Comparing 3D Plots of One and Two Variable Functions

Plots > 3D Plots > Comparing 3D Plots of One and Two Variable Functions

It is possible to plot a function of one or two variables in a 3D plot region.

This topic compares the plots of such functions.

Plot a Function of One Variable

Define and plot a one variable function f1(x):

The default 3D plot of f1(x) is on the left, and it shows the trace of a line. The plot on the right is the same plot but rotated to verify that the trace is indeed a line.

Default Axes Position	Rotated Axes
<region id="ID0EY6MJB" actualWidth="288" actualHeight="291.79999999999995" top="57.600000000000009" left="0" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="174" num-of-points="41"> <traceStyle color="#FFED1D2F" automaticStyle="true">curve</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-1" end="1"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f1</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>	<region id="ID0EGANJB" actualWidth="288" actualHeight="291.79999999999995" top="57.600000000000009" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="177" num-of-points="41"> <traceStyle color="#FF2E3192" automaticStyle="true">curve</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-1" end="1"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f1</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>0.97392039271312958</Cell> <Cell>0.0042824428894279939</Cell> <Cell>0.22684957425660363</Cell> <Cell>0</Cell> <Cell>-0.070292004431413369</Cell> <Cell>0.95632519314437225</Cell> <Cell>0.28372726176804614</Cell> <Cell>0</Cell> <Cell>-0.21572691712096964</Cell> <Cell>-0.29227347748346583</Cell> <Cell>0.93168562916318742</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>

Default Axes Position

Rotated Axes

<region id="ID0EY6MJB" actualWidth="288" actualHeight="291.79999999999995" top="57.600000000000009" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="174" num-of-points="41">
          <traceStyle color="#FFED1D2F" automaticStyle="true">curve</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-1" end="1">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f1</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>

<region id="ID0EGANJB" actualWidth="288" actualHeight="291.79999999999995" top="57.600000000000009" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="177" num-of-points="41">
          <traceStyle color="#FF2E3192" automaticStyle="true">curve</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-1" end="1">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f1</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>0.97392039271312958</Cell>
          <Cell>0.0042824428894279939</Cell>
          <Cell>0.22684957425660363</Cell>
          <Cell>0</Cell>
          <Cell>-0.070292004431413369</Cell>
          <Cell>0.95632519314437225</Cell>
          <Cell>0.28372726176804614</Cell>
          <Cell>0</Cell>
          <Cell>-0.21572691712096964</Cell>
          <Cell>-0.29227347748346583</Cell>
          <Cell>0.93168562916318742</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>

Plotting the one variable function f1(x) in a 3D region is equivalent to plotting a two variable function f2(u,v) where you only specify one variable, as shown below:

Default Axes Position	Rotated Axes
<region id="ID0ENDNJB" actualWidth="288" actualHeight="291.79999999999995" top="499.20000000000005" left="0" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="182" num-of-points="41"> <traceStyle color="#FFED1D2F" automaticStyle="true">surface</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-1" end="1"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f2</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>	<region id="ID0E2DNJB" actualWidth="288" actualHeight="291.79999999999995" top="499.20000000000005" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="185" num-of-points="41"> <traceStyle color="#FF2E3192" automaticStyle="true">surface</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-1" end="1"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f2</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>0.97392039271312958</Cell> <Cell>0.0042824428894279939</Cell> <Cell>0.22684957425660363</Cell> <Cell>0</Cell> <Cell>-0.070292004431413369</Cell> <Cell>0.95632519314437225</Cell> <Cell>0.28372726176804614</Cell> <Cell>0</Cell> <Cell>-0.21572691712096964</Cell> <Cell>-0.29227347748346583</Cell> <Cell>0.93168562916318742</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>

Default Axes Position

Rotated Axes

<region id="ID0ENDNJB" actualWidth="288" actualHeight="291.79999999999995" top="499.20000000000005" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="182" num-of-points="41">
          <traceStyle color="#FFED1D2F" automaticStyle="true">surface</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-1" end="1">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f2</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>

<region id="ID0E2DNJB" actualWidth="288" actualHeight="291.79999999999995" top="499.20000000000005" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="185" num-of-points="41">
          <traceStyle color="#FF2E3192" automaticStyle="true">surface</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-1" end="1">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f2</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>0.97392039271312958</Cell>
          <Cell>0.0042824428894279939</Cell>
          <Cell>0.22684957425660363</Cell>
          <Cell>0</Cell>
          <Cell>-0.070292004431413369</Cell>
          <Cell>0.95632519314437225</Cell>
          <Cell>0.28372726176804614</Cell>
          <Cell>0</Cell>
          <Cell>-0.21572691712096964</Cell>
          <Cell>-0.29227347748346583</Cell>
          <Cell>0.93168562916318742</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>

Plot a Function of Two Variables

Define a two variable function f3(x,y) in terms of one variable:

The default 3D plot of f3(x,y) is on the left, and it shows a plane. The plot on the right is the same plot but rotated to verify that the trace is indeed a plane.

Default Axes Position	Rotated Axes
<region id="ID0EQHNJB" actualWidth="288" actualHeight="291.79999999999995" top="921.60000000000014" left="0" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="190" num-of-points="41"> <traceStyle color="#FFED1D2F" automaticStyle="true">surface</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f3</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>	<region id="ID0E5HNJB" actualWidth="288" actualHeight="291.79999999999995" top="921.60000000000014" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="193" num-of-points="41"> <traceStyle color="#FF2E3192" automaticStyle="true">surface</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f3</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>0.97392039271312958</Cell> <Cell>0.0042824428894279939</Cell> <Cell>0.22684957425660363</Cell> <Cell>0</Cell> <Cell>-0.070292004431413369</Cell> <Cell>0.95632519314437225</Cell> <Cell>0.28372726176804614</Cell> <Cell>0</Cell> <Cell>-0.21572691712096964</Cell> <Cell>-0.29227347748346583</Cell> <Cell>0.93168562916318742</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>

Default Axes Position

Rotated Axes

<region id="ID0EQHNJB" actualWidth="288" actualHeight="291.79999999999995" top="921.60000000000014" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="190" num-of-points="41">
          <traceStyle color="#FFED1D2F" automaticStyle="true">surface</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f3</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>

<region id="ID0E5HNJB" actualWidth="288" actualHeight="291.79999999999995" top="921.60000000000014" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="193" num-of-points="41">
          <traceStyle color="#FF2E3192" automaticStyle="true">surface</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f3</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>0.97392039271312958</Cell>
          <Cell>0.0042824428894279939</Cell>
          <Cell>0.22684957425660363</Cell>
          <Cell>0</Cell>
          <Cell>-0.070292004431413369</Cell>
          <Cell>0.95632519314437225</Cell>
          <Cell>0.28372726176804614</Cell>
          <Cell>0</Cell>
          <Cell>-0.21572691712096964</Cell>
          <Cell>-0.29227347748346583</Cell>
          <Cell>0.93168562916318742</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>

Plotting the one variable function f3(x,y) in a 3D region is equivalent to plotting a two variable function f4(u,v) where you specify two variables, as shown below:

Default Axes Position	Rotated Axes
<region id="ID0EFLNJB" actualWidth="288" actualHeight="291.79999999999995" top="1324.8000000000002" left="0" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="198" num-of-points="41"> <traceStyle color="#FFED1D2F" automaticStyle="true">surface</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f4</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>	<region id="ID0ETLNJB" actualWidth="288" actualHeight="291.79999999999995" top="1324.8000000000002" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50"> <plot background-type="white"> <plot3D> <traces> <trace3D resultRef="201" num-of-points="41"> <traceStyle color="#FF2E3192" automaticStyle="true">surface</traceStyle> </trace3D> </traces> <graph-size width="230.39999999999998" height="230.39999999999998" /> <axes> <xAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </xAxis> <yAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </xyDomain> </yAxis> <zAxis rank="1" start="-10" end="10"> <axisLine /> <axisGrid> <gridFrequency>5</gridFrequency> <gridLabels display="true" /> <gridLines /> <tickMarks display="true" /> </axisGrid> <axisLabel /> <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear"> <startValue /> <endValue /> </zDomain> </zAxis> </axes> <plotEquations> <plotEquation> <math> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f4</id> </math> <math> <placeholder xmlns="http://schemas.mathsoft.com/math50" /> </math> </plotEquation> </plotEquations> <CameraTransformMatrix Columns="4" Rows="4"> <Cells> <Cell>0.97392039271312958</Cell> <Cell>0.0042824428894279939</Cell> <Cell>0.22684957425660363</Cell> <Cell>0</Cell> <Cell>-0.070292004431413369</Cell> <Cell>0.95632519314437225</Cell> <Cell>0.28372726176804614</Cell> <Cell>0</Cell> <Cell>-0.21572691712096964</Cell> <Cell>-0.29227347748346583</Cell> <Cell>0.93168562916318742</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>0</Cell> <Cell>1</Cell> </Cells> </CameraTransformMatrix> </plot3D> </plot> </region>

Default Axes Position

Rotated Axes

<region id="ID0EFLNJB" actualWidth="288" actualHeight="291.79999999999995" top="1324.8000000000002" left="0" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="198" num-of-points="41">
          <traceStyle color="#FFED1D2F" automaticStyle="true">surface</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f4</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>

<region id="ID0ETLNJB" actualWidth="288" actualHeight="291.79999999999995" top="1324.8000000000002" left="326.40000000000003" xmlns="http://schemas.mathsoft.com/worksheet50">
  <plot background-type="white">
    <plot3D>
      <traces>
        <trace3D resultRef="201" num-of-points="41">
          <traceStyle color="#FF2E3192" automaticStyle="true">surface</traceStyle>
        </trace3D>
      </traces>
      <graph-size width="230.39999999999998" height="230.39999999999998" />
      <axes>
        <xAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </xAxis>
        <yAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <xyDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </xyDomain>
        </yAxis>
        <zAxis rank="1" start="-10" end="10">
          <axisLine />
          <axisGrid>
            <gridFrequency>5</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
          </axisGrid>
          <axisLabel />
          <zDomain auto-scale="true" allow-negative-values="true" scale-type="linear">
            <startValue />
            <endValue />
          </zDomain>
        </zAxis>
      </axes>
      <plotEquations>
        <plotEquation>
          <math>
            <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true" xmlns="http://schemas.mathsoft.com/math50">f4</id>
          </math>
          <math>
            <placeholder xmlns="http://schemas.mathsoft.com/math50" />
          </math>
        </plotEquation>
      </plotEquations>
      <CameraTransformMatrix Columns="4" Rows="4">
        <Cells>
          <Cell>0.97392039271312958</Cell>
          <Cell>0.0042824428894279939</Cell>
          <Cell>0.22684957425660363</Cell>
          <Cell>0</Cell>
          <Cell>-0.070292004431413369</Cell>
          <Cell>0.95632519314437225</Cell>
          <Cell>0.28372726176804614</Cell>
          <Cell>0</Cell>
          <Cell>-0.21572691712096964</Cell>
          <Cell>-0.29227347748346583</Cell>
          <Cell>0.93168562916318742</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>0</Cell>
          <Cell>1</Cell>
        </Cells>
      </CameraTransformMatrix>
    </plot3D>
  </plot>
</region>