1. Cylindrical and Spherical Coordinates
Azmal Thahireen John Thai

2. (r, ) First, a review of Polar Coordinates:
Angles are measured from the positive x axis.
Points are represented by a radius and an angle
radius
angle
(r, )
To plot the point
First find the angle
Then move out along the terminal side 5

3. Now, a Review of 3D Coordinatesz
(3,2,4) 4 y
(3,2,0) 2 x 3

4. Representing 3D points in Cylindrical Coordinates.
Now combine polar representations… r 

5. Representing 3D points in Cylindrical Coordinates.
With 3D Coordinates! r 

6. Representing 3D points in Cylindrical Coordinates.
With 3D Coordinates!  r

7. Representing 3D points in Cylindrical Coordinates.
With 3D Coordinates!  r

8. Representing 3D points in Cylindrical Coordinates.
With 3D Coordinates!  r

9. Representing 3D points in Cylindrical Coordinates.
With 3D Coordinates!  r

10. Representing 3D points in Cylindrical Coordinates.
With 3D Coordinates!  r

11. Representing 3D points in Cylindrical Coordinates.
(r,,z)  r  r

12. Converting between rectangular and Cylindrical Coordinates
Cylindrical to rectangular
No real surprises here!  r
(r,,z)
Rectangular to Cylindrical

13. Converting Points Converting between Cylindrical and Rectangular is
Similar to Polar to Rectangular

14. Converting Equations Similarly, entire equations can be converted
using the aforemented rules.

15. 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. 

16. Representing 3D points in Spherical Coordinates
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.

17. Representing 3D points in Spherical Coordinates
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.

18. Representing 3D points in Spherical Coordinates
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.

19. Representing 3D points in Spherical Coordinates
And “drop it down” onto the xy-plane.

20. Representing 3D points in Spherical Coordinates
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 .
Note that this is the same angle as the  in cylindrical coordinates!

21. 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. 

22. Converting Between Rectangular and Spherical Coordinates
First 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
(x,y,z) r  z 

23. Converting Between Rectangular and Spherical Coordinates
(x,y,z) r
Spherical to rectangular  z 

24. Converting from Spherical to Rectangular Coordinates
Rectangular to Spherical
(x,y,z) r  z 

25. Convert the following from Rectangular to Spherical
Conversions
Convert the following from Rectangular to Spherical

26. Special Thanks To Kenyon University Harvard University
Stanford University
Arizona State
Purdue University
Michigan State
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