Overview
"ELK" calculates Gaussian curvature KG & mean curvature KM of an ellipsoid (E): x2 / a2 + y2 / b2 + z2 / c2 = 1 at a point M ( x , y , z )
Formulae:
KG = 1 / [ a2 b2 c2 ( x2 / a4 + y2 / b4 + z2 / c4 )2 ]
KM = (1/2) Ñ.n where n = ( x2 / a2 , y2 / b2 , z2 / c2 ) / ( x2 / a4 + y2 / b4 + z2 / c4 )2 & Ñ = ( ¶/¶x , ¶/¶y , ¶/¶z ) ¶ = partial derivative.
Program Listing
-The semi-axes of the triaxial ellipsoid are to be stored in registers R01-R02-R03
Data Registers: R00: temp ( Registers R01 thru R03 are to be initialized before executing "ELK" )
• R01 = a R04 = x R07 to R09: temp
• R02 = b R05 = y
• R03 = c R06 = z
Flags: /
Subroutines: /
| 01 LBL "ELK" 02 STO 04 03 RCL 01 04 X^2 05 / 06 X^2 07 STO 07 08 X<>Y 09 STO 05 10 RCL 02 11 X^2 12 / |
13 X^2 14 STO 08 15 + 16 X<>Y 17 STO 06 18 RCL 03 19 X^2 20 / 21 X^2 22 STO 09 23 + 24 STO 00 |
25 RCL 01 26 RCL 02 27 RCL 03 28 * 29 * 30 * 31 X^2 32 1/X 33 RCL 01 34 X^2 35 1/X 36 ST* 07 |
37 RCL 02 38 X^2 39 1/X 40 ST* 08 41 RCL 03 42 X^2 43 1/X 44 ST* 09 45 + 46 + 47 RCL 00 48 * 49 RCL 07 |
50 - 51 RCL 08 52 - 53 RCL 09 54 - 55 RCL 00 56 1.5 57 Y^X 58 / 59 2 60 / 61 X<>Y 62 END |
( 75 bytes / SIZE 010 )
| STACK | INPUTS | OUTPUTS |
|
Z |
z |
/ |
| Y | y | KM |
| X | x | KG |
Where KM = mean curvature & KG = Gaussian curvature
Example: Ellipsoid: x2 / 49 + y2 / 36 + z2 / 25 = 1 M ( 3.896191634 , 3.511764224 , 2.948002121 )
7 STO 01
6 STO 02
5 STO 03
2.948002121 ENTER^
3.511764224 ENTER^
3.896191634 XEQ "ELK" >>>> KG = 0.025631779
RDN KM = 0.163109816
Note:
-"ELK" does not check that M really belongs to (E)
Reference:
[1] https://mathworld.wolfram.com/Ellipsoid.html