Hypergeometric function – Perimeter of an Ellipse and other applications (wp34s)

[Gerson W. Barbosa]

Here is still another approximation to the perimeter of the ellipse:

p ≈ 2π[(aˢ + bˢ)/2]¹ᐟˢ

where

s = 3/2 + 1/(32/h² - ⅗h² - 4)

and where

h = [(a - b)/(a + b)]

That’s Thomas Muir’s approximation, I’ve only added an empirical correction term to his s = 3/2 parameter. The precision quite matches that of Ramanujan’s second approximation, the results being slightly above or below depending on the h parameter. In the case of the orbit of Halley’s comet, for instance, the absolute value of the difference is about three times less: 920.378 km. That’s about the flight distance from Warsaw to Stuttgart.


The wp34s program should be smaller (I’ll see to that later), but for convenience I will shift to Free42:

00 { 61-Byte Prgm }
01▸LBL "MB"
02 COMPLEX
03 ENTER
04 COMPLEX
05 ENTER
06 X<> ST Z
07 STO- ST Z
08 +
09 ÷
10 X↑2
11 32
12 X<>Y
13 ÷
14 0.6
15 RCL× ST L
16 -
17 4
18 -
19 1/X
20 1.5
21 +
22 X<>Y
23 COMPLEX
24 RCL ST Z
25 Y↑X
26 X<>Y
27 LASTX
28 Y↑X
29 +
30 2
31 ÷
32 X<>Y
33 1/X
34 Y↑X
35 STO+ ST X
36 PI
37 ×
38 END

Examples:

9 ENTER 8 XEQ MB ->

53.45328500297186507413258774441006

2 ENTER 3 XEQ MB ->

15.86543958923382742479398274914387

4 ENTER 1 XEQ MB ->

17.15684512619729903456675762188867

For exact results, use the ELP and 2F1 programs below, the latter copied and pasted from Jean-Marc Baillard in the Old HP-41C Software Library

00 { 38-Byte Prgm }
01▸LBL "ELP"
02 RCL- ST Y
03 X<>Y
04 RCL+ ST L
05 STO 04
06 ÷
07 X↑2
08 -0.5
09 RCL ST X
10 1
11 R↑
12 XEQ " 2F1"
13 RCL× 04
14 PI
15 ×
16 END

00 { 58-Byte Prgm }
01▸LBL " 2F1"
02 STO "Z"
03 R↓
04 STO 03
05 R↓
06 STO 02
07 R↓
08 STO 01
09 CLST
10 SIGN
11 ENTER
12 ENTER
13▸LBL 01
14 R↓
15 X<>Y
16 RCL 01
17 R↑
18 STO+ ST Y
19 R↓
20 ×
21 RCL 02
22 R↑
23 STO+ ST Y
24 R↓
25 ×
26 RCL 03
27 R↑
28 STO+ ST Y
29 ISG ST X
30 CLX
31 STO× ST Y
32 R↓
33 ÷
34 RCL× "Z"
35 STO ST Z
36 X<>Y
37 STO+ ST Y
38 X≠Y?
39 GTO 01
40 END

——-

PS: 01-15-2020, 00:17 AM

wp34s

001:LBL A
002:©ENTER
003:STO- Y
004:RCL+ T
005:/
006:x²
007:# 032
008:x⇆ Y
009:/
010:# 003
011:RCL× L
012:SDR 001
013:STO+ X
014:-
015:# 004
016:-
017:1/x
018:# 015
019:SDR 001
020:+
021:yᵡ
022:x⇆ Y
023:RCL L
024:yᵡ
025:STO+ Y
026:x⇆ L
027:# 002
028:STO/ Z
029:R↓
030:ᵡ√y
031:STO+ X
032:# π  
033:×
034:END

If a = b, that is, when h = 0, the program will return “+∞ Error” (“Divide by 0” on the HP-42S). If this is an issue, just test for a = b in the beginning of the program and jump to current line # 31 if true. Or use this equivalent expression for s:

s = [h²(9h² + 50) - 480]/[h²(6h² + 40) - 320]