, s.p.r.

I have worked out curves for speed as a function of slope with various
headwinds and tailwinds for a "standard" bicycle. I haven't posted them
anywhere.

I have analysed route elevation data for real routes on real terrain
with mapping software and Mathcad. I have decomposed the elevation
readings into Fourier series.

I am sure that any sample of terrain between points A and B at the
diagonals of a rectangle can be decomposed into a 2D Fourier transform
of the altitude within the limits of that rectangle, and that doing so
will help efficiently further characterize the terrain as having a
probable distribution of slope. Multiplying the speed-by-slope curve by
the probability-by-slope curve will give, with integration, an average
speed for a bit of terrain between A and B.

For now, though, I need to pick two different routes through local
terrain, take the Fourier transform, find the slope distribution and
the average speed, and compare my model to a real ride.

I will report on this later.

If you are interested in this kind of work, please post or mail.

Doug Goncz
Replikon Research
Falls Church, VA 22044-0394
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DGoncz@aol.com
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 122

PostPosted: Thu Jun 15, 2006 9:22 pm    Post subject: Re: Distribution of slope in a terrain sample Reply with quote

Hello, again, s.p.r.

Here is a spell checked letter I wrote to my Statistics instructor
summarizing the smart bike math. It's a little more readable than my
earlier post, which follows the letter.

Doug

-----

Hello, again, Dr. Wahl.

Here's a quantum mechanical statistics problem.

In general, in QM, the expected variable of an observable is E(x), the
integral of the product of a variable as a function of position times
its associated probability over all space: x, y, and z from -oo
(-infinity) to oo (infinity). I think it's the square of the wave
function that is equal to the probability....

Consider a trip from point A to point B on mapped terrain. The bounding
rectangle including A and B at corners on a map is a patch of terrain
having a distribution of altitude and a distribution of slope. For any
path from A to B within that rectangle, the path also has a
distribution of slope.

Would it be right to think that the integral of the product of the
slope times the expected speed of a bicycle as a function of slope,
over a path, would equal the expected average speed of a bicycle along
that path? Yes, the speed lags the expected speed due to inertia. Can
we say that will average out?

Do you see the advantage? Simulation of bicycle motion is a
multivariable differential equation. I have simulated bicycle travel.
It takes a long time to run. This method provides a quick estimate of
route speed, making processing by a route optimizer possible. Software
could choose the fastest path from A to B within the patch, usually on
mapped streets.

This is what I have been working on with Dr. Majestic for several
years: the smart bike equipped the energy or time "eikonal" between
points A and B. There is also the possibility that a time*energy
product would be the metric of a nearly universal "bike-eikonal".

Doug


Doug Goncz (I) wrote:
Quote:
Hello, s.p.r.

I have worked out curves for speed as a function of slope with various
headwinds and tailwinds for a "standard" bicycle. I haven't posted them
anywhere.

I have analysed route elevation data for real routes on real terrain
with mapping software and Mathcad. I have decomposed the elevation
readings into Fourier series.

I am sure that any sample of terrain between points A and B at the
diagonals of a rectangle can be decomposed into a 2D Fourier transform
of the altitude within the limits of that rectangle, and that doing so
will help efficiently further characterize the terrain as having a
probable distribution of slope. Multiplying the speed-by-slope curve by
the probability-by-slope curve will give, with integration, an average
speed for a bit of terrain between A and B.

For now, though, I need to pick two different routes through local
terrain, take the Fourier transform, find the slope distribution and
the average speed, and compare my model to a real ride.

I will report on this later.

If you are interested in this kind of work, please post or mail.

Doug Goncz
Replikon Research
Falls Church, VA 22044-0394
Back to top
DGoncz@aol.com
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 122

PostPosted: Thu Jun 22, 2006 6:42 am    Post subject: Re: Distribution of slope in a terrain sample Reply with quote

Hello, all.

I had written about slope of terrain and bicycle performance.

I have a distribution function for the slope of a unit sinusoid in
hand. It is:

1 / d/dx ( asin( slope ))=
1 / sqrt( 1 - x^2 )

I just have to wrap my head around integrating the product of this
end-spikey curve with the curve I have for speed as a function of
slope. That function is a root function in Mathcad. It matters little
that the distribution curve is end-spikey.

I have some trials at solving the cubic form of this speed function. I
found in an earlier paper for Dr. Majewski, which I couldn't post here
because it was all Mathcad, some facts about the derviative of this
speed-slope curve. I now have Mathcad 13, which can read and translate
that paper to various HTML flavors.

Are there any avid bicyclists here?

Doug

Doug Goncz (I) wrote:

Quote:
In general, in QM, the expected variable of an observable is E(x), the
integral of the product of a variable as a function of position times
its associated probability over all space: x, y, and z from -oo
(-infinity) to oo (infinity). I think it's the square of the wave
function that is equal to the probability....

Consider a trip from point A to point B on mapped terrain. The bounding
rectangle including A and B at corners on a map is a patch of terrain
having a distribution of altitude and a distribution of slope. For any
path from A to B within that rectangle, the path also has a
distribution of slope.

Would it be right to think that the integral of the product of the
slope times the expected speed of a bicycle as a function of slope,
over a path, would equal the expected average speed of a bicycle along
that path? Yes, the speed lags the expected speed due to inertia. Can
we say that will average out?

Yes, provisionally. The choice of maximum slope will tend to produce a
certain central tendency. Study is needed. I have study worksheets that
generate arbitrary terrain, pick points A and B, and serialize the
points on the perimeter of the rectangle bounding A and B. I have a
distribution of slope for such a path. And as I wrote before, I have a
bicycle simulator.

Doug
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DGoncz@aol.com
science forum Guru Wannabe


Joined: 25 Oct 2005
Posts: 122

PostPosted: Mon Jun 26, 2006 10:47 pm    Post subject: Re: Distribution of slope in a terrain sample Reply with quote

Doug Goncz (I) wrote:

Quote:
I have study worksheets that
generate arbitrary terrain, pick points A and B, and serialize the
points on the perimeter of the rectangle bounding A and B. I have a
distribution of slope for such a path. And as I wrote before, I have a
bicycle simulator.

OK, some ways of testing this are becoming clear.

In Statistics we learned to test the hypothesis that two population
parameters differ in a particular way. One such test is for the
difference of the means. Another might be for any bias from zero in
paired differences.

A way we didn't study would be to compare travel time on perimeter
paths with that for the direct path *by forming a ratio* instead of
taking a difference, in pairs. If there is a statistically significant
deviation in travel times ration from the distance ratio (the square
root of two for points A and B at the diagonals of a square), that
would be a finding.

Paired left-turning and right-turnings paths can also be compared with
a ratio test, the deviation from a ratio of one (1) being of interest.

What's really interesting is the curvy path. It doesn't take much
effort to *steer* a bicycle, but much is gained by an informed choice
of direction.

I'll read up on the eikonal problem.

Doug
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