The Vhs Triangle
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THE VHS TRIANGLE (Vertical, Horizontal, and
Slope distances)
Math : (opposite (a), adjacent (b), hypotenuse (c) sides)
zenith 0± reference direction for zenith angles PYTHAGORAS: S2 = H2 + V2
(right triangle)
z S V D >Horizon 0±
H Nadir
Zenith Nadir line, a vertical line defined by gravity
reference direction for D
A vertical angle is an angle observed in a vertical plane
D = SLOPE ANGLE = A vertical angle measured from the horizon z
(0± to 90±) Convention: If above the horizon, an elevation angle and +ve If below the horizon, a depression angle and –ve = ZENITH ANGLE = A vertical angle measured from the zenith ( 0± to 180±, most instruments show complete circle: 0± to 360± but can be Reduced to between 0± to 180± i.e. if greater than 180± subtract from 360±)
In the VHS triangle, which internal angle is = to z ∠ ?
sketch it in.
a and z are complementary angles ( their sum should = 90±) V/H = Tan(a) = Cot(z)
V/S = Sin(a) = Cos(z)
H=S*Cos(a)=S*Sin(z)=◊(S2 – V2)
H/S = Cos(a) = Sin(z)
V=S*Sin(a)=S*Cos(z)=◊( S2 – H2)
S V
S
a P 100
H Remember from similar triangles: Proportional relationship! \ The ratio V/H = P/100 = slope (gradient or rate of grade) \ If P represents Percentage gradient or percent slope (i.e. slope times 100%) P: The amount of units of rise per 100 horizontal units (-P for fall) Then: P/100 = Tan(a)
Then: S*Cos(a) = S*Cos(Atan(P/100)) = H Then: S*Sin(a) = S*Sin(Atan(P/100)) = V
If slope is negative (falling) Then P, a and V are also negative and z > 90± Also: Σ = ◊(P2 + 1002)
and S/ Σ = H/100
\ H= (100)*S/ Σ = (100)*S/◊(P2 + 1002) And so forth.
Slope distances)
Math : (opposite (a), adjacent (b), hypotenuse (c) sides)
zenith 0± reference direction for zenith angles PYTHAGORAS: S2 = H2 + V2
(right triangle)
z S V D >Horizon 0±
H Nadir
Zenith Nadir line, a vertical line defined by gravity
reference direction for D
A vertical angle is an angle observed in a vertical plane
D = SLOPE ANGLE = A vertical angle measured from the horizon z
(0± to 90±) Convention: If above the horizon, an elevation angle and +ve If below the horizon, a depression angle and –ve = ZENITH ANGLE = A vertical angle measured from the zenith ( 0± to 180±, most instruments show complete circle: 0± to 360± but can be Reduced to between 0± to 180± i.e. if greater than 180± subtract from 360±)
In the VHS triangle, which internal angle is = to z ∠ ?
sketch it in.
a and z are complementary angles ( their sum should = 90±) V/H = Tan(a) = Cot(z)
V/S = Sin(a) = Cos(z)
H=S*Cos(a)=S*Sin(z)=◊(S2 – V2)
H/S = Cos(a) = Sin(z)
V=S*Sin(a)=S*Cos(z)=◊( S2 – H2)
S V
S
a P 100
H Remember from similar triangles: Proportional relationship! \ The ratio V/H = P/100 = slope (gradient or rate of grade) \ If P represents Percentage gradient or percent slope (i.e. slope times 100%) P: The amount of units of rise per 100 horizontal units (-P for fall) Then: P/100 = Tan(a)
Then: S*Cos(a) = S*Cos(Atan(P/100)) = H Then: S*Sin(a) = S*Sin(Atan(P/100)) = V
If slope is negative (falling) Then P, a and V are also negative and z > 90± Also: Σ = ◊(P2 + 1002)
and S/ Σ = H/100
\ H= (100)*S/ Σ = (100)*S/◊(P2 + 1002) And so forth.
