1. Transformations Dr. Hugh Blanton ENTC 3331

2. Dr. Blanton - ENTC 3331 - Coordinate Transformations 2 / 29 It is important to compare the units that are used in Cartesian coordinates with the units that are used in cylindrical coordinates and spherical coordinates.

3. Dr. Blanton - ENTC 3331 - Coordinate Transformations 3 / 29 In Cartesian coordinates, (x, y, z), all three coordinates measure length and, thus, are in units of length. In cylindrical coordinates, ( r, , z ), two of the coordinates – r and z -- measure length and, thus, are in units of length but the coordinate  measures angles and is in "units" of radians.

4. Dr. Blanton - ENTC 3331 - Coordinate Transformations 4 / 29 The most important part of the preceding slide is the quotation marks around the word "units" – radians are a dimensionless quantity – radians are a dimensionless quantity That is, they do not have associated units.

5. Dr. Blanton - ENTC 3331 - Coordinate Transformations 5 / 29 The formulas below enable us to convert from cylindrical coordinates to Cartesian coordinates. Notice the units work out correctly. The right side of each of the first two equations is a product in which the first factor is measured in units of length and the second factor is dimensionless.

6. Dr. Blanton - ENTC 3331 - Coordinate Transformations 6 / 29 Cylindrical-to-Cartesian z y x  r (x,y,z) = (r, ,z)

7. Dr. Blanton - ENTC 3331 - Coordinate Transformations 7 / 29 Cartesian-to-Cylindrical z y x  r (x,y,z) = (r, ,z) x y z = z

8. Dr. Blanton - ENTC 3331 - Coordinate Transformations 8 / 29 Find the cylindrical coordinates of the point whose Cartesian coordinates are (1, 2, 3)

9. Dr. Blanton - ENTC 3331 - Coordinate Transformations 9 / 29 Cylindrical Coordinates -- Answer 1

10. Dr. Blanton - ENTC 3331 - Coordinate Transformations 10 / 29 Find the Cartesian coordinates of the point whose cylindrical coordinates are (2,  /4, 3)

11. Dr. Blanton - ENTC 3331 - Coordinate Transformations 11 / 29 Cylindrical Coordinates -- Answer 2

12. Dr. Blanton - ENTC 3331 - Coordinate Transformations 12 / 29 Spherical coordinates consist of the three quantities (R 

13. Dr. Blanton - ENTC 3331 - Coordinate Transformations 13 / 29 First there is R. This is the distance from the origin to the point. Note that R  0.

14. Dr. Blanton - ENTC 3331 - Coordinate Transformations 14 / 29 Next there is . This is the same angle that we saw in cylindrical coordinates. It is the angle between the positive x- axis and the line denoted by r (which is also the same r as in cylindrical coordinates). There are no restrictions on 

15. Dr. Blanton - ENTC 3331 - Coordinate Transformations 15 / 29 Finally there is . This is the angle between the positive z- axis and the line from the origin to the point. We will require 0 ≤  ≤ .

16. Dr. Blanton - ENTC 3331 - Coordinate Transformations 16 / 29 In summary, R is the distance from the origin to the point,  is the angle that we need to rotate down from the positive z-axis to get to the point and  is how much we need to rotate around the z-axis to get to the point.

17. Dr. Blanton - ENTC 3331 - Coordinate Transformations 17 / 29 We should first derive some conversion formulas. Let’s first start with a point in spherical coordinates and ask what the cylindrical coordinates of the point are.

18. Dr. Blanton - ENTC 3331 - Coordinate Transformations 18 / 29 Spherical-to-Cylindrical z y x  r (R  ) = (r, ,z) x y R   = 

19. Dr. Blanton - ENTC 3331 - Coordinate Transformations 19 / 29 Cylindrical-to-Spherical z y x  r (R  ) = (r, ,z) x y R   = 

20. Dr. Blanton - ENTC 3331 - Coordinate Transformations 20 / 29 Cartesian-to-Spherical z y x  r (R  ) = (r, ,z) x y R   =  Recall from Cartesian-to- cylindrical transformations:

21. Dr. Blanton - ENTC 3331 - Coordinate Transformations 21 / 29 Cartesian-to-Spherical z y x  r (R  ) = (r, ,z) x y R 

22. Dr. Blanton - ENTC 3331 - Coordinate Transformations 22 / 29 Spherical-to-Cartesian z y x  r (R  ) = (r, ,z) x y R 

23. Dr. Blanton - ENTC 3331 - Coordinate Transformations 23 / 29 Converting points from Cartesian or cylindrical coordinates into spherical coordinates is usually done with the same conversion formulas. To see how this is done let’s work an example of each.

24. Dr. Blanton - ENTC 3331 - Coordinate Transformations 24 / 29 Perform each of the following conversions. (a) Convert the point from cylindrical to spherical coordinates. (b) Convert the point from Cartesian to spherical coordinates.

25. Dr. Blanton - ENTC 3331 - Coordinate Transformations 25 / 29 Solution (a) Convert the point from cylindrical to spherical coordinates. We’ll start by acknowledging that is the same in both coordinate systems.

26. Dr. Blanton - ENTC 3331 - Coordinate Transformations 26 / 29 Next, let’s find R.

27. Dr. Blanton - ENTC 3331 - Coordinate Transformations 27 / 29 Finally, let’s get . To do this we can use either the conversion for r or z. We’ll use the conversion for z.

28. Dr. Blanton - ENTC 3331 - Coordinate Transformations 28 / 29 So, the spherical coordinates of this point will are

29. Dr. Blanton - ENTC 3331 - Coordinate Transformations 29 / 29