Example: Contour Plots with Singularities

Plots > Contour Plots > Example: Contour Plots with Singularities

Example: Contour Plots with Singularities

1. Define a 2-variable function with a known singularity.

Click to copy this expression

2. Plot F in a contour plot.

Click to copy this expression

<region id="ID0E6UZGB" top="192.00000000000003" left="38.400000000000006" actualWidth="263.52" actualHeight="245.20000000000002" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-32" end="32">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>9</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="20" />
      <graph-size width="192" height="192" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

3. Set the z-axis range to -25 to 25:

Click to copy this expression

<region id="ID0EQWZGB" top="537.60000000000014" left="38.400000000000006" actualWidth="263.52" actualHeight="245.20000000000002" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-25" end="25">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="21">
              <real xmlns="http://schemas.mathsoft.com/math50">-25</real>
            </startValue>
            <endValue resultRef="22">
              <real xmlns="http://schemas.mathsoft.com/math50">25</real>
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="23" />
      <graph-size width="192" height="192" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

4. Zoom in on the singularity by modifying the y-axis tick marks:

Click to copy this expression

<region id="ID0EBYZGB" top="902.40000000000009" left="38.400000000000006" actualWidth="284.51999999999953" actualHeight="243.03333333333336" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.1" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="24">
              <real xmlns="http://schemas.mathsoft.com/math50">-0.1</real>
            </startValue>
            <endValue resultRef="25">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-25" end="25">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="26">
              <real xmlns="http://schemas.mathsoft.com/math50">-25</real>
            </startValue>
            <endValue resultRef="27">
              <real xmlns="http://schemas.mathsoft.com/math50">25</real>
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="28" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

The red and the blue areas represent regions where the z-axis values fall outside the range of -25 to +25.

5. Restore the default z-axis values by deleting all its user-defined tick mark values.

Click to copy this expression

<region id="ID0EM1ZGB" top="1267.2000000000003" left="38.400000000000006" actualWidth="284.51999999999953" actualHeight="243.03333333333336" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.1" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="29">
              <real xmlns="http://schemas.mathsoft.com/math50">-0.1</real>
            </startValue>
            <endValue resultRef="30">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-3200" end="3200">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>9</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="31" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

6. View the upper-half of the plot by setting the y-axis minimum tick mark to zero.

Click to copy this expression

<region id="ID0E42ZGB" top="1747.2000000000003" left="38.400000000000006" actualWidth="284.51999999999953" actualHeight="243.03333333333336" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="0" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="32">
              <real xmlns="http://schemas.mathsoft.com/math50">0</real>
            </startValue>
            <endValue resultRef="33">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-3250.00000000003" end="3250.00000000003">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="34" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

The contour plot fails to compute the data because there is a singularity at y=0.

7. Set the y-axis lower tick mark value to a value that is slightly smaller than zero.

Click to copy this expression

<region id="ID0EI5ZGB" top="2131.2000000000003" left="38.400000000000006" actualWidth="284.51999999999953" actualHeight="243.03333333333336" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="transparent">
    <contourPlot>
      <axes>
        <xAxis rank="1" legend-position="PlotBoundaryBottom" start="-10" end="10">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </xyDomain>
        </xAxis>
        <yAxis rank="1" legend-position="PlotBoundaryLeft" start="-0.01" end="0.1">
          <axisLine position="ticknumberlock" positionticmark="0" legendWidth="0" />
          <axisGrid>
            <gridFrequency>12</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <xyDomain scale-type="linear" auto-scale="true">
            <startValue resultRef="35">
              <real xmlns="http://schemas.mathsoft.com/math50">-0.01</real>
            </startValue>
            <endValue resultRef="36">
              <real xmlns="http://schemas.mathsoft.com/math50">0.1</real>
            </endValue>
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-7500" end="7500">
          <axisLine position="origin" positionticmark="5" legendWidth="0" />
          <axisGrid>
            <gridFrequency>11</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <markers />
          <numberFormat>
            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
          </numberFormat>
          <zDomain scale-type="linear" auto-scale="true">
            <startValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </startValue>
            <endValue>
              <placeholder xmlns="http://schemas.mathsoft.com/math50" />
            </endValue>
          </zDomain>
        </zAxis>
      </axes>
      <contour color="#FF000000" show-lables="false" resultRef="37" />
      <graph-size width="212.99999999999955" height="190" />
      <plotEquation>
        <math>
          <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">F</id>
        </math>
        <math>
          <placeholder xmlns="http://schemas.mathsoft.com/math50" />
        </math>
      </plotEquation>
    </contourPlot>
  </plot>
</region>

The contour plot is now able to compute the data.