1. The Shadow Curve
BY: STEPHEN EBERT
ADVISOR: DR. CHUNGWU HO

2. Purpose
Determine the shadow curve of a rod as the sun moves
given the latitude of the location and the inclination of
the sun.

3. Geometry – Earth & Tangent plane
Z
P
Sun
t
180 - a
latitude) 

Sun
x
y
a
Z
z

4. Concepts, Definitions & Variables
•Angle rho (ρ) is the latitude of the Earth
When it descends from the Celestial pole. b = 90 – ρ
•The sun is denoted with a dot in a circle.
•P is the Celestial pole
•Delta (δ) is the angle of the sun measured from the equatorial
plane.
•δ & ρ are constant.
Z
P
Sun
t
180 - a
latitude) 

o

5. Tangent Plane
•X is the North(+) and South(-); while Y is East(-) and West(+).
•z(lower case) is the distance from the sun to Z.
•Angle a is the angle the shadow casts onto the xy plane. This is
measured from the X axis.
•The shadow’s projection from the sun rays can be approximated on
a plane tangent to the Earth.
Sun
x
y
a
Z
z

6. Concepts, Definitions & Variables(Cont.)
•a is the angle between the shadow of the rod and the North
(i.e. X axis).
•Z is the zenith of the rod. Z is independent from the Sun.
•t = time
•z and t are variables
Z
P
Sun
t
180 - a
latitude) 


7. Spherical Trigonometry
Sine:
𝑆𝑖𝑛(𝐴)
𝑆𝑖𝑛(𝑎)
=
𝑆𝑖𝑛(𝐵)
𝑆𝑖𝑛(𝑏)
=
𝑆𝑖𝑛(𝐶)
𝑆𝑖𝑛(𝑐)
Cosine: Cos(a) = Cos(b)Cos(c) + Sin(b)Sin(c)Cos(A)
Cotangent: Cos(b)Cos(C) = Cot(a)Sin(b) – Cot(A)Sin(C)

8. Spherical Trigonometry
Law of Sines: Sin(a)Sin(z) = Sin(p)Sin(t) … [1]
Law of Cosines: Cos(z) = Cos(b)Cos(p) +Sin(b)Sin(p)Cos(t) … [2]
Law of Cotangents: -Cos(b)Cos(a) = Sin(b)Cot(z)- Sin(a)Cot(t) … [3]
Z
P
Sun
t
latitude) 
180 - a
o
z

9. Law of Sines: Sin(a)Sin(z) = Sin(p)Sin(t) [1]
Law of Cosines: Cos(z) = Cos(b)Cos(p) +Sin(b)Sin(p)Cos(t) [2]
Law of Cotangents: -Cos(b)Cos(a) = Sin(b)Cot(z)- Sin(a)Cot(t) [3]
Recall:
[1]
[2]
=
𝑆𝑖𝑛 𝑎 𝑆𝑖𝑛(𝑍)
𝐶𝑜𝑠(𝑍)
=
𝑆𝑖𝑛 𝑃 𝑆𝑖𝑛(𝑡)
𝐶𝑜𝑠 𝑏 𝐶𝑜𝑠 𝑃 + 𝑆𝑖𝑛 𝑏 Sin p Cos(t)
[1]
[3]
1
𝐶𝑜𝑠(𝑃)
1
𝐶𝑜𝑠(𝑃)
:
𝑇𝑎𝑛 𝑃 𝑆𝑖𝑛(𝑡)
𝐶𝑜𝑠 𝑡 𝑇𝑎𝑛 𝑃 𝑆𝑖𝑛 𝑏 +𝐶𝑜𝑠(𝑏)
→ Ψ(psi)
𝐶𝑜𝑠 𝑏 = 𝐶𝑜𝑠(90 − ρ) = Sin(ρ)
Sin 𝑏 = 𝑆𝑖𝑛(90 − ρ) = Cos(ρ)
b = 90 −ρ
o
o

10. [1]
[2]
=
𝑆𝑖𝑛 𝑎 𝑆𝑖𝑛(𝑧)
𝐶𝑜𝑠(𝑍)
=
𝑆𝑖𝑛 𝑃 𝑆𝑖𝑛(𝑡)
𝐶𝑜𝑠 𝑏 𝐶𝑜𝑠 𝑃 + 𝑆𝑖𝑛 𝑏 𝑆𝑖𝑛 𝑃 𝐶𝑜𝑠(𝑡)
Recall:
Γ becomes the Following: Sin(ρ)Cos(a)Tan(z) = -Cos(ρ) + Cot(t)Tan(z)Sin(a)
X = Cos(a)Tan(z)
Y = Sin(a)Tan(z)
i = Sin(ρ)
o = Cos(ρ)
Q = Tan(p)
Γ = iX = -o + YCot(t)

11. Wcan transform This equation to Y =
𝑄𝑆𝑖𝑛(𝑡)
𝑖+𝑜𝑄𝐶𝑜𝑠(𝑡)
Need to relate X & Y to a function
Γ = Y = Tan(t)(o+iX) =
𝑄𝑆𝑖𝑛(𝑡)
𝑖 + 𝑜𝑋𝐶𝑜𝑠𝑡(𝑡)
𝑄𝐶𝑜𝑠 𝑡 = (𝑜 + 𝑖𝑋)(𝑖 + 𝑜𝑄𝐶𝑜𝑠(𝑡))
QCos(t) =
𝑜+𝑖𝑋
𝑖 −𝑜𝑋

12. QSin(t) = Y( i + oQCos(t))
=
𝑌{ 𝑖 + 𝑜 𝑜 + 𝑖𝑋 }
𝑖 − 𝑜𝑋
=
𝑌{ 𝑖2
+ 𝑜2
}
𝑖 − 𝑜𝑋 i = Sin(ρ)
o = Cos(ρ)
∴ QSin(t) =
𝑌
𝑖 −𝑜𝑋

13. Now apply 𝑆𝑖𝑛2
(𝑡)+ 𝐶𝑜𝑠2
(𝑡) = 1
Since QCos(t) =
𝑜+𝑖𝑋
𝑖 −𝑜𝑋
& 𝑄𝑆𝑖𝑛 𝑡 =
𝑌
𝑖 −𝑜𝑥
(𝑜+𝑖𝑥)2
𝑄2(𝑖 −𝑜𝑥)2 +
𝑌2
𝑄2(𝑖 −𝑜𝑋2)
= 1  𝑌2 = 𝑄2(𝑖 − 𝑜𝑋)2 − (𝑜 + 𝑖𝑋)2

14. Equation
𝑌2
= 𝑄2
(𝑖 − 𝑜𝑋)2
− (𝑜 + 𝑖𝑋)2

15. Alternative Equation
By substituting the values:
i = Sin(ρ)
o = Cos(ρ)
Q = Tan(p)
We obtain
𝑌2 = 2Tan(p)X – (1 - -
𝐶𝑜𝑠2(ρ)
𝐶𝑜𝑠2(𝑝)
) 𝑋2

16. Meaning
𝑌2
= 2Tan(p)X – (1 - -
𝐶𝑜𝑠2(ρ)
𝐶𝑜𝑠2(𝑝)
) 𝑋2
◦Case I: ρ > 𝑝  Ellipse
◦Case II: ρ = p  Parabola
◦Case III: ρ < p  Hyperbola