SPHERICAL AND CYLINDRICAL COORDINATES
MATH 200 WEEK 8 - FRIDAY
MATH 200
GOALS
▸ Be able to convert between the three different coordinate systems in 3-Space: rectangular, cylindrical, spherical
▸ Develop a sense of which surfaces are best represented by which coordinate systems
MATH 200
CYLINDRICAL COORDINATES▸ Cylindrical coordinates are
basically polar coordinates plus z
▸ Coordinates: (r,θ,z)
▸ x = rcosθ
▸ y = rsinθ
▸ z = z
▸ r2 = x2 + y2
▸ tanθ = y/x
x
y
z
θ
θ
r
r
JUST LIKE 2D POLAR
MATH 200
SURFACES▸ Let’s look at the types of
surfaces we get when we set polar coordinates equal to constants.
▸ Consider the surface r = 1
▸ This is the collection of all points 1 unit from the z-axis
▸ Or, using our transformation equations, it’s the same as the surface x2+y2=1
MATH 200
▸ How about θ=c?
▸ This is the set of all points for which the θ component is fixed, but r and z can be anything.
▸ Or, since tanθ = c, we have y/x = c
▸ y = cx is a plane
θ
MATH 200
SPHERICAL COORDINATES▸ Coordinates: (ρ, θ, φ)
▸ ρ: distance from origin to point
▸ θ: the usual θ (measured off of positive x-axis)
▸ φ: angle measured from positive z-axis
x
y
z
ρ
φ
θ
θ
MATH 200
CONVERTING ▸ Let’s start with ρ:
▸ From the distance formula/Pythagorus we get ρ2=x2+y2+z2
▸ We already know that tanθ=y/x
▸ Lastly, since z = ρcosφ, we have x
y
z
ρ
φ
θ
θ
cosφ =z!
x2 + y2 + z2
z
For φ, z is the adjacent side
MATH 200
▸ Going the other way around is a little trickier…
▸ From cylindrical/polar, we have
x
y
z
ρ
r
φ
θ
θ
!x = r cos θ
y = r sin θ
▸ Notice that r = ρsinφ. So,⎧⎪⎨
⎪⎩
x = ρ sinφ cos θ
y = ρ sinφ sin θ
z = ρ cosφ
r is the opposite side to φ
MATH 200
SURFACES IN SPHERICAL▸ Let’s start with ρ=constant
▸ What does ρ=2 look like?
▸ It’s all points 2 units from the origin
▸ Also, if ρ=2, then ρ2=4. So, x2+y2+z2=4
▸ It’s a sphere!
MATH 200
▸ How about φ=constant?
▸ Let φ = π/3.
▸ From the conversion formula we have
cos�
3=
z�x2 + y2 + z2
1
2=
z�x2 + y2 + z2
▸ Let’s simplify some
�x2 + y2 + z2 = 2z
x2 + y2 + z2 = 4z2
x2 + y2 = 3z2
z2 =1
3x2 +
1
3y2
▸ Recall: z2=x2+y2 is a double cone
▸ Multiplying the right-hand side by 1/3 just stretches it
MATH 200
▸ For spherical coordinates, we restrict ρ and φ
▸ ρ≥0 and 0≤φ≤π
▸ So, φ=π/3 is just the top of the cone
���
��
x = � sin � cos �
y = � sin � sin �
z = � cos �
=�
���
��
x = 5 sin 2�3 cos �
3
y = 5 sin 2�3 sin �
3
z = 5 cos 2�3
=�
����
���
x = 5��
32
� �12
�
y = 5��
32
���3
2
�
z = 5�� 1
2
�
MATH 200
EXAMPLE 1: CONVERTING POINTS▸ Consider the point (ρ,θ,φ) = (5, π/3, 2π/3)
▸ Convert this point to rectangular coordinates
▸ Convert this point to cylindrical coordinates
▸ Rectangular
▸ In rectangular coordinates, we have
(x, y, z) =
!5√3
4,15
4,−5
2
"
MATH 200
▸ Polar:
r2 = x2 + y2
r2 =
�5�
3
4
�2
+
�15
4
�2
r2 =75
16+
225
16
r2 =300
16
r =10
�3
4
r =5�
3
2
▸ We already have z and θ:
(r, �, z) =
�5�
3
2,�
3, �5
2
�
MATH 200
ρ
θ
φ
MATH 200
EXAMPLE 2: CONVERTING SURFACES▸ Express the surface x2+y2+z2=3z in
both cylindrical and spherical coordinates
▸ Cylindrical
▸ Using the fact that r2=x2+y2, we have r2+z2=3z
▸ Spherical
▸ Using the facts that ρ2=x2+y2+z2 and z = ρcosφ, we get that ρ2=3ρcosφ
▸ More simply, ρ=3cosφ
MATH 200
EXAMPLE 3: CONVERTING MORE SURFACES▸ Express the surface
ρ=3secφ in both rectangular and cylindrical coordinates
▸ We can rewrite the equation as ρcosφ=3
▸ This is just z = 3 (a plane)
▸ Conveniently, this is exactly the same in cylindrical!