1 / 26

290 likes | 666 Vues

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.

Télécharger la présentation
## Cylindrical and Spherical Coordinates

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

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

More Related