cylindrical and spherical coordinates
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
First, a review of Polar Coordinates:
Angles are measured from the positive x axis.
Points are represented by a radius and an angle
(r, )
radius angle
To plot the point
4,5
First find the angle
Then move out along the terminal side 5
Converting between rectangular and Cylindrical Coordinates
r
r
(r,,z)cos( )
sin( )
x r
y r
z z
Rectangular to Cylindrical
2 2 2
tan( )
r x y
y
xz z
Cylindrical to rectangularNo real surprises here!
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
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
Note:
all angles are measured in radians, as always.
0 .
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
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 .
0 2 .
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
(x,y,z)
z
rFirst 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
sin( ) cos( ) tan( )r z r
z
Converting Between Rectangular and Spherical Coordinates
(x,y,z)
z
r
cos( ) sin( )cos( )
sin( ) sin( )sin( )
cos( )
x r
y r
z
Spherical to rectangular
Converting from Spherical to Rectangular Coordinates
(x,y,z)
z
rRectangular to Spherical
2 2 2
2 2
2 2 2
tan( )
tan( )
cos( )
x y z
y
x
x yr
z zz z
x y z