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 radius angle (r, ) To plot the point First find the angle Then move out along the terminal side 5

Now, a Review of 3D Coordinates z (3,2,4) 4 y (3,2,0) 2 x 3

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

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

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

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

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

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

Converting Equations Similarly, 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. 

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.

Representing 3D points in Spherical Coordinates Note: all angles are measured in radians, as always. 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 

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.

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

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!

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 • 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 

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

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

Conversions Convert the following from Rectangular to Spherical

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

Cylindrical and Spherical Coordinates