How to get more width for a tarp....

[GrizzlyAdams]

Sister Mary Augustine says that if the hex truly has 8' parallel to the ground with a 14' ridgeline, then the width of the side has to be 70/11 feet, which makes the tarp width on the ground closer to 4.95' on one side rather than 4.8'. She thought that was enough to go after you with her ruler, but I pointed out that these days she needs a passport to get back into the USA, and so I think I dissuaded her.

But I got to thinking that since not all "covered space" is created equal (your metric of coverage is bird's eye---straight down) what is really wanted here is a utility function that, for a given tarp, ascribes a "coverage" value to every point in the x-y plane. For example, the value could be the fraction of all rays in the positive z dimension with that point as the origin, which intercept the tarp. This is motivated by the idea of rain coming down. But then not all rays are equal in terms of threat, so the rays that are more likely to correspond to rain would be given more weight. Then to get the utility coverage you'd sum the coverage value over all points in the x-y plane.

There's a simpler way I'm sure. If any ideas come to me that can be easily conveyed (unlike the babble above) I'll jot 'em down.

Grizz

[Peter_pan]

What do you guys think of the Gosamer tarp?

http://www.gossamergear.com/cgi-bin/...inn-Twinn.html

It looks like it might be a bit short for a hammock at 117".

For the shorter hammocks HH BULA size, ENO , Travel pod, Byers and Claytor exped and mosquito it should be fine... taper on the foot end just means a little less side coverage, probably no big deal... little pricy though

Pan

[Rapt]

Yes, I pointed out that the hex was merely close and slightly short on area rather than exact. (To simplify the math as 6.25 is easier to use than 70/11.) I felt safe doing that as I knew it would give me a conservative result.

And yes I understand the thinking that not all covered space is equal, the closer and edge is to the ground the less wind penetration. That said, if you assume a fixed angle of rainfall being driven by a wind, this isn't that hard to calculate. I suspect that we'll find the resulting superiority of one shape over another is still basically the same, at least as long as the ridgeline is perpendicular to the wind direction.

I haven't yet worked through the ridgeline parallel to the wind direction case but my suspicion is that it'll come out the same as well, since any loss of coverage at one end will be matched by an equivalent gain at the other.

Most interesting might be to work out coverage if we assume a given rain angle and a wind that shifts a full 360 degrees. (I'd calculate one side with a perpendicular situation and one with a parallel and then take it off twice as a reasonable estimate.)

[GrizzlyAdams]

Yes, I pointed out that the hex was merely close and slightly short on area rather than exact. (To simplify the math as 6.25 is easier to use than 70/11.) I felt safe doing that as I knew it would give me a conservative result.

And yes I understand the thinking that not all covered space is equal, the closer and edge is to the ground the less wind penetration. That said, if you assume a fixed angle of rainfall being driven by a wind, this isn't that hard to calculate. I suspect that we'll find the resulting superiority of one shape over another is still basically the same, at least as long as the ridgeline is perpendicular to the wind direction.

I haven't yet worked through the ridgeline parallel to the wind direction case but my suspicion is that it'll come out the same as well, since any loss of coverage at one end will be matched by an equivalent gain at the other.

Most interesting might be to work out coverage if we assume a given rain angle and a wind that shifts a full 360 degrees. (I'd calculate one side with a perpendicular situation and one with a parallel and then take it off twice as a reasonable estimate.)

Devil's advocate here. If the rain is coming in at the angle normal to the tarp surface, then I think your coverage area is just exactly that---area, which would be the same for the rectangle or hex, right? If the rain comes in at an angle that is not normal to the tarp surface, but is orthogonal to the ridgeline, then the spread of the tarp is going to make a difference, but the only difference will be for points that are very close to the edge. Seems a small effect overall.

I just know there's a way of doing this with rotation matrices (to describe point of view) and matrix-matrix (or maybe matrix-vector) multiplications....this is just computer graphics calculations....but I got real work to do, play time will have to wait

Grizz

[GrizzlyAdams]

In thinking about rain pounding on tarps at normal angles and the like, I realized that to get the coverage from rain falling straight down, you just multiply the tarp area by the sin of the angle formed by the side of the tarp, and the vertical. Does not matter what shape the tarp is, so long as the whole side is at that angle. In fact, so long as the rain is coming in orthogonal to the ridgeline, a similar calculation works, but you need to break it up into two parts to account for the two sides of the tarp. But the transformation in both cases is a matter of multiplying the given area by the sin of an angle.

This means that the farther out the tarp is staked (given a fixed height pitch), the more coverage you get.

Which means that an isosceles triangle whose area is the same as the 14x10 rectangle or hex with same area will have larger coverage.

Which led me to wonder why the triangle in your example came up short---and it is because a 10x10 tarp has less area than a 10x14 tarp. Which you knew, the point was to choose among tarps with common 14' ridgelines, not necessarily tarps with common surface area.

To complete the thought, given your figures, the length of the tarp from ridge to corner is 10', giving 127 sq ft of coverage. A tarp that large, with that shape---which maximizes coverage of a certain type---accentuate the point that not all coverage is created equal.

Grizz

[TiredFeet]

Grizz you lost me (which is easy to do ).

But

.....Does not matter what shape the tarp is, so long as the whole side is at that angle. ....

This means that the farther out the tarp is staked (given a fixed height pitch), the more coverage you get.

Which means that an isosceles triangle whose area is the same as the 14x10 rectangle or hex with same area will have larger coverage.

If the shape doesn't matter, how does that triangle give larger coverage?

[GrizzlyAdams]

Grizz you lost me (which is easy to do ).

But



If the shape doesn't matter, how does that triangle give larger coverage?

With the height pitch fixed (say at 4 ft) and the width pitch (of the furthest point) fixed, then it doesn't matter what the shape is. The whole surface is at the same angle. If that angle was zero (sides hanging flat) then there is no coverage. If that angle was 90 degrees (of course it can't be) then the covered area is just the area of the side, no matter what shape. The same is true whatever that angle is. In geometric terms you'd say that the coverage is the "projection" of the area onto the plane. (Quick, call your mother and thrill her with your big new words ).

Here's another way to look at it. Think of the surface of the tarp as being the assemblage of many many tiny squares. If you can figure out how much coverage you get from one tiny square, then you get the total coverage by adding up the coverages from all the tiny squares because they are the same. So if every tiny square is at the same angle with respect to the ground, and you get the total coverage by adding up the coverages of all the tiny squares, then any way you shuffle those squares around will give you the same answer, provided that each is at the same angle. That's why shape doesn't matter for a fixed pitch.

So why should angle matter? If you imagine looking straight down at a tiny square and imagine how much of the ground is covered from view as a function of the angle, you see nothing is covered when the angle is zero, and as much as possible is covered when the angle is 90 degrees. The closer the angle is to 90, the more is covered.

So coverage is driven by this angle, and for a fixed tarp height pitch the angle is determined by how far out you can pitch the sides of the tarp. Of the three (rectangle, hex, triangle), the farthest pitch is the triangle, so it has the angle closest to 90 degrees, and the largest coverage (as defined above).

here ends the reading of the First Lesson
Grizz

[pure_mahem]

Man I sold my freaking graphing calculator when I was done with college, this stuff is giving me a headache. I'm gonna go play sodoku it's less complicated!

[Rapt]

I think this thread needs pictures... oh wait... That'd mean I'm probably the one to make them...

Anyways, I think I understand what you're saying Grizz, and if you're saying what I think then I agree... While its possible with matrices, I was thinking more in line with your follow up post. To my mind the fairly simple trig is the way to go. However we can't neglect the effect of the drip line on the leeward side acting to reduce the "covered" area for that half... With winds orthogonal to the ridgeline.... So its not actually the whole area... However its still in the same proportions as the projected areas.

[GrizzlyAdams]

Man I sold my freaking graphing calculator when I was done with college, this stuff is giving me a headache. I'm gonna go play sodoku it's less complicated!

You are new here, and maybe haven't explored the site well enough to know that this sort of thing does happen. As a reminder, I've attached a screen-dump of the page with the required-by-law alert.

Grizz