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\(f(x)=x^2-1\)
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Real Domain Polar ColoringIn a domain coloring plot, each point \(x\) in the domain is marked with a color corresponding to \(f(x)\) (top). Specifically, \(|f(x)|\) is plotted as lightness (from black → white as \(|f(x)|\) goes from \(0\to\infty\)), and \(\text{sign}(f(x))\) is plotted as hue (from positive → red, negative → cyan).Complex outputs are also supported and appear as colors other than cyan or red. Because roots are colored black and poles are colored white, this type of plot is particularly useful for polynomials and rational functions. |
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Real Function Range ColoringIn this range coloring plot, each point \(x+yi\) in the domain is marked with a color (top). Then that point is transformed, so that \(f(x+yi)\) is plotted with the same color as the original \(x+yi\) that it came from (bottom). \(x+yi\) points in the domain might be mapped to the same points in the range. These are plotted on top of each other, and some information is lost. | ||||||
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\(f(z)=z^2-1\)
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Polar Domain ColoringOne of the more standard complex function plotting styles, along with Rectangular Grid.In a domain coloring plot, each point \(x+yi\) in the domain is marked with a color corresponding to \(f(x+yi)\). Specifically, \(r=|f(x+yi)|\) is plotted as lightness (from black → white as \(r\) goes from \(0\to\infty\)), and \(\theta=\arg(f(x+yi))\) is plotted as hue (from red → orange → yellow → green → cyan → blue → magenta → red as \(\theta\) goes from \(0\to2\pi\)). This plot doesn't have multifunction support. Because each point in the domain gets exactly one color, only the principal branch is graphed. Because roots are colored black and poles are colored white, this type of plot is particularly useful for polynomials and rational functions. This color plot is also particularly useful for seeing the effects of \(\exp(z)\). |
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\(f(z)=z^2-1\)
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Rectangular Domain ColoringIn this domain coloring plot, each point \(x+yi\) in the domain is marked with a color corresponding to \(f(x+yi)\). Specifically, \(u=\Re(f(x+yi))\) is plotted as red (from black → red as \(u\) goes from \(-\infty\to\infty\)), and \(v=\Im(f(x+yi))\) is plotted as cyan (from black → cyan as \(v\) goes from \(-\infty\to\infty\)).The combination of red and cyan colors makes white. As a result, each quadrant is colored in a different color. Roots are colored grey, and as we move towards \(\infty\) each point gets more color:
Because each axis gets compressed to the boundary of two quadrant colors, this plot can be useful for examining the transformations of the axes, and function end behavior. This color plot is also particularly useful for seeing the effects of \(\log(z)\). |
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\(f(z)=z^2-1\)
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Rectangular Domain Coloring 2In this alternate domain coloring plot, each point \(x+yi\) in the domain is marked with a color corresponding to \(f(x+yi)\). Specifically, \(|u|=|\Re(f(x+yi))|\) is plotted as red (from black → red as \(|u|\) goes from \(0\to\infty\)), \(|v|=|\Im(f(x+yi))|\) is plotted as blue (from black → blue as \(|v|\) goes from \(0\to\infty\)). Sign is plotted in green, (from black → green as \(9)u+v\) goes from \(-\infty\to\infty\)) The result is that positive real components are colored more yellow, negative real components more red positive imaginary components more cyan, and negative imaginary components more blue.This plot doesn't have multifunction support. Because each point in the domain gets exactly one color, only the principal branch is graphed. Because each axis gets compressed a single pair of positive and negative colors, this plot can be useful for examining the transformations of the axes, and function end behavior. Roots in this coloring are marked medium green, so this plot can also be useful for studying polynomials and rational functions. |
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\(f(z)=z^2-1\)
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Polar Grid TransformationIn this plot, a grid is drawn marking certain points \(x+yi\) with a color (left). Then that point is transformed, so that \(f(x+yi)\) is plotted with the same color as the original \(x+yi\) that it came from (right).For the polar grid: \(r=|z|=c\) circles are colored grey with intensity based on their distance from zero (from black → white as \(r\) goes from \(0\to\infty\)), with r=1 plotted black to make it stand out. \(\theta=\arg(z)=c\) lines are plotted as hue (from red → orange → yellow → green → cyan → blue → magenta → red as \(\theta\) goes from \(0\to2\pi\)). \(x+yi\) points in the domain might be mapped to the same points in the range. These may be plotted on top of each other, and some information is lost. In better but more inaccurate cases, rounding error causes the points to be plotted next to each other, so both are visible. Grid plots are particularly useful for seeing how a specific region or shape is transformed by the function. This is a particularly good plotter for seeing the effects of \(\log(z)\). |
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\(f(z)=z^2-1\)
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Rectangular Grid TransformationOne of the more standard complex function plotting styles, along with Polar Domain Coloring.In this plot, a grid is drawn marking certain points \(x+yi\) with a color (left). Then that point is transformed, so that \(f(x+yi)\) is plotted with the same color as the original \(x+yi\) that it came from (right). For the rectangular grid: \(x=\Re(z)=c\) vertical lines are colored blue with intensity based on their distance from zero (from black/blue → white/cyan as \(x\) goes from \(0\to\infty\)). x=0 is colored dark to make it stand out. \(y=\Im(z)=c\) horizontal lines are colored red with intensity based on their distance from zero (from black/red → white/orange as \(y\) goes from \(0\to\infty\)). y=0 is colored dark to make it stand out. \(x+yi\) points in the domain might be mapped to the same points in the range. These may be plotted on top of each other, and some information is lost. In better but more inaccurate cases, rounding error causes the points to be plotted next to each other, so both are visible. Grid plots are particularly useful for seeing how a specific region or shape is transformed by the function. This is a particularly good plotter for seeing the effects of \(\exp(z)\). |
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\(f(z)=z^2-1\)
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Polar Range ColoringThis is a polar grid plot taken to the extreme.In a range coloring plot, each point \(x+yi\) in the domain is marked with a color (left). Then that point is transformed, so that \(f(x+yi)\) is plotted with the same color as the original \(x+yi\) that it came from (right). For the polar coloring: \(r=|z|\) is plotted with intensity based on its distance from zero (from black → white as \(r\) goes from \(0\to\infty\)), and \(\theta=\arg(z)\) is plotted as hue (from red → orange → yellow → green → cyan → blue → magenta → red as \(\theta\) goes from \(0\to2\pi\)). \(x+yi\) points in the domain might be mapped to the same points in the range. These are plotted on top of each other, and some information is lost. In the default (Graph Bottom Up), points in the bottom right corner are plotted first, and can be overwritten by points higher up or farther left This is a particularly good plotter for seeing the effects of \(\log(z)\). |
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\(f(z)=z^2-1\)
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Rectangular Range ColoringThis is a rectangular grid plot taken to the extreme.In a range coloring plot, each point \(x+yi\) in the domain is marked with a color (left). Then that point is transformed, so that \(f(x+yi)\) is plotted with the same color as the original \(x+yi\) that it came from (right). For the rectangular coloring: \(x=\Re(z)\) is plotted with red (from black → red as \(x\) goes from \(-\infty\to\infty\)), and \(y=\Im(z)\) is plotted as cyan (from black → cyan as \(y\) goes from \(-\infty\to \infty\)). For the alternative rectangular coloring: \(|x|=|\Re(z)|\) is plotted with red (from black → red as \(|x|\) goes from \(0\to\infty\)), and \(|y|=|\Im(z)|\) is plotted as blue (from black → blue as \(|y|\) goes from \(0\to \infty\)). Sign is indicated in green (from black → green as \(x+y\) goes from \(-\infty\to \infty\)). \(x+yi\) points in the domain might be mapped to the same points in the range. These are plotted on top of each other, and some information is lost. In the default (Graph Bottom Up), points in the bottom right corner are plotted first, and can be overwritten by points higher up or farther left This is a particularly good plotter for seeing the effects of \(\exp(z)\). |
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