Transformations Dr Hugh Blanton ENTC 3331 It is
Transformations Dr. Hugh Blanton ENTC 3331
• 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 3331 - Coordinate Transformations 2 / 29
• In Cartesian coordinates, (x, y, z), all three coordinates measure length and, thus, are in units of length. • In cylindrical coordinates, (r, f, z), two of the coordinates – r and z -- measure length and, thus, are in units of length but • the coordinate f measures angles and is in "units" of radians. Dr. Blanton - ENTC 3331 - Coordinate Transformations 3 / 29
• The most important part of the preceding slide is the quotation marks around the word "units" – • radians are a dimensionless quantity – • That is, they do not have associated units. Dr. Blanton - ENTC 3331 - Coordinate Transformations 4 / 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 3331 - Coordinate Transformations 5 / 29
Cylindrical-to-Cartesian z (x, y, z) = (r, f, z) y f r x Dr. Blanton - ENTC 3331 - Coordinate Transformations 6 / 29
Cartesian-to-Cylindrical z z=z (x, y, z) = (r, f, z) y x x f r y Dr. Blanton - ENTC 3331 - Coordinate Transformations 7 / 29
• Find the cylindrical coordinates of the point whose Cartesian coordinates are (1, 2, 3) Dr. Blanton - ENTC 3331 - Coordinate Transformations 8 / 29
Cylindrical Coordinates -- Answer 1 Dr. Blanton - ENTC 3331 - Coordinate Transformations 9 / 29
• Find the Cartesian coordinates of the point whose cylindrical coordinates are (2, p/4, 3) Dr. Blanton - ENTC 3331 - Coordinate Transformations 10 / 29
Cylindrical Coordinates -- Answer 2 Dr. Blanton - ENTC 3331 - Coordinate Transformations 11 / 29
• Spherical coordinates consist of the three quantities (R, q, f). Dr. Blanton - ENTC 3331 - Coordinate Transformations 12 / 29
• First there is R. • This is the distance from the origin to the point. • Note that R 0. Dr. Blanton - ENTC 3331 - Coordinate Transformations 13 / 29
• Next there is f. • This is the same angle that we saw in cylindrical coordinates. • It is the angle between the positive xaxis and the line denoted by r (which is also the same r as in cylindrical coordinates). • There are no restrictions on f. Dr. Blanton - ENTC 3331 - Coordinate Transformations 14 / 29
• Finally there is q. • This is the angle between the positive zaxis and the line from the origin to the point. • We will require 0 ≤q ≤p. Dr. Blanton - ENTC 3331 - Coordinate Transformations 15 / 29
• In summary, • R is the distance from the origin to the point, • q is the angle that we need to rotate down from the positive z-axis to get to the point and • f is how much we need to rotate around the z-axis to get to the point. Dr. Blanton - ENTC 3331 - Coordinate Transformations 16 / 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 3331 - Coordinate Transformations 17 / 29
Spherical-to-Cylindrical z R (R, q, f) = (r, f, z) q f = f x x y f r y Dr. Blanton - ENTC 3331 - Coordinate Transformations 18 / 29
Cylindrical-to-Spherical z f = f R (R, q, f) = (r, f, z) q f = f x x y f r y Dr. Blanton - ENTC 3331 - Coordinate Transformations 19 / 29
Cartesian-to-Spherical z Recall from Cartesian-tocylindrical transformations: f = f R (R, q, f) = (r, f, z) q f = f x x y f r y Dr. Blanton - ENTC 3331 - Coordinate Transformations 20 / 29
Cartesian-to-Spherical z R (R, q, f) = (r, f, z) q y x x f r y Dr. Blanton - ENTC 3331 - Coordinate Transformations 21 / 29
Spherical-to-Cartesian z R (R, q, f) = (r, f, z) q y x x f r y Dr. Blanton - ENTC 3331 - Coordinate Transformations 22 / 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 3331 - Coordinate Transformations 23 / 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 3331 - Coordinate Transformations 24 / 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 3331 - Coordinate Transformations 25 / 29
• Next, let’s find R. Dr. Blanton - ENTC 3331 - Coordinate Transformations 26 / 29
• Finally, let’s get q. • To do this we can use either the conversion for r or z. • We’ll use the conversion for z. Dr. Blanton - ENTC 3331 - Coordinate Transformations 27 / 29
• So, the spherical coordinates of this point will are Dr. Blanton - ENTC 3331 - Coordinate Transformations 28 / 29
Dr. Blanton - ENTC 3331 - Coordinate Transformations 29 / 29
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