CoordinateSystemsA

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

Introduction

The purpose of this Notebook is to illustrate the use of the three major coordinate systems which are used in electromagnetics, namely cartesian, cylindrical and spherical.  A movement from the origin to a specific ™end point∫ will be shown in each of these coordinate systems.   In two dimensions the end point  will be the Cartesian point (3,2), while in three dimensions the end point will be (3,2,1).  Hopefully, you already have a good feeling for the Cartesian coordinate system;  therefore, it is included here more for the sake of consistency than for illustration.
       
     Unlike most of the other Notebooks, you will not find the Mathematica commands in this Notebook that generated the output.  At this stage, the inclusion of these commands would merely be confusing.  
       
     All of the illustrations in this notebook can be animated.  Information about the use of animations can be found in the NoteBook entitled Introduction.
     
      Since the animations are large, this Notebook was split into three sections.  You are currently reading CoordinateSystemsA, which contains illustrations of the Cartesian coordinate system.  CoordinateSystemsB contains the polar and cylindrical coordinate systems and CoordinateSystemsC contains the spherical coordinate system.

Cartesian Coordinates

Two dimensions

Our goal is to move from the origin, point (0,0), to the point (3,2).  Shown below is a pair of axes where the horizontal axis is the x axis and the vertical axis is the y axis.  To specify a point in two-dimensional cartesian coordinates we need to specify two numbers, one corresponding to the x value and one corresponding to the y value.  If only a single value is specified, say an x value, while letting y equal anything, then we have described a vertical line.  For example, if x = 1 and y equals anything, we have specified a line that is parallel to the y axis and passes through the x axis at the point x = 1.  By specifying a y value and leaving x unspecified, we describe a horizontal line.  If one line is specified by a x value and another line is given by a y value, these two lines intersect at a single point.  This intersection is designated the point (x,y).
       
     We are interested in moving from the origin to the point (3,2).  This movement can be accomplished several ways.  We choose to move the entire x distance first and then move the entire y distance.  While x increases, y is treated as zero.  The point specified by the intersection of the two lines described by x = number and y = number is indicated by a large dot.  The lines which are specified by the values of x and y taken alone are also shown.  The current values for x and y are indicated in the lower left corner of the diagram.
       
     Now, see if you can get the animation of the following graphics cell going.

Three dimensions

A third dimension is now added to the two previously considered.  Instead of moving from the origin to the point (3,2), we now move to the point (3,2,1).  This new point is a distance of one ``above'' the point we considered for the two-dimensional case.  If  the value of x is given while y and z are unspecified, a plane is obtained.  This plane intersects the x axis at the value given and never intersect the y or z axes (unless, of course, the plane is specified by x = 0, in which case the plane completely contains the y and z axes).  This plane is perpendicular to the x axis.  Specifying a y value alone describes a plane perpendicular to the y axis.  The intersection of the planes described by an x and a y value is a line parallel to the z axis.  Finally, specifying a z value describes a plane perpendicular to the z axis.  The intersection of the plane described by a constant z and the line defined by an x and y value is a single point.  This point is unique and is designated the point (x,y,z).
       
     Again, there are several ways that we can move from the origin to a point (x,y,z).  However, first the entire x distance will be moved (to a final x value of 3), then the entire y distance (to a final y value of 2) and then the entire z distance (to a final z value of 1).  This is broken into three distinct animations.   While x increases, y and z are treated as zero.  While y increases, x is held at 3 and z is fixed at zero.  If a value is zero, the plane which corresponds to it is not shown.  Once again, a large dot will move in accordance with the current values of x, y and z.   You will have to open the particular cell or cells that you want to see before beginning.  The labels for the axes may be a little confusing.  The x label is next to a edge of the surrounding box which is parallel to the x axis, the y label is next to an edge parallel to the y axis, and likewise for the z label.  This means that the x axis runs approximately left to right with positive x being to the left.  The positive y axis is roughly out of the computer screen and positive z is up.

Increasing x

Increasing y

Increasing z

Increasing x, y, and z

There is a way to tie all of the above animations together.  To do this you must have at least one of the cells above open that shows the three dimensional graphics.  (It doesn't matter which one or if there are more than one.)  You then 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.  By selecting the line which is the second furthest to the right, you have told Mathematica to animate all the cell contained within this section.  It doesn't matter if some of the cells are closed or not – they all get included in the animation.