Transformations Dr. Hugh Blanton ENTC 3331

Dr. Blanton - ENTC 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.

Dr. Blanton - ENTC 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.

Dr. Blanton - ENTC 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.

Dr. Blanton - ENTC 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.

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

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

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

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

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

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

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

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

Dr. Blanton - ENTC 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 

Dr. Blanton - ENTC 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 ≤  ≤ .

Dr. Blanton - ENTC 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.

Dr. Blanton - ENTC 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.

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

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

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

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

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

Dr. Blanton - ENTC 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.

Dr. Blanton - ENTC 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.

Dr. Blanton - ENTC 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.

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

Dr. Blanton - ENTC 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.

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

Dr. Blanton - ENTC Coordinate Transformations 29 / 29