CoordinateSystemsB

Version 0.92
9/12/91
by John B. Schneider

Polar and Cylindrical Coordinates

Two dimensions (Polar)

Again, our goal is to move from the point (0,0) to the point (3,2) (as specified in Cartesian coordinates).  In polar coordinates two numbers must be given to locate a point.  These numbers are often written as an ordered pair, such as (rho,phi), where rho is the distance from the origin, and phi is the inverse tangent of y/x.  Actually, the value of phi must take into account the quadrant – so you cannot just plug arctan(y/x)  into your calculator.  You should be aware that the point (3,2)(Cartesian) could be specified directly in cylindrical coordinates as (3.61,.59)(polar, phi expressed in radians).

     If rho is given and phi is left unspecified, a circle with a radius of rho is obtained.  If phi is given and rho is unspecified, a ray that starts at the origin and ™points∫ in the phi direction is obtained.  In the cell below, the circle that corresponds to rho, the ray which corresponds to phi, and the point which is uniquely determined by the intersection of the circle and ray are shown as rho and phi vary.
               
     In the following cell the horizontal axis is the x axis and the vertical is the y axis.  There are no rho and phi axes.  In this animation, the ™current∫ point is moved away from the origin by increasing rho while leaving phi equal to zero.  When phi is zero, the point moves along the positive x axis.  Rho increases until it obtains a value equal to the square root of 13 (13 = 3^2 + 2 ^2).  Next, phi increases from zero to a value of inverse tangent of 1.5.  Notice that increases in phi sweep the ray in the counter-clockwise direction.

Three dimensions (Cylindrical)

Now, we will move from the origin to the point (3,2,1) (as specified in Cartesian coordinates).  The ™third∫ dimension in cylindrical coordinates is just like the third dimension in Cartesian coordinates.  If a point (x,y,z) in Cartesian coordinates is equivalent to the point (rho,phi,z) in cylindrical coordinates, the z's are equal.  If  a z value is given and rho and phi are unspecified, we describe a plane.  Specifying only a rho value describes a circular cylinder.  This cylinder has radius rho and is concentric with the z axis.  Specifying only a phi value describes a half plane (sometimes called a semi-infinite plane).  This half plane has an edge along the z axis and is an angle phi away from the positive x axis (measured in the counter-clockwise direction and along the shortest arc from the positive x axis to the phi plane).
       
     The intersection of the rho cylinder, the phi plane and the z plane is a point.  As before, the following animations show the point given by the current intersections of these surfaces.  However, the surface is not shown if the value that specifies that surface is zero (e.g., the plane specifed by the value of z is not shown if z is zero).  Now, on to the animations.

Increasing rho

Increasing phi

Increasing z

Increasing rho, phi, and z

This is just a reminder that, as was the case with the Cartesian coordinates, you can tie all of the three-dimensional cylindrical coordinate system animations together.  To do this you must have at least one of the cells above open that shows the three dimensional cylindrical graphics.  (It doesn't matter which one or if there are more than one.)  You must click on the line that is the second furthest to the right (keep in mind that you shouldn't include the edge of the window as a line).  Now hit Command-y.