Rhumb Lines and Great Circles

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Rhumb Lines and Great Circles
« on: July 19, 2007, 10:15:54 PM »
Since sailors have been using map and compass, they have known that (except in very special cases) you cannot hold a constant bearing (travel on a rhumb line, or loxodrome) while traveling on a geodesic (path of least distance). On a flat Earth, rhumb lines and geodesics would be the same. How do the FE'ers explain this?

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CommonCents

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Re: Rhumb Lines and Great Circles
« Reply #1 on: July 20, 2007, 06:35:29 AM »
I like rum, does that count?
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Ferdinand Magellen

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Re: Rhumb Lines and Great Circles
« Reply #2 on: July 20, 2007, 06:41:01 AM »
http://en.wikipedia.org/wiki/Rhumb_line

yay wikipedia clarification!
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Re: Rhumb Lines and Great Circles
« Reply #3 on: July 20, 2007, 10:34:11 PM »
http://en.wikipedia.org/wiki/Rhumb_line

yay wikipedia clarification!

Sorry, I totally forgot about this. I even found this link to put here...

Re: Rhumb Lines and Great Circles
« Reply #4 on: July 22, 2007, 01:45:09 AM »
Would some FE'ers please reply to this?

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divito the truthist

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Re: Rhumb Lines and Great Circles
« Reply #5 on: July 22, 2007, 02:31:22 AM »
You want an explanation on why they are the same in both FE and RE?
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Re: Rhumb Lines and Great Circles
« Reply #6 on: July 22, 2007, 06:04:36 AM »
You want an explanation on why they are the same in both FE and RE?

Rhumb lines and geodesics are different on spherical Earth but the same on flat Earth. Evidence shows that they are different. This suggests that the Earth is round. That is the point I am making.

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divito the truthist

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Re: Rhumb Lines and Great Circles
« Reply #7 on: July 22, 2007, 09:19:19 AM »
Rhumb lines and geodesics are different on spherical Earth but the same on flat Earth. Evidence shows that they are different. This suggests that the Earth is round. That is the point I am making.

There are geodesics on a flat Earth?
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Re: Rhumb Lines and Great Circles
« Reply #8 on: July 22, 2007, 09:20:43 AM »
Rhumb lines and geodesics are different on spherical Earth but the same on flat Earth. Evidence shows that they are different. This suggests that the Earth is round. That is the point I am making.

There are geodesics on a flat Earth?

Geodesics are paths of (locally) shortest length.

http://en.wikipedia.org/wiki/Geodesic

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divito the truthist

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Re: Rhumb Lines and Great Circles
« Reply #9 on: July 22, 2007, 09:49:51 AM »
Geodesics are paths of (locally) shortest length.

http://en.wikipedia.org/wiki/Geodesic

Then they would be curves as stated in the article you posted. Therefore they are not the same, and you still wouldn't be able to follow a Rhumb line on a flat Earth with "geodesics".
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Re: Rhumb Lines and Great Circles
« Reply #10 on: July 22, 2007, 09:55:00 AM »
Geodesics are paths of (locally) shortest length.

http://en.wikipedia.org/wiki/Geodesic

Then they would be curves as stated in the article you posted. Therefore they are not the same, and still both not possible on a flat Earth just like a spherical one.

Geodesics on a plane are straight lines. (Straight lines are curves too.) Rhumb lines on a plane are straight lines, and so are always geodesics.

Geodesics on a sphere are great circles. Rhumb lines are sometimes great circles (the equator and lines of longitude are the only ones), but are more generally spirals. Hence they are not always geodesics.

In real life, sailors and pilots have noticed that rhumb lines are not always geodesics. This is evidence that the Earth is not flat.

Do you understand the initial post? Please let me know if you do not and I can send it to you as a message, so that the thread doesn't get clogged.

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divito the truthist

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Re: Rhumb Lines and Great Circles
« Reply #11 on: July 22, 2007, 10:05:55 AM »
Geodesics on a plane are straight lines. (Straight lines are curves too.) Rhumb lines on a plane are straight lines, and so are always geodesics.

Yes, curves are straight lines in math.

In real life, sailors and pilots have noticed that rhumb lines are not always geodesics. This is evidence that the Earth is not flat.

Why are you thinking it'd be any different on a flat Earth?
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Re: Rhumb Lines and Great Circles
« Reply #12 on: July 22, 2007, 10:27:11 AM »
Geodesics on a plane are straight lines. (Straight lines are curves too.) Rhumb lines on a plane are straight lines, and so are always geodesics.

Yes, curves are straight lines in math.

I am saying three things here:
F1: Geodesics on a plane are equivalent to (the same as) straight lines.
F2: Rhumb lines on a plane are straight lines.
F3: Hence, Rhumb lines, on a plane are always geodesics. (This is a logical consequence of F1 and F2.)

In real life, sailors and pilots have noticed that rhumb lines are not always geodesics. This is evidence that the Earth is not flat.

Why are you thinking it'd be any different on a flat Earth?

Now I say another three things:
R1: Geodesics on a sphere are equivalent to (the same as) great circles.
R2: Rhumb lines on a sphere are SOMETIMES great circles, and SOMETIMES spirals.
R3: Hence, Rhumb lines, on a sphere are NOT always geodesics. (This is a logical consequence of R1 and R2.)

Now, the penultimate step is:
P: On Earth, Rhumb lines are not ALWAYS geodesics. Hence, the Earth cannot be flat. (This is the contrapositive of the statement "The Earth is flat --> Rhumb lines are always geodesics".)

Note that I did NOT prove that the Earth is spherical, but I did prove that it is not flat.

Re: Rhumb Lines and Great Circles
« Reply #13 on: July 22, 2007, 10:33:23 AM »
Geodesics on a plane are straight lines. (Straight lines are curves too.) Rhumb lines on a plane are straight lines, and so are always geodesics.

Yes, curves are straight lines in math.

I don't quite understand this. In general, curves are not straight lines (on a general manifold, what is a straight line?), but in a Euclidean space, straight lines are curves. So I don't quite understand your statement.

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Tom Bishop

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Re: Rhumb Lines and Great Circles
« Reply #14 on: July 22, 2007, 01:31:18 PM »
When in doubt consult Earth Not a Globe by Dr. Samuel Birley Robothem.

From Chapter 14, Section 16:

GREAT CIRCLE SAILING

AMONG landsmen a great amount of misconception prevails as to what is really meant by the so-called "great circle sailing;" and notwithstanding that the subject is very imperfectly understood, the "project" or hypothesis--for it is nothing more--is often very earnestly advanced as an additional proof of the earth's rotundity. But, like all the other "proofs" which have been given, there is no necessary connection between the facts adduced and the theory sought to be proved. Although professional mariners are familiar with several modes of navigation--"parallel sailing," "plane sailing," "traverse sailing," "current sailing," "middle latitude sailing," "Mercator sailing," and "great circle sailing," the "Mercator" and "great circle" methods are now the favorites. Nearly all the above systems necessitated the sailing by, or in relation to, Rhumb-lines, or lines at right angles to the meridian lines; and whether the earth is a plane or a globe, these are not geometrically at right angles to lines of latitude, except at the equator. Hence Mercator's projection, on account of its lines of latitude and longitude being square to each other, has been almost universally employed. But previous to the general adoption of Mercator's plan, many leading navigators saw that Rhumb-line sailing upon a globe was practically a series of small circles, and conceived of a method very similar to that which is now called the "great circle" system. As early as 1495 Sebastian Cabot suggested the adoption of this method. It was also advocated in 1537 by Numez, and in 1561, and subsequently by Cortez, Zamarano, and others. After lying dormant for a long time, the system was revived by Mr. Towson, of Devonport, who read a paper before the Society of Arts, in May, 1850, and afterwards presented his "tables to facilitate the practice of great circle sailing," to the Lords Commissioners of the Admiralty, who "ordered them to be printed for the use of all mariners."

Many persons suppose that the words "great circle sailing" simply mean that the mariner, instead of sailing in a direct line from one place to another, on the same latitude, takes a circuitous path to the south or north of this direct line, where the degrees of longitude being smaller, the distance passed over, although apparently greater, is actually less. It is then falsely argued that as "the greatest distance round is the nearest path," the degrees of longitude must be smaller, and therefore the earth must be a globe. This is another instance of the self-deception practised by many of the advocates of rotundity. It is really painful to reflect upon the manner in which a merely fanciful hypothesis has reduced its advocates to mental prostitution. The poor dawdling creature, who vaguely wanders in search of anything or everything which will satisfy her longings, is only a type of the philosophical wanderer who seeks for, and pounces upon, whatever will prove, or only seem to prove, his one idea--his uncontrolled and often uncontrollable longing for something to confirm his notions, and satisfy his desire to be wise and great. The motive which actuates the greater number of modern philosophers, cannot be less or other than the love of distinction. If it were a love of truth and of human progress and welfare they would scrupulously examine the premises on which their theories are founded. But this the advocates of the earth's rotundity and motion have seldom or never done. There is no single instance recorded where even the necessity for doing so is admitted. Hence it is that whilst to question the groundwork is forbidden, they abruptly seize upon everything which gives colour to their assumptions, although in many cases neither pertinent nor logically consistent. In the case before us the contraction or convergence of the degrees of longitude beyond the equator is unproved; and again if they were convergent there could not be a single inch of gain in taking a so-called great circle course between any two places east and west of each other. Let the following experiment be tried in proof of this statement. On an artificial globe mark out a great circle path, between say Cape Town and Sydney, or Valparaiso and Cape Town. Take a strip of sheet lead, and bend it to the form of this path; and after making it straight measure its length as compared with the parallel of latitude between the places. The result will fully satisfy the experimenter that this view of great circle sailing is contrary to known geometrical principles. Strictly speaking, it is not "great circle sailing" at all which Mr. Towson and the Lords of the Admiralty have recommended. The words great circle are only used in comparison with the small circles which are described in sailing upon a Rhumb-line track.

"The fundamental principle of this method is that axiom of spherical geometry, that the shortest distance between any two points on the surface of a sphere lies on the line of a great circle; or, in other words, of a circle passing through the centre of a sphere. But maps and charts, being flat representations of the surface of a globe, are of necessity distorted, and are only correct near the equator, the distortion increasing as the poles are approached; and hence it follows that the course which on the globe is the shortest, is on the chart made to appear very much the longest, and the reverse. This was clearly shown to be the case by the comparison on a chart and on a globe of the course between Van Dieman's Land and Voldivia, on the western coast of South America: the course, which by the chart appeared to be a straight line, when laid down upon the. globe was found to be very circuitous, whilst the line of a great circle, cutting the two points, appeared on the chart as a loop of great length." 1

"Mercator and parallel sailing conduct the ship by a circuitous route when compared with the track of a great circle." 2

In nautical language Rhumb-line sailing, which was almost universally practised before the recent introduction of great circle sailing, consists in following parallels at right angles to the meridian lines, and as these meridian lines are supposed to be convergent, it is evident that the course of a ship so navigated is not the most direct; a great circle path is one at angles less than 90 north and south of the meridian. If the reader will draw a series of Rhumb-lines on a map of "the globe," he will at once see that the course is circuitous. But if he draws lines at a slight angle north in the northern, and south in the southern region, to the above-named Rhumb-lines, he will readily notice that the ship's course is more direct, and therefore the mariner adopting the so-called "great circle'' method, must of necessity save both time and distance, but only in comparison with the Rhumb-line path. It is not absolutely the shortest route; as the earth is a plane, the degrees of longitude in the south must diverge or expand, and spread out as the latitude increases; and the parallels or lines of latitude must be circles concentric with the northern centre. Hence there is in reality a still shorter path than either the Rhumb-line or the great circle course.

This will at once be evident on trying the following simple experiment. Place a light, to represent the sun, at an elevation of say two feet on the centre of a round table. Draw lines from the centre to the circumference to represent meridian lines. Mark any two places to represent Cape Town and Melbourne; now take any small object to represent a ship sailing from one of these places to the other, and, on moving it forward, keeping the light at the same altitude all the way the line of latitude or path of the ship will be seen to be an arc of a circle, which practically is a great circle route, whilst the Rhumb-line and greater route would be represented by a series of tangents to the meridian lines between the two places. The nearest route geometrically possible is the chord or straight line joining the ends of the arc which forms the line of latitude. Let this line or chord be drawn, and all argument will be superfluous, the proposition will be immediately self-evident.

Thus we have seen that great circle sailing is not the shortest route possible, but merely shorter than several other routes, which have been theoretically suggested and adopted; and to affirm that the results are confirmatory or demonstrative of the earth's rotundity, is in the highest degree illogical.

Footnotes

1 "From "A Paper on the Principles of Great Circle Sailing," by Mr. J. T. Towson, of Devonport, in the "Journal of the Society of Arts," for May, 1850.

2 "Treatise on Navigation," p. 50. By. J. Greenwood, Esq., of Jesus College, Cambridge. Weale, 59, High Holborn, London.
« Last Edit: July 22, 2007, 01:40:59 PM by Tom Bishop »

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sokarul

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Re: Rhumb Lines and Great Circles
« Reply #15 on: July 22, 2007, 01:35:14 PM »
You copy pasted all that and it still does not disprove the great circle. 
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Tom Bishop

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Re: Rhumb Lines and Great Circles
« Reply #16 on: July 22, 2007, 01:41:13 PM »
You copy pasted all that and it still does not disprove the great circle. 

Actually, it does.

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sokarul

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Re: Rhumb Lines and Great Circles
« Reply #17 on: July 22, 2007, 01:44:24 PM »
You copy pasted all that and it still does not disprove the great circle. 

Actually, it does.
Actually it doesnt.
He says



Quote
Thus we have seen that great circle sailing is not the shortest route possible, but merely shorter than several other routes, which have been theoretically suggested and adopted; and to affirm that the results are confirmatory or demonstrative of the earth's rotundity, is in the highest degree illogical.


If the circle is short than other routes it does prove the earth is round.  The shortest distance on a flat surface is a straight line and yet we see there are short ways to get there.  Thus the earth is round. 
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divito the truthist

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Re: Rhumb Lines and Great Circles
« Reply #18 on: July 22, 2007, 05:27:33 PM »
I don't quite understand this. In general, curves are not straight lines (on a general manifold, what is a straight line?), but in a Euclidean space, straight lines are curves. So I don't quite understand your statement.

"In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments."


I'm acknowledging that the there is more than one definition and use of the word; even if you may or may not be taking it out of context in your transitive property that you used. I'm still not convinced all Rhumb lines are geodesics on a plane.
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Re: Rhumb Lines and Great Circles
« Reply #19 on: July 22, 2007, 07:59:03 PM »
I don't quite understand this. In general, curves are not straight lines (on a general manifold, what is a straight line?), but in a Euclidean space, straight lines are curves. So I don't quite understand your statement.

"In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments."


I'm acknowledging that the there is more than one definition and use of the word; even if you may or may not be taking it out of context in your transitive property that you used. I'm still not convinced all Rhumb lines are geodesics on a plane.

I'm sorry but you said that "curves are straight lines" which is what confused me. I think what you meant is "straight lines are curves"...

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The Communist

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Re: Rhumb Lines and Great Circles
« Reply #20 on: July 31, 2007, 01:17:14 PM »
Regarding the OP, why is the perimeter calculated as (pi*r^2)/4?  I thought perimeters were measured as pi*r^2
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Re: Rhumb Lines and Great Circles
« Reply #21 on: July 31, 2007, 05:15:03 PM »
I don't quite understand this. In general, curves are not straight lines (on a general manifold, what is a straight line?), but in a Euclidean space, straight lines are curves. So I don't quite understand your statement.

"In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments."


I'm acknowledging that the there is more than one definition and use of the word; even if you may or may not be taking it out of context in your transitive property that you used. I'm still not convinced all Rhumb lines are geodesics on a plane.

A rhumb line is a line of constant bearing. Hence, in $\bf{R}^3$, the equation of a rhumb line is
$$\overline{x}(t) = t \cdot \overline{x}_0 + \overline{x_1}$$ where $\overline{x}_1$ is the position at time $t = 0$ and $\overline{x}_0$ is the tangent vector at any time. This is the equation of a straight line.

(To be rigorous, there are a few steps in between (solving simple vector DE's) that I proved in my differential geometry class.)

Re: Rhumb Lines and Great Circles
« Reply #22 on: July 31, 2007, 05:24:05 PM »
You copy pasted all that and it still does not disprove the great circle. 

Actually, it does.

Please summarize what this work says in relation to the topic in question...

Re: Rhumb Lines and Great Circles
« Reply #23 on: July 31, 2007, 05:35:10 PM »
I don't quite understand this. In general, curves are not straight lines (on a general manifold, what is a straight line?), but in a Euclidean space, straight lines are curves. So I don't quite understand your statement.

"In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments."


I'm acknowledging that the there is more than one definition and use of the word; even if you may or may not be taking it out of context in your transitive property that you used. I'm still not convinced all Rhumb lines are geodesics on a plane.

A rhumb line is a line of constant bearing. Hence, in $\bf{R}^3$, the equation of a rhumb line is
$$\overline{x}(t) = t \cdot \overline{x}_0 + \overline{x_1}$$ where $\overline{x}_1$ is the position at time $t = 0$ and $\overline{x}_0$ is the tangent vector at any time. This is the equation of a straight line.

(To be rigorous, there are a few steps in between (solving simple vector DE's) that I proved in my differential geometry class.)
Sorry, but that's gibberish. cdot is undefined, for example.

Re: Rhumb Lines and Great Circles
« Reply #24 on: July 31, 2007, 05:52:09 PM »
I don't quite understand this. In general, curves are not straight lines (on a general manifold, what is a straight line?), but in a Euclidean space, straight lines are curves. So I don't quite understand your statement.

"In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. A simple example is the circle. In everyday use of the term "curve", a straight line is not curved, but in mathematical parlance curves include straight lines and line segments."


I'm acknowledging that the there is more than one definition and use of the word; even if you may or may not be taking it out of context in your transitive property that you used. I'm still not convinced all Rhumb lines are geodesics on a plane.

A rhumb line is a line of constant bearing. Hence, in $\bf{R}^3$, the equation of a rhumb line is
$$\overline{x}(t) = t \cdot \overline{x}_0 + \overline{x_1}$$ where $\overline{x}_1$ is the position at time $t = 0$ and $\overline{x}_0$ is the tangent vector at any time. This is the equation of a straight line.

(To be rigorous, there are a few steps in between (solving simple vector DE's) that I proved in my differential geometry class.)
Sorry, but that's gibberish. cdot is undefined, for example.

Opps.. I'm writing in LaTeX. I'd post a picture of it, but I can't figure out how...
cdot means center dot.. It's just a dot. Overline puts a line over everything; it should be overrightarrow which will put a right arrow over everything, which most people use to denote vectors. Dollar signs are for math mode.

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divito the truthist

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Re: Rhumb Lines and Great Circles
« Reply #25 on: July 31, 2007, 06:01:02 PM »
Hit the print screen button on your keyboard. Then open paint or another graphics program you have and paste it in, save and upload.
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Re: Rhumb Lines and Great Circles
« Reply #26 on: July 31, 2007, 06:28:05 PM »
Hit the print screen button on your keyboard. Then open paint or another graphics program you have and paste it in, save and upload.

I already have the picture, but I couldn't figure out how to download it.

*Shuffles noobishly away*

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divito the truthist

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Re: Rhumb Lines and Great Circles
« Reply #27 on: July 31, 2007, 06:35:52 PM »
You mean upload?

http://imageshack.us
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The Communist

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Re: Rhumb Lines and Great Circles
« Reply #28 on: August 01, 2007, 08:29:21 AM »
Regarding the OP, why is the perimeter calculated as (pi*r^2)/4?  I thought perimeters were measured as pi*r^2

I now understood why he placed the 1/4 factor.  He converted radius, r, into diamter, d/2 = r, thus [(d/2)^2]*pi = [pi * (d^2)]/4
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narcberry

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Re: Rhumb Lines and Great Circles
« Reply #29 on: August 01, 2007, 10:09:05 AM »
Hit the print screen button on your keyboard. Then open paint or another graphics program you have and paste it in, save and upload.

I already have the picture, but I couldn't figure out how to download it.

*Shuffles noobishly away*

e-mail it to the mods.