Cylindrical and Spherical Coordinates

Cylindrical and Spherical Coordinates

Azmal Thahireen John Thai

First, a review of Polar Coordinates:

Angles are measured from the positive x axis.

Points are represented by a radius and an angle

(r, )

radius angle

To plot the point

4,5

First find the angle

Then move out along the terminal side 5

Now, a Review of 3D Coordinates

x

y

3

2

z

(3,2,0)

4(3,2,4)

Representing 3D points in Cylindrical Coordinates.

r

Now combine polar representations…


r

With 3D Coordinates!

r


r

(r,,z)

Converting between rectangular and Cylindrical Coordinates

r

r

(r,,z)cos( )

sin( )

x r

y r

z z

Rectangular to Cylindrical

2 2 2

tan( )

r x y

y

xz z

Cylindrical to rectangularNo real surprises here!

Converting PointsConverting between Cylindrical and Rectangular is

Similar to Polar to Rectangular

Converting EquationsSimilarly, entire equations can be converted

using the aforemented rules.

Representing 3D points in Spherical Coordinates

(x,y,z)

We start with a point (x,y,z) given in rectangular coordinates.

Then, measuring its distance from the origin, we locate it on a sphere of radius centered at the origin.

Next, we have to find a way to describe its location on the sphere.


We use a method similar to the method used to measure latitude and longitude on the surface of the Earth.

We find the great circle that goes through the “north pole,” the “south pole,” and the point.


We measure the latitude or polar angle starting at the “north pole” in the plane given by the great circle.

This angle is called . The range of this angle is

Note:

all angles are measured in radians, as always.

0 .


We use a method similar to the method used to measure latitude and longitude on the surface of the Earth.

Next, we draw a horizontal circle on the sphere that passes through the point.


And “drop it down” onto the xy-plane.


We measure the longitude or azimuthal angle on this circle, starting at the positive x-axis and rotating toward the positive y-axis.

The range of the angle is

Angle is called .

0 2 .

Note that this is the same angle as the in cylindrical coordinates!

Finally, a Point in Spherical Coordinates!

( , ,)

Our designated point on the sphere is indicated by the three spherical coordinates ( , , ) ---(radial distance, latitude angle, polar angle).

Please note that this notation is not at all standard and varies from author to author and discipline to discipline.

Converting Between Rectangular and Spherical Coordinates

(x,y,z)

z

rFirst note that if r is the usual cylindrical coordinate for (x,y,z)

we have a right triangle with •angle , •hypotenuse , and •legs r and z.

It follows that

sin( ) cos( ) tan( )r z r

z

Converting Between Rectangular and Spherical Coordinates

(x,y,z)

z

r

cos( ) sin( )cos( )

sin( ) sin( )sin( )

cos( )

x r

y r

z

Spherical to rectangular

Converting from Spherical to Rectangular Coordinates

(x,y,z)

z

rRectangular to Spherical

2 2 2

2 2

2 2 2

tan( )

tan( )

cos( )

x y z

y

x

x yr

z zz z

x y z

ConversionsConvert the following from Rectangular to Spherical

Special Thanks To

Kenyon UniversityHarvard UniversityStanford University

Arizona State Purdue University

Michigan StateMathXL.com

Cylindrical and Spherical Coordinates

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