SPHERICAL AND CYLINDRICAL COORDINATES MATH 200 GOALS Be able to - - PowerPoint PPT Presentation

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

• ff 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

cos φ = 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 sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ

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 π 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 = ρ sin φ cos θ

y = ρ sin φ sin θ z = ρ cos φ = ⇒

• x = 5 sin 2π

3 cos π 3

y = 5 sin 2π

3 sin π 3

z = 5 cos 2π

3

= ⇒

• x = 5

√

3 2

1

2

• y = 5

√

3 2

√

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, θ, z) =

• 5

√ 3 2 , π 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!