Spherical coordinates

[Bazil]

Hi all,
I am just trying to get my head around spherical coordinates. How do you calculate the theta coords along the edge between (0, 2a) (72, 2a) a=arctan(1/phi), and the internal vertices, for any given v? I can do it with 2d trig but it seems pretty convoluted. Is there a more elegant way?
Thx, bryan

[Hector Alfredo Hernández Hdez.]

Can you make a draw?
Thanks

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[Gerry in Quebec]

Good suggestion, Hector!

 

Bryan, if you have a copy of Hugh Kenner's book Geodesic Math and How to Use It, check out chapter 12, which starts on page 74. Referring to the spherical coordinates of method 1 domes, classes I and II, Kenner writes: "...the process by which they are obtained is almost ridiculously simple."

 

- Gerry in Québec

[Bryan L]

Oh!!! I have it. Just re read it... I must have missed that bit.... Will look again tomorrow.

[Gerry in Quebec]

Kenner's book has quite a few errors, including at least two in chapter 12. Here's the errata list, including a correction to equation 12.15. Also, I believe the "f" in equation 12.6 refers to frequency; elsewhere Kenner uses "v", in italics, to represent frequency.

 

http://bobwb.tripod.com/synergetics/hugh/index.html

 

- Gerry

[Bryan L]

Hector, I am using my phone so can't create a drawing.
I am just trying to work out the spherical coords of a given angular subdivision on the icosa symmetry triangle.

Gerry thx for the errata link, I had read it when I first got the book.

I see why Hugh has f in the z1 equation. f = x + y + z = frequency of breakdown and he uses it in the ch 12 appendix.

I didn't bother too much with chapter 12 because I wanted to start from angular division not class I method I. I should have mentioned that in my op.

I see what Hugh is doing in ch 12. I have similar integer based formulas for the length of any a' along an edge and the height of that point from the origin. All based on arctan(1/phi).

This is where I am at:

a= arctan(1/phi)
v=4
(theta, phi) : theta polar, phi azimuth

Given an icosa triangle one point at the zenith, one edge on the xz plane,
I have correct coords from (0,0) to (4,0) : (0, 0), (a/2, 0), (a, 0), (3a/2, 0), and (2a, 0).
And from (1,1) to (4,4) : (a/2, 72), (a, 72), (3a/2, 72) and (2a, 72).
I worked these out intuitively not from a formula.
I believe phi will be 36, 24, 48 for (2,1), (3,1), (3,2) and 18, 36, 54 for (4,1) to (4,3).
I need to work out theta for those other points and derive formulas for phi. If I use the ch 12 appendix (projection planes) I will hopefully nut it out...

[Paul Kranz]

Bazil:

What will spherical coordinates do for you?

My dome designs use algebra and trig to get the face angles of the triangles. The dihedral angles are supplied by the spreadsheets/calculators that have been floating around GeodesicHelp. If you know the dimensions (face angles and side lengths) of the triangles, where they go, which way to place them, and their orientation to one another (dihedrals), you are pretty much home-free.

Paul sends...

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Very high regards,

 

Paul sends...

ReviveTheFlame.com

[Bryan L]

Hi Paul, I have used algebra and trig successfully too.
I now want to do some programming to play with the Mexican method and other angular divisions. It is time to progress to 3d space for this as I can see it becomes much easier to derive vertices.

[Hector Alfredo Hernández Hdez.]

For draws you can hnd draw and take photo and sent it  by email .

By the way, there another special coordinates type, more usefull for geodesic domes.... including sphericals coordinates :)

El ene 19, 2016 8:09 PM, "Bryan L" <bhla...@gmail.com> escribió:

Hi Paul, I have used algebra and trig successfully too.
I now want to do some programming to play with the Mexican method and other angular divisions. It is time to progress to 3d space for this as I can see it becomes much easier to derive vertices.

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[Gerry in Quebec]

Bryan,
If you wish to divide each edge of the icosa symmetry triangle into four equal geodesics, then your coordinates are correct for pathways (0,0) to (4,0) and (0,0) to (4,4), but the phi coordinates are incorrect for pathway (4,0) to (4,4).

 

We know the following:

Arc (4,0) to (4,4) = arc (0,0) to (4,0) = theta coordinate of (4,0) = 2a = 63.43495 degrees
Arc (4,0) to (4,1) = 63.43495 / 4 = 15.85874 degrees
Spherical angle (0,0)-(4,0)-(4,1) = (0,0)-(4,0)-(4,4) = 72 degrees

 

Kenner's vertex labels are a bit cumbersome in trig equations, so I've used simple letter labels in the attached diagram:
Vertex 0,0 = A
Vertex 4,0 = B
Vertex 4,1 = C

 

Here's one of a few ways to find the coordinates:
Theta coordinate of (4,1) = arc AC = arcos [ (cos spherical angle ABC * sin arc BC * sin arc AB) + (cos arc BC * cos arc AB) ] = 59.62078 degrees

Phi coordinate of (4,1) = spherical angle BAC = arcos { [cos arc BC - (cos arc AC * arc AB) ] / [sin arc AC * sin arc AB] } = 17.53300 degrees.

 

The same equations are used to find the coordinates of (4,2) and (4,3).

 

To obtain the coordinates of internal vertices (2,1), (3,1) and (3,2), your calculations need to take into account the nature of the subdivision method, i.e., a set of conditions. For example, with the Mexican Method in which you expressed an interest, you want all arcs/chords on a pathway of a given colour to be of equal length. Unfortunately, this set of conditions is hard to express mathematically (for me a least), so I have simply used Excel's Solver feature to find the coordinates for domes and spheres of various frequencies. This works well and has demonstrated that at frequency 6 and above, the Mexican Method breaks down. Slight changes in radial values are needed to maintain arc/chord equality on each type of pathway. However, this is not a problem for dome construction because the fine-tuning is miniscule. Hector may have more to say about this.

 

Here are two other examples -- 16v icosa, class II, method 3, and 3v icosa, class I, Fuller-Kruschke lesser-circle subdivision -- of how spherical trig in Excel can be used to find phi and theta coordinates:

 

https://groups.google.com/forum/#!searchin/geodesichelp/method$203/geodesichelp/zG4Mm__cVHI/5WbjH5gTLgIJ
May 25, 2013

 

https://groups.google.com/forum/#!searchin/geodesichelp/method$203/geodesichelp/GCD6Gis0-U0/gjzede7KzXQJ
Oct. 22, 2011.

 

- Gerry

[Bazil]

The links Gerry posted don't seem to work for me - at least in chrome on Android. It just takes me to google groups home page.

Could someone post a direct link to the topics?

There is a way to get a link to a topic within Google groups. If you click on the little down arrow under the topic heading (beside where it says x posts by y authors), there is a menu entry "link to this topic".

Cheers,

Bryan

[Gerry in Quebec]

Bryan,

Here's the direct link to the class I, method 2 discussion and the Excel spreadsheet showing the derivation of spherical coordinates and chord factors of the 4v icosa.

 

https://groups.google.com/d/topic/geodesichelp/vXPqZw7Wyq4/discussion

[Gerry in Quebec]

 

And...

 

https://groups.google.com/d/topic/geodesichelp/zG4Mm__cVHI/discussion
May 25, 2013. Excel spreadsheet, derivation of coordinates & chord factors, 3v icosa, class I, Fuller-Kruschke subdivision

 

https://groups.google.com/d/topic/geodesichelp/GCD6Gis0-U0/discussion

Oct. 22, 2011. Excel spreadsheet, derivation of coordinates & chord factors, 16v icosa, class II, method 3

 

 

 

On Monday, January 25, 2016 at 11:05:22 PM UTC-5, Bazil wrote:

[Bryan L]

Hi Hector,

For a drawing, look at any of Gerry's spreadsheets where he has the PPT anchored at (0,0), (a,0), (a,72), a = arctan(1/phi).
That is what I tried to describe.
But from Gerry's help, I can now calculate spherical coords for various breakdowns.

As a point of interest, can you tell me what other coordinate types are more useful for geodesic domes?

Cheers,
Bryan

[Hector Alfredo Hernández Hdez.]

It is useful to add to the spherical coordinates these others:

Let be the angle alpha POR, red point moves in the plane containing P, O and R.  (O is in center of dome)

Let be the angle beta POG and green point moves in the plane containing P, O and G.




These coordinates are used to relocate these issues in a more simple way,
you can use spherical coorrdenadas but it is  more complicated.

See you.

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