spiral maths Project Page 2

talking with rsgpitt in "Spiral Formula" post

http://oupower.com/phpBB2/viewtopic.php?t=481&postdays=0&postorder=asc&start=0
http://oupower.com/phpBB2/viewtopic.php?t=481&postdays=0&postorder=asc&start=15

rsgpitt wasnt convinced my numbers were right, and i know that i used a summing tangents method to calculate the inter-hole mounting distances, rather than a nice 100% accurate mathematical method, so i wrote another part to my code

the fullsize picture to the left when printed at 300dpi will be a lifesize depiction of rsgpitt's numbers
http://oupower.com/index.php?dir=_Other_Peoples_Projects/AntDavison/spiral%20maths&PageNum=2&FullSizeImg=_Other_Peoples_Projects/AntDavison/spiral%20maths/rspitt.jpg

shim = 0.004" * 25.4 = 0.1016mm
washers = 1/8" * 25.4 = 3.1750mm
spacing = 3.2766mm
id = 0.935" = 23.749mm

///////////////////////////////

id = 23.749
spacing = 3.2766
dpmm = 300 / 25.4

turns = 3.75
plates = 2

decrement = plates * 2 * spacing
od = (turns * decrement) + id
pxls = od * dpmm

spiral = bitmap ((pxls + 0.5) as integer) ((pxls + 0.5) as integer) color:white
pxlcolour = #(color 255 0 255)

circum = z = zz = theta = sn = cs = pxlD = pxlR = sx = cy = lastx = lasty = deltax = deltay = 0.0

deg = 360 / plates

for zz = 0 to (360-deg) by deg do
(
pxlcolour.h = (zz / 360.) * 255
pxlD=pxls

for z = zz to ((turns*360) + zz) by .03125 do
(
if z > 0 then (lastx = sx ; lasty = cy)
theta = mod z 360
sn = sin theta
cs = cos theta
pxlD = pxls - (((z-zz) / (360.)) * decrement * dpmm)
pxlR = pxlD / 2
sx = (pxls / 2) + (sn * pxlR)
cy = (pxls / 2) + (cs * pxlR)
if z == 0 then (lastx = sx ; lasty = cy)
setPixels spiral [((sx + 0.5) as integer),((cy + 0.5) as integer)] pxlcolour
if zz == 0 then
(
deltax = sx - lastx
deltay = cy - lasty
circum = sqrt((deltax * deltax) + (deltay * deltay)) + circum
if ((mod z 90.) == 0) then print (((circum / dpmm) as string) +"mm = " +  (((circum / dpmm)/25.4) as string))
)
)
display spiral
)
display spiral

setPixels spiral [((pxls / 2) as integer),((pxls / 2) as integer)] #(color 0 0 0)
lngth = circum / dpmm
print lngth
;id = od - (turns * decrement)
print od

lngth = 0

print "confidence\n==========="
print "0.0mm = 0.0"

for z = (od - (decrement / 8.)) to (id + (decrement / 8.)) by (-decrement / 4.) do
(
lngth += (z * pi / 4.)
print ((lngth as string) +"mm = " +  ((lngth/25.4) as string))
)

///////////////////////////////

id in mm = 23.749
spacing = 3.2766
turns = 3.75
plates = 2
od in mm = 72.898
pixels dimensions for bitmap @ 300dpi = 861.0
plate mounting offset in degrees = 180

"0.0mm = 0.0"
"55.9912mm = 2.20438"
"109.41mm = 4.30749"
"160.257mm = 6.30934"
"208.532mm = 8.20992"
"254.235mm = 10.0092"
"297.366mm = 11.7073"
"337.926mm = 13.3042"
"375.914mm = 14.7998"
"411.332mm = 16.1942"
"444.18mm = 17.4874"
"474.457mm = 18.6794"
"502.165mm = 19.7703"
"527.305mm = 20.7601"
"549.878mm = 21.6487"
"569.884mm = 22.4364"
OK
OK
true
569.884
569.884
569.884
23.749
72.898
72.898
0

"confidence
==========="

"0.0mm = 0.0"
"55.9672mm = 2.20343"
"109.361mm = 4.30555"
"160.181mm = 6.30635"
"208.428mm = 8.20584"
"254.102mm = 10.004"
"297.202mm = 11.7009"
"337.728mm = 13.2964"
"375.682mm = 14.7906"
"411.061mm = 16.1835"
"443.868mm = 17.4751"
"474.101mm = 18.6654"
"501.76mm = 19.7543"
"526.846mm = 20.742"
"549.359mm = 21.6283"
"569.298mm = 22.4133"
OK
OK
1

thus the second set of numbers will be 100% correct

========================================

so you dont have 3dstudio, and dont want a bitmap printed out

then translate the following to whatever programming language you use ... maybe even eXcel

the theory is

circumference = 2 * pi * radius  = pi * diameter

therefor

length of a 90 degree arc = pi * diameter / 4

with a constantly increasing radius/diameter a 90 degree arc component of a spiral would have a length of

((pi * od / 4) + (pi * id / 4)) / 2

or

pi * ((od + id ) / 2) / 4

or

pi * ((od/2) + (id/2)) / 4

my country is metric, so the below dimensions are in millimetres

-------------------------

id = 23.749
spacing = 3.2766
turns = 3.75
plates = 2

decrement = plates * 2 * spacing
od = (turns * decrement) + id

lngth = 0

print ("outside diameter = " + (od as string)  +"mm = " +  ((od/25.4) as string))

print "0.0mm = 0.0"

for z = (od - (decrement / 8.)) to (id + (decrement / 8.)) by (-decrement / 4.) do
(
lngth += (z * pi / 4.)
print ((lngth as string) +"mm = " +  ((lngth/25.4) as string))
)

==============================

when i run this code with my kettle dimensions i get the following numbers

id = 28.
spacing = 2.
turns = 5
plates = 4

outside diameter = 108.0mm = 4.25197"

0.0mm = 0.0"
83.2522mm = 3.27765"
163.363mm = 6.43161"
240.332mm = 9.46188"
314.159mm = 12.3685"
384.845mm = 15.1514"
452.389mm = 17.8106"
516.792mm = 20.3461"
578.053mm = 22.758"
636.172mm = 25.0462"
691.15mm = 27.2106"
742.987mm = 29.2514"
791.681mm = 31.1686"
837.234mm = 32.962"
879.646mm = 34.6317"
918.916mm = 36.1778"
955.044mm = 37.6002"
988.031mm = 38.8989"
1017.88mm = 40.0739"
1044.58mm = 41.1252"
1068.14mm = 42.0528"

compare this with the numbers in item #2 of this directory, and you will see that the "tangental summing" method is only 99.9% accurate