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!