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+--- Thread: (12C Platinum) Length of An Ellipse (/thread-13242.html)
(12C Platinum) Length of An Ellipse - Gamo - 07-07-2019 05:44 AM
Calculate the Length of an ellipse according to this series formula:
[attachment=7449]
[attachment=7450]
-------------------------
Example: FIX 5
a = 50
b = 10
50 [ENTER] 10 [R/S] display 210.08595
------------------------
Program: RPN mode: "Program run this formula in three terms"
Code:
001 STO 2
002 R↓
003 STO 1
004 RCL 2
005 -
006 RCL 1
007 RCL 2
008 +
009 STO 3
010 ÷
------------
011 STO 0
012 X^2
013 4
014 ÷
015 1
016 +
017 RCL 0
018 4
019 Y^X
020 6
----------
021 4
022 ÷
023 +
024 RCL 0
025 6
026 Y^X
027 2
028 5
029 6
030 ÷
---------
031 +
032 3
033 5
034 5
035 ENTER
036 1
037 1
038 3
039 ÷
040 RCL 3
041 x
042 xGamo
RE: (12C Platinum) Length of An Ellipse - Leviset - 07-07-2019 10:35 AM
Attached a number of approximations & infinite series to calculate the perimeter of an Ellipse.
RE: (12C Platinum) Length of An Ellipse - Albert Chan - 07-07-2019 04:06 PM
(07-07-2019 05:44 AM)Gamo Wrote: Program: RPN mode: "Program run this formula in three terms" ...
You might consider Ramanujan's approximation, less code and almost 2 extra terms!
Replace steps 011-031 with this:
Quote:011 Enter
012 x ; h = ((a-b)/(a+b))^2
013 3
014 x
015 Enter
016 Enter
017 4
018 -
019 CHS
020 SQRT
021 1
022 0
023 +
024 /
025 1
026 + ; correction = 1 + 3h / (10 + sqrt(4-3h))
Using your example, a=50, b=10, FIX 5, we get ellipse perimeter = 210.10028, absolute error = -0.00017
RE: (12C Platinum) Length of An Ellipse - Gamo - 07-08-2019 04:51 AM
Thanks Albert Chan
That formula is much simple as stated in the link
"The simplest formula possible for the circumference of a general ellipse"
Gamo
RE: (12C Platinum) Length of An Ellipse - Albert Chan - 07-08-2019 02:53 PM
(07-07-2019 04:06 PM)Albert Chan Wrote: a=50, b=10, FIX 5, we get ellipse perimeter = 210.10028, absolute error = -0.00017
FYI, absolute error based on ellipse perimeter AGM2 method:
(50, 10) repeated AGM:
→ (30 , √(500) ≅ 22.36067977), gap = 7.63932023
→ (26.18033989, 25.90020064), gap = 0.28013825
→ (26.04027027, 26.03989355), gap = 0.00037672
→ (26.04008191, 26.04008191)
effective radius = (30² - (0.5)(7.63932023²) - (1)(0.28013825²) - (2)(0.00037672²)) / 26.04008191 = 33.43852445
ellipse perimeter = 2 * pi * 33.43852445 = 210.1004455 ≅ 210.10045
RE: (12C Platinum) Length of An Ellipse - John Keith - 07-23-2019 05:06 PM
I have tried several of these methods and I believe that Ramanujan's approximation is the clear winner. It gives 6 - 9 digits of accuracy for reasonable ellipses and is small and fast. For exact solutions, Albert's AGM2 program is fastest by far, almost 200* as fast as numerical integration on the HP 50! The Gauss-Kummer infinite series (series 2 from the PDF in post #2) looks good but the terms are very slow to compute and the program is large and complicated.
At the risk of trolling this thread I am posting HP-48 versions of the programs for those who may be interested.
Ramanujan:
Code:
\<< DUP2 - ROT ROT + DUP \pi \->NUM * ROT ROT / SQ 3. * 4. OVER - \v/ 10. + / 1. + *
\>>For the HP 50, ROT ROT can be replaced by UNROT.
AGM2:
Code:
\<< DUP2 SQ SWAP SQ + .5 * .25 0. 0. \-> a b s t k s1
\<<
WHILE b a - 'k' STO s t k SQ * - 's1' STO s s1 \=/
REPEAT a b * \v/ 'b' STO 'a' k .5 * STO+ s1 's' STO t 't' STO+
END \pi \->NUM 2. * s * k .5 * a + /
\>>
\>>This is just a straightforward translation of Albert's Lua code which could be made smaller and faster by moving local variables to the stack at the cost of readability (and effort!).
RE: (12C Platinum) Length of An Ellipse - Albert Chan - 01-16-2020 10:18 PM
For ellipse perimeter using integrals, we can use Weierstrass Substitution.
\(t = \tan(x/2)\quad\quad\quad → dt = sec^2(x/2)/2\;dx\quad\quad\quad\quad\quad\quad
\; → dx = \left({2\over 1+t^2}\right)\;dt \)
Ellipse perimeter = \(4a \int _0 ^{\pi \over 2} \sqrt{1 - e^2\sin^2 x}\;dx
= 4a \int _0 ^1 \sqrt{1 - e^2\left({2t \over 1+t^2}\right)^2}
\left({2\over 1+t^2}\right)\;dt \)
Example, Casio FX-115ES Plus, a=50, b=10 → e² = 1-(b/a)² = 24/25
Perimeter = 210.100445396890 // 4*50*EllipticE(24/25)
dx integral = 210.100445053320 // time = 4.0 sec
dt integral = 210.100445398816 // time = 2.8 sec
Turns out dx integral work better with simple Trapezoidal rule !
John D Cook's blog: Three surprises with the trapezoid rule
Quote:2. Although the trapezoid rule is inefficient in general, it can be shockingly efficient for periodic functions.
Redo above example for ellipse perimeter. (n = number of trapezoids)
Code:
n dx trapezoid dx simpson dx romberg
2 207.519513599 213.860831727
4 209.945722803 210.754459204 210.547367702
8 210.098446673 210.149354630 210.102056335
16 210.100444351 210.101110244 210.097700423
32 210.100445397 210.100445745 210.100454618
64 210.100445397 210.100446082
128 210.100445394
256 210.100445397