Swiss Micros DM32: Spherical Triangle ft.
Law of Cosines
Introduction
The
state file spheretri.d32 is about solving triangles on the spherical
space.
The
programs solve spherical triangles in two common problems: SSS
(side-side-side, really arc lengths) and SAS (side-angle-side). All
the inputs are in decimal degrees.
Also
calculated are the surface area and perimeter, both in radians. The
radius is assumed to be 1.
Surface
Area = ( A° + B° + C° ) * π / 180 – π = A + B + C – π
Perimeter
= ( X° + Y° + Z°) * π / 180 = X + Y + Z
The
sum of the angles (A, B, C) must be greater than 180° (π radians).
Due to this requirement, in solving for angles and sides, the Law of
Cosines will be used in each instance. The Law of Sines is only
advised to check ratios.
Equation
listing
Law
of Sines – can be used as a check on triangles:
SIN(A)÷SIN(X)=SIN(B)÷SIN(Y)
Law
of Cosines – two equations:
COS(Z)=COS(X)×COS(Y)+SIN(X)×SIN(Y)×COS(C)
COS(C)=-COS(A)×COS(B)+SIN(A)×SIN(B)×COS(Z)
Here
the variables are general place holders.
Program
Listing
Labels:
Label
H: help program
Label
I: Initialization routine. Sets the angles mode to degrees and
clears the variables.
Label
C: Starts the solve spherical triangle routine: given the three arc
lengths X, Y, and Z.
Label
Z: Starts the solve spherical triangle routine: given the arc
lengths X and Y and and the angle between the arcs, angle C
Label
F: Routine to solve for angles A and B, perimeter, and area
General
Instructions
To
start a new problem, execute program I.
To
solve a spherical triangle given the sides (arc lengths), execute
program C. (SSS)
To
solve a spherical triangle given two sides and the internal angle,
execute program Z. (SAS)
This
program also solves for the surface area, assuming a radius of 1, and
perimeter of the triangle.
Program
Code
H01
LBL H
H02
SF 10
H03
EQN: A N G L E _ A _ B _ C
H04
EQN: S I D E S _ X _ Y _ Z
H05
EQN: X E Q _ C _ S S S
H06
EQN: X E Q _ Z _ S A S
H07
EQN: E Q N S _ A R E _ S I N E
H08
EQN: A N D _ C O S I N E _ L A W S
H09
CF 10
H10
RTN
I01
LBL I
I02
DEG
I03
CLVARS
I04
CLx
I05
RTN
C01
LBL C
C02
INPUT X
C03
INPUT Y
C04
INPUT Z
C05
RCL Z
C06
COS
C07
RCL X
C08
COS
C09
RCL Y
C10
COS
C11
×
C12
–
C13
RCL X
C14
SIN
C15
RCL Y
C16
SIN
C17
×
C18
÷
C19
ACOS
C20
STO C
C21
VIEW C
C22
XEQ F
C23
RTN
Z01
LBL Z
Z02
INPUT X
Z03
INPUT Y
Z04
INPUT C
Z05
RCL X
Z06
COS
Z07
RCL Y
Z08
COS
Z09
×
Z10
RCL X
Z11
SIN
Z12
RCL Y
Z13
SIN
Z14
×
Z15
RCL C
Z16
COS
Z17
×
Z18
+
Z19
ACOS
Z20
STO Z
Z21
VIEW Z
Z22
XEQ F
Z23
RTN
F01
LBL F
F02
RCL X
F03
COS
F04
RCL Z
F05
COS
F06
RCL Y
F07
COS
F08
×
F09
–
F10
RCL Z
F11
SIN
F12
RCL Y
F13
SIN
F14
×
F15
÷
F16
ACOS
F17
STO A
F18
RCL Y
F19
COS
F20
RCL X
F21
COS
F22
RCL Z
F23
COS
F24
×
F25
–
F26
RCL X
F27
SIN
F28
RCL Z
F29
SIN
F30
×
F31
÷
F32
ACOS
F33
STO B
F34
RCL A
F35
RCL+ B
F36
RCL+ C
F37
→RAD
F38
π
F39
–
F40
STO R
F41
RCL X
F42
RCL+ Y
F43
RCL+ Z
F44
→RAD
F45
STO P
F46
VIEW A
F47
VIEW B
F48
VIEW R
F49
VIEW P
F50
RTN
You
can download the DM32 state file here:
https://drive.google.com/file/d/1qX-y2G5sCOmm4ktmZbnGoPI3uzrx6IfF/view?usp=sharing
Examples (FIX 5)
SSS
Problem (LBL C)
X =
18.66°
Y =
20.49°
Z =
19.95°
Results:
C =
62.04726°
A =
55.92702°
B =
64.98954°
R =
0.05173 radians (surface area)
P =
1.03149 radians (perimeter)
SAS
Problem (LBL Z)
X =
17.00 °
Y =
23.32°
C =
64.55°
Results:
Z =
21.88733°
A =
45.08768°
B =
73.51096 °
R =
0.05495 radians (surface area)
P =
1.08572 radians (perimeter)
Sources
Wikipedia.
“Spherical Triangle” Updated April 9, 2024. Retrieved April
11, 2024.
https://en.wikipedia.org/wiki/Spherical_trigonometry#:~:text=Spherical%20trigonometry%20is%20the%20branch,sphere%2C%20geodesics%20are%20great%20circles.
Gray,
Glen. “Spherical Trigonometry – An Introduction and Basic
Theorems” Video. February 12, 2023. Retrieved April 11, 2024.
Note:
The blog will be posted on Saturdays only on June and July 2024.
Eddie
All
original content copyright, © 2011-2024. Edward Shore.
Unauthorized use and/or unauthorized distribution for commercial
purposes without express and written permission from the author is
strictly prohibited. This blog entry may be distributed for
noncommercial purposes, provided that full credit is given to the
author.
