Parameterizing the Boundary

If you have the coordinates of a list of points from around the boundary of R, you can construct a parameterization of the boundary. It just so happens that Mathematica lets you pick off a list of points from any graphics cell. Let's start with a circle of radius 1. Here is the graph.


  Plot[{Sqrt[1-x^2],-Sqrt[1-x^2]}, {x,-1,1},
    AspectRatio->Automatic];

Click on the graph. A box will appear enclosing the graph. Now hold down the Command key (the one with the picture on an apple on it. If you move the mouse cursor into the graph's bounding box, the cursor will turn into a cross hairs and the coordinates of the point will appear in the bottom left corner of the notebook window. Hold down the "apple" button and click on the circle. Continue to keep the "apple" button depressed and click around the circle counterclockwise until you get back to where you started. Now release the "apple" button. Select Copy under the Edit menu. Click in the following input cell and then select Paste under the Edit menu. You should see a list of all of the data points that you selected. Evaluate the input cell to assign the list of data points to the variable, data.


  data = {{0.992195, -0.00374408}, {0.984707, -0.160998}, 
  {0.932289, -0.363181}, {0.827453, -0.557876}, 
  {0.722617, -0.692664}, {0.610294, -0.790012}, 
  {0.45304, -0.894847}, {0.250857, -0.96973}, 
  {0.0936033, -0.992195}, {-0.00374409, -0.999683}, 
  {-0.190951, -0.977218}, {-0.400622, -0.917312}, 
  {-0.58034, -0.812477}, {-0.760059, -0.647735}, 
  {-0.879871, -0.468016}, {-0.954754, -0.295786}, 
  {-0.984707, -0.168486}, {-0.992195, 0.116068}, 
  {-0.939777, 0.333228}, {-0.887359, 0.460528}, 
  {-0.834941, 0.557876}, {-0.707641, 0.722618}, 
  {-0.572852, 0.819965}, {-0.468016, 0.879871}, 
  {-0.273322, 0.962242}, {-0.153509, 0.984707}, 
  {0.0861151, 0.992195}, {0.205927, 0.977218}, 
  {0.370669, 0.924801}, {0.527923, 0.84243}, 
  {0.685176, 0.722618}, {0.812477, 0.58034}, 
  {0.924801, 0.393134}, {0.96973, 0.250857}};

Let's take a look at these points to verify that they go around the circle. 


  ListPlot[data, AspectRatio->Automatic];

Good enough? If not, you can always start over again and pick a new set of data points.

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