spherical-trig good.ppt
Post on 02-Jan-2016
63 Views
Preview:
DESCRIPTION
TRANSCRIPT
A
B
C
a
b
c
Rule 1.
The sum of the lengths of a spherical triangle's sides is always less than 360º.
A
B
C
a
b
c
Rule 3.
The sum of the lengths of any two sides is greater than the length of the third side.
A
B
C
a
b
c
Rule 4.
If a side (or angle) differs from 90º by more than another side (or angle), then it is in the same quadrant as its opposite angle (or side).
In other words, they are either both greater than 90º or both less than 90º.
A
B
C
a
b
c
Rule 5.
Half the sum of two sides of a spherical triangle must be in the same quadrant as half the sum of the two opposite angles.
A
B
C
a
b
c
Cosine Law
The Fundamental LawOf Spherical Trigonometry
cos cos cos sin sin cosc a b a b C
A
B
C
a
b
c
Cosine Law
The Fundamental LawOf Spherical Trigonometry
cos cos cos sin sin cosc a b a b C Acbcba cossinsincoscoscos
cos cos cos sin sin cosb a c a c B
A
a
b
B
aAbB sin/sinsinsin
sin / sin sin / sin sin / sina A b B c C
But… Is B acute or obtuse??? Appeal to Rules 4 and 5
Most Difficult Case
P
Z
S
az
t
Cosine law gives us declination from altitude, azimuth and latitude.
Then sine law gives us hour angle from declination, azimuth and altitude.
P
Z
S
az
t
Alternatively, use the Four Element equation to obtain the hour angle directly from altitude, azimuth and latitude.
Then use the sine law or the cosine law to find the declination.
Reducing A PlaneTo The EquivalentHorizontal
Begin with a horizontal plane at latitude Spike the celestial sphere
Reducing A PlaneTo The EquivalentHorizontal
Incline the plane by 80d, dragging the spike within the sphere’s surface.
i
Reducing A PlaneTo The EquivalentHorizontal
Decline the plane by 40 d. This is a rotation about the vertical at your site.
i
d
d
Reducing A PlaneTo The EquivalentHorizontal
i
d
d
h
C
Lat. 42
Inc. 80
Dec. 40
Eq. Lat. -26.4
Inc. Merid. 44.9
Slope -32.2
top related