--- title: "SphericalPlot3D" language: "en" type: "Symbol" summary: "SphericalPlot3D[r, \\[Theta], \\[Phi]] generates a 3D plot with a spherical radius r as a function of spherical coordinates \\[Theta] and \\[Phi]. SphericalPlot3D[r, {\\[Theta], \\[Theta]min, \\[Theta]max}, {\\[Phi], \\ \\[Phi]min, \\[Phi]max}] generates a 3D spherical plot over the specified ranges of spherical coordinates. SphericalPlot3D[{r1, r2, ...}, {\\[Theta], \\[Theta]min, \\[Theta]max}, {\\[Phi], \\ \\[Phi]min, \\[Phi]max}] generates a 3D spherical plot with multiple surfaces." keywords: - azimuth - latitude - longitude - polar plot 3d - spherical - spherical coordinates - spherical eigenfunctions - spherical plot 3D - spherical plot3 - spherical symmetry - zenith - azimuthal plot canonical_url: "https://reference.wolfram.com/language/ref/SphericalPlot3D.html" source: "Wolfram Language Documentation" related_guides: - title: "Function Visualization" link: "https://reference.wolfram.com/language/guide/FunctionVisualization.en.md" - title: "Angles and Polar Coordinates" link: "https://reference.wolfram.com/language/guide/AnglesAndPolarCoordinates.en.md" related_functions: - title: "RevolutionPlot3D" link: "https://reference.wolfram.com/language/ref/RevolutionPlot3D.en.md" - title: "ParametricPlot3D" link: "https://reference.wolfram.com/language/ref/ParametricPlot3D.en.md" - title: "PolarPlot" link: "https://reference.wolfram.com/language/ref/PolarPlot.en.md" - title: "Sphere" link: "https://reference.wolfram.com/language/ref/Sphere.en.md" - title: "RotationMatrix" link: "https://reference.wolfram.com/language/ref/RotationMatrix.en.md" related_tutorials: - title: "Some Special Plots" link: "https://reference.wolfram.com/language/tutorial/GraphicsAndSound.en.md#31896" --- # SphericalPlot3D SphericalPlot3D[r, θ, ϕ] generates a 3D plot with a spherical radius r as a function of spherical coordinates θ and ϕ. SphericalPlot3D[r, {θ, θmin, θmax}, {ϕ, ϕmin, ϕmax}] generates a 3D spherical plot over the specified ranges of spherical coordinates. SphericalPlot3D[{r1, r2, …}, {θ, θmin, θmax}, {ϕ, ϕmin, ϕmax}] generates a 3D spherical plot with multiple surfaces. ## Details and Options * The angles ``θ`` and ``ϕ`` are measured in radians. * $\pi /2-\theta$ corresponds to "latitude"; $\theta$ is 0 at the "north pole", and $\pi$ at the "south pole". * ``ϕ`` corresponds to "longitude", varying from 0 to $2 \pi$ counterclockwise looking from the north pole. * ``SphericalPlot3D[r, θ, ϕ]`` takes ``θ`` to have range 0 to $\pi$, and ``ϕ`` to have range 0 to $2 \pi$. * The $x$, $y$, $z$ position corresponding to $r$, $\theta$, $\phi$ is $r \sin (\theta ) \cos (\phi )$, $r \sin (\theta ) \sin (\phi )$, $r \cos (\theta )$. The variables $\theta$ and $\phi$ can have any values. The surfaces they define can overlap radially. * Holes are left at positions where the $r_i$ etc. evaluate to ``None``, or anything other than real numbers. * ``SphericalPlot3D`` treats the variables ``θ`` and ``ϕ`` as local, effectively using ``Block``. * ``SphericalPlot3D`` has attribute ``HoldAll``, and evaluates the $r_i$ only after assigning specific numerical values to variables. * In some cases it may be more efficient to use ``Evaluate`` to evaluate the $r_i$ symbolically before specific numerical values are assigned to variables. * ``SphericalPlot3D`` has the same options as ``Graphics3D``, with the following additions and changes: [] | | | | | :------------------------- | :---------------- | :------------------------------------------------------ | | Axes | True | whether to draw axes | | BoundaryStyle | Automatic | how to draw boundary lines for surfaces | | ColorFunction | Automatic | how to determine the color of curves and surfaces | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | θ, ϕ curves to exclude | | ExclusionsStyle | None | what to draw at excluded points or curves | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | Automatic | how many mesh divisions in each direction to draw | | MeshFunctions | {#4&, #5&} | how to determine the placement of mesh divisions | | MeshShading | None | how to shade regions between mesh divisions | | MeshStyle | Automatic | the style for mesh divisions | | Method | Automatic | the method to use for refining surfaces | | NormalsFunction | Automatic | how to determine effective surface normals | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLegends | None | legends for surfaces | | PlotPoints | Automatic | the initial number of sample points in each parameter | | PlotStyle | Automatic | graphics directives for the style for each object | | PlotTheme | \$PlotTheme | overall theme for the plot | | RegionFunction | (True&) | how to determine whether a point should be included | | ScalingFunctions | None | how to scale individual coordinates | | TextureCoordinateFunction | Automatic | how to determine texture coordinates | | TextureCoordinateScaling | True | whether to scale arguments to TextureCoordinateFunction | | WorkingPrecision | MachinePrecision | the precision used in internal computations | * Interactive labeling can be specified for curves or surfaces using ``Tooltip``, ``StatusArea``, or ``Annotation``. * ``SphericalPlot3D[Tooltip[{r1, r2, …}], …]`` specifies that the $r_i$ should be displayed as tooltip labels for the corresponding surfaces. * ``Tooltip[r, label]`` specifies an explicit tooltip label for a surface. * ``SphericalPlot3D`` initially evaluates each function at a number of equally spaced sample points specified by ``PlotPoints``. Then it uses an adaptive algorithm to choose additional sample points, subdividing in each parameter at most ``MaxRecursion`` times. * You should realize that with the finite number of sample points used, it is possible for ``SphericalPlot3D`` to miss features in your functions. To check your results, you should try increasing the settings for ``PlotPoints`` and ``MaxRecursion``. * ``On[SphericalPlot3D::accbend]`` makes ``SphericalPlot3D`` print a message if it is unable to reach a certain smoothness of curve. * The arguments supplied to functions in ``MeshFunctions`` and ``RegionFunction`` are $x$, $y$, $z$, $\theta$, $\phi$, and $r$. Functions in ``ColorFunction`` and ``TextureCoordinateFunction`` are by default supplied with scaled versions of these arguments. * The functions are evaluated all over each surface. * By default, surfaces are treated as uniform white diffuse reflectors, corresponding to ``ColorFunction -> (White&)``. * ``SphericalPlot3D`` returns ``Graphics3D[GraphicsComplex[data]]``. * Themes that affect 3D surfaces include: | | | | | ------- | -------------- | --------------------------------- | | [image] | "DarkMesh" | dark mesh lines | | [image] | "GrayMesh" | gray mesh lines | | [image] | "LightMesh" | light mesh lines | | [image] | "ZMesh" | vertically distributed mesh lines | | [image] | "ThickSurface" | add thickness to surfaces | * Possible settings for ``ScalingFunctions`` include: | | | | -------------------- | ---------------------------------- | | {sx, sy, sz} | scale the x, y and z coordinates | | {sx, sy, sz, sθ, sϕ} | scale the θ and ϕ parameter spaces | * Common built-in scaling functions ``s`` include: | | | | | ----------- | ------- | --------------------------------------------------- | | "Log" | [image] | log scale with automatic tick labeling | | "Log10" | [image] | base-10 log scale with powers of 10 for ticks | | "SignedLog" | [image] | log-like scale that includes 0 and negative numbers | | "Reverse" | [image] | reverse the coordinate direction | | "Infinite" | [image] | infinite scale | * Scaling the ``θ`` or ``ϕ`` parameter space affects how the plot is sampled, but not the overall visual appearance of the plot. ### List of all options | | | | | ------------------------- | ----------------- | ---------------------------------------------------------------------------------- | | AlignmentPoint | Center | the default point in the graphic to align with | | AspectRatio | Automatic | ratio of height to width | | Axes | True | whether to draw axes | | AxesEdge | Automatic | on which edges to put axes | | AxesLabel | None | axes labels | | AxesOrigin | Automatic | where axes should cross | | AxesStyle | {} | graphics directives to specify the style for axes | | Background | None | background color for the plot | | BaselinePosition | Automatic | how to align with a surrounding text baseline | | BaseStyle | {} | base style specifications for the graphic | | BoundaryStyle | Automatic | how to draw boundary lines for surfaces | | Boxed | True | whether to draw the bounding box | | BoxRatios | Automatic | bounding 3D box ratios | | BoxStyle | {} | style specifications for the box | | ClipPlanes | None | clipping planes | | ClipPlanesStyle | Automatic | style specifications for clipping planes | | ColorFunction | Automatic | how to determine the color of curves and surfaces | | ColorFunctionScaling | True | whether to scale arguments to ColorFunction | | ContentSelectable | Automatic | whether to allow contents to be selected | | ControllerLinking | False | when to link to external rotation controllers | | ControllerPath | Automatic | what external controllers to try to use | | Epilog | {} | 2D graphics primitives to be rendered after the main plot | | EvaluationMonitor | None | expression to evaluate at every function evaluation | | Exclusions | Automatic | θ, ϕ curves to exclude | | ExclusionsStyle | None | what to draw at excluded points or curves | | FaceGrids | None | grid lines to draw on the bounding box | | FaceGridsStyle | {} | style specifications for face grids | | FormatType | TraditionalForm | default format type for text | | ImageMargins | 0. | the margins to leave around the graphic | | ImagePadding | All | what extra padding to allow for labels, etc. | | ImageSize | Automatic | absolute size at which to render the graphic | | LabelStyle | {} | style specifications for labels | | Lighting | Automatic | simulated light sources to use | | MaxRecursion | Automatic | the maximum number of recursive subdivisions allowed | | Mesh | Automatic | how many mesh divisions in each direction to draw | | MeshFunctions | {#4&, #5&} | how to determine the placement of mesh divisions | | MeshShading | None | how to shade regions between mesh divisions | | MeshStyle | Automatic | the style for mesh divisions | | Method | Automatic | the method to use for refining surfaces | | NormalsFunction | Automatic | how to determine effective surface normals | | PerformanceGoal | \$PerformanceGoal | aspects of performance to try to optimize | | PlotLabel | None | a label for the plot | | PlotLegends | None | legends for surfaces | | PlotPoints | Automatic | the initial number of sample points in each parameter | | PlotRange | All | range of values to include | | PlotRangePadding | Automatic | how much to pad the range of values | | PlotRegion | Automatic | final display region to be filled | | PlotStyle | Automatic | graphics directives for the style for each object | | PlotTheme | \$PlotTheme | overall theme for the plot | | PreserveImageOptions | Automatic | whether to preserve image options when displaying new versions of the same graphic | | Prolog | {} | 2D graphics primitives to be rendered before the main plot | | RegionFunction | (True&) | how to determine whether a point should be included | | RotationAction | "Fit" | how to render after interactive rotation | | ScalingFunctions | None | how to scale individual coordinates | | SphericalRegion | Automatic | whether to make the circumscribing sphere fit in the final display area | | TextureCoordinateFunction | Automatic | how to determine texture coordinates | | TextureCoordinateScaling | True | whether to scale arguments to TextureCoordinateFunction | | Ticks | Automatic | specification for ticks | | TicksStyle | {} | style specification for ticks | | TouchscreenAutoZoom | False | whether to zoom to fullscreen when activated on a touchscreen | | ViewAngle | Automatic | angle of the field of view | | ViewCenter | Automatic | point to display at the center | | ViewMatrix | Automatic | explicit transformation matrix | | ViewPoint | {1.3, -2.4, 2.} | viewing position | | ViewProjection | Automatic | projection method for rendering objects distant from the viewer | | ViewRange | All | range of viewing distances to include | | ViewVector | Automatic | position and direction of a simulated camera | | ViewVertical | {0, 0, 1} | direction to make vertical | | WorkingPrecision | MachinePrecision | the precision used in internal computations | --- ## Examples (130) ### Basic Examples (3) Plot a spherical surface: ```wl In[1]:= SphericalPlot3D[1 + 2Cos[2θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Plot several spherical surfaces: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}] Out[1]= [image] ``` --- Style the resulting surface: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 5, {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotStyle -> Directive[Orange, Opacity[0.7], Specularity[White, 10]], Mesh -> None, PlotPoints -> 30] Out[1]= [image] ``` ### Scope (18) #### Sampling (9) More points are sampled when the function changes quickly: ```wl In[1]:= SphericalPlot3D[1 + Sin[5θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> All] Out[1]= [image] ``` --- The plot range is selected automatically: ```wl In[1]:= SphericalPlot3D[1 / θ, {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Ranges where the function becomes nonreal are excluded: ```wl In[1]:= SphericalPlot3D[Sqrt[Sin[θ]], {θ, -Pi / 2, Pi / 2}, {ϕ, 0, 3Pi / 2}] Out[1]= [image] ``` --- The surface is split when there are discontinuities in the function: ```wl In[1]:= SphericalPlot3D[Floor[ϕ], {θ, 0, Pi}, {ϕ, 0, 3Pi}, Mesh -> None, ExclusionsStyle -> {None, Red}] Out[1]= [image] ``` --- Use ``PlotPoints`` and ``MaxRecursion`` to control adaptive sampling: ```wl In[1]:= Grid[Table[SphericalPlot3D[1 + Sin[5θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotPoints -> pp, MaxRecursion -> mr, PlotRange -> All, Mesh -> None], {mr, {0, 2}}, {pp, {5, 10}}]] Out[1]= | | | | ------- | ------- | | [image] | [image] | | [image] | [image] | ``` --- Use ``PlotRange`` to focus in on areas of interest: ```wl In[1]:= {SphericalPlot3D[1 + Sin[5θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}], SphericalPlot3D[1 + Sin[5θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotRange -> 1]} Out[1]= [image] ``` --- Use ``Exclusions`` to remove points or split the resulting surface: ```wl In[1]:= SphericalPlot3D[Round[θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, Exclusions -> {θ == 0.5, θ == 1.5, θ == 2.5}, Axes -> None, Mesh -> None] Out[1]= [image] ``` --- Plot multiple surfaces: ```wl In[1]:= SphericalPlot3D[{Sin[θ], 2Sin[θ]}, {θ, 0, Pi}, {ϕ, 0, Pi}, PlotStyle -> {Blue, Red}] Out[1]= [image] ``` --- Use ``ScalingFunctions`` to reverse the direction of the ``x`` axis: ```wl In[1]:= SphericalPlot3D[Floor[ϕ], {θ, 0, Pi}, {ϕ, 0, 3Pi}, ScalingFunctions -> {"Reverse", None, None}] Out[1]= [image] ``` #### Presentation (9) Provide explicit styling to different surfaces: ```wl In[1]:= SphericalPlot3D[{ϕ, -ϕ}, {θ, -Pi / 2, Pi / 2}, {ϕ, 0, 3Pi / 2}, PlotStyle -> {Directive[Red, Specularity[White, 20]], Directive[Cyan, Specularity[White, 20]]}] Out[1]= [image] ``` --- Add labels: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesLabel -> {x, y}, PlotLabel -> r == 1, AxesEdge -> {{-1, -1}, {1, -1}, {-1, -1}}] Out[1]= [image] ``` --- Use legends for multiple surfaces: ```wl In[1]:= SphericalPlot3D[{ϕ, -ϕ}, {θ, -Pi / 2, Pi / 2}, {ϕ, 0, 3Pi / 2}, PlotLegends -> "Expressions"] Out[1]= [image] ``` --- Provide an interactive ``Tooltip`` for each surface: ```wl In[1]:= SphericalPlot3D[{Tooltip[ϕ, r == ϕ], Tooltip[-ϕ, r == -ϕ]}, {θ, -Pi / 2, Pi / 2}, {ϕ, 0, 3Pi / 2}, PlotStyle -> {Green, Yellow}] Out[1]= [image] ``` --- Create an overlay mesh: ```wl In[1]:= SphericalPlot3D[Sin[θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 10, MeshStyle -> {Cyan, Directive[Green, Dashed]}, PlotStyle -> Gray] Out[1]= [image] ``` --- Style the areas between mesh levels: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 5, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 25, MeshShading -> {{Red, Yellow}, {Pink, Orange}}] Out[1]= [image] ``` --- Color by parameter values: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 5, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 25, ColorFunction -> Function[{x, y, z, t, θ, r}, Hue[θ ]]] Out[1]= [image] ``` Use named color schemes: ```wl In[2]:= SphericalPlot3D[1 + Sin[5ϕ] / 5, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 25, ColorFunction -> (ColorData["Rainbow"][#6]&)] Out[2]= [image] ``` --- Remove portions of a curve or surface: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, RegionFunction -> Function[{x, y, z, t, θ, r}, Sin[8t]Sin[8θ] > 0.1], Mesh -> None, PlotStyle -> FaceForm[Red, Cyan], PlotPoints -> 35, BoundaryStyle -> Black] Out[1]= [image] ``` --- Use a highly stylized theme: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 5, {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotTheme -> "Marketing"] Out[1]= [image] ``` ### Options (93) #### Axes (3) By default, ``Axes`` are drawn: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Use ``Axes -> False`` to turn off axes: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Axes -> False] Out[1]= [image] ``` --- Turn on each axis individually: ```wl In[1]:= {SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Axes -> {False, False, True}], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Axes -> {False, True, False}], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Axes -> {True, False, False}]} Out[1]= [image] ``` #### AxesLabel (3) No axes labels are drawn by default: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Place a label on the $z$ axis: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesLabel -> z] Out[1]= [image] ``` --- Specify axes labels: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesLabel -> {x, y, z}] Out[1]= [image] ``` #### AxesOrigin (2) The position of the axes is determined automatically: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Specify an explicit origin for the axes: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesOrigin -> {.9, -.9, .9}] Out[1]= [image] ``` #### AxesStyle (4) Change the style for the axes: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesStyle -> Red] Out[1]= [image] ``` --- Specify the style of each axis: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesStyle -> {{Thick, Brown}, {Thick, Blue}, {Thick, Green}}] Out[1]= [image] ``` --- Use different styles for the ticks and the axes: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesStyle -> Green, TicksStyle -> Black] Out[1]= [image] ``` --- Use different styles for the labels and the axes: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AxesStyle -> Green, LabelStyle -> Black] Out[1]= [image] ``` #### BoundaryStyle (4) ``BoundaryStyle`` automatically matches ``MeshStyle`` : ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, MeshStyle -> Red] Out[1]= [image] In[2]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, MeshStyle -> None] Out[2]= [image] ``` --- Use a thick red boundary: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, BoundaryStyle -> Directive[Thick, Red]] Out[1]= [image] ``` --- Boundaries are drawn where the surface is clipped by ``RegionFunction`` : ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, BoundaryStyle -> Red, Mesh -> None, RegionFunction -> Function[{x, y, z, θ, ϕ}, Sin[5ϕ]Sin[5θ] > 0]] Out[1]= [image] ``` --- Boundaries are not drawn where the surface is clipped by ``Exclusions`` : ```wl In[1]:= SphericalPlot3D[Floor[ϕ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, BoundaryStyle -> Red, Exclusions -> Table[ϕ == k, {k, 0, 6}]] Out[1]= [image] ``` #### BoxRatios (2) The default ``BoxRatios`` preserves the natural scale of the surface: ```wl In[1]:= SphericalPlot3D[Sin[θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Use specific ``BoxRatios`` : ```wl In[1]:= SphericalPlot3D[Sin[θ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, BoxRatios -> {1, 1, 1}] Out[1]= [image] ``` #### ColorFunction (5) Color a surface by $x$, $y$, $z$, $\theta$, $\phi$, and $r$ parameters: ```wl In[1]:= Table[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, Evaluate[f]], PlotLabel -> f, Axes -> None], {f, {Hue[x], Hue[y], Hue[z], Hue[θ], Hue[ϕ], Hue[r]}}] Out[1]= [image] ``` --- Use ``ColorData`` for predefined color gradients: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, ColorData["Rainbow"][r]]] Out[1]= [image] ``` --- Named color gradients color in the $z$ direction: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ColorFunction -> "Rainbow"] Out[1]= [image] ``` --- ``ColorFunction`` has higher priority than ``PlotStyle`` : ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ColorFunction -> "Rainbow", PlotStyle -> Directive[Opacity[0.5], Red]] Out[1]= [image] ``` --- ``ColorFunction`` has lower priority than ``MeshShading`` : ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, Hue[z]], MeshShading -> {{Automatic, None}, {None, Automatic}}] Out[1]= [image] ``` #### ColorFunctionScaling (1) Use scaled coordinates in the $\theta$ direction and unscaled coordinates in the $\phi$ direction: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, Hue[ϕ / (2Pi), 1, 1, 1 - θ]], ColorFunctionScaling -> {True, True, True, True, False, True}] Out[1]= [image] ``` #### EvaluationMonitor (2) Show where ``RevolutionPlot3D`` samples a function in $\{\theta ,\phi \}$ coordinates: ```wl In[1]:= ListPlot[Reap[SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, EvaluationMonitor :> Sow[{θ, ϕ}]]][[-1, 1]]] Out[1]= [image] ``` --- Count the number of sample points on the surface: ```wl In[1]:= Block[{k = 0}, SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, EvaluationMonitor :> k++];k] Out[1]= 2214 ``` #### Exclusions (5) This uses automatic methods to compute exclusions, in this case from branch cuts: ```wl In[1]:= SphericalPlot3D[Im[θ Sqrt[Exp[I ϕ]]], {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotRange -> All] Out[1]= [image] ``` --- Indicate that no exclusions should be computed: ```wl In[1]:= SphericalPlot3D[Im[θ Sqrt[Exp[I ϕ]]], {θ, 0, Pi}, {ϕ, 0, 2Pi}, Exclusions -> None, PlotRange -> All] Out[1]= [image] ``` --- Give a set of exclusions as an equation: ```wl In[1]:= SphericalPlot3D[Tan[ϕ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, Exclusions -> {Cos[ϕ] == 0}, BoxRatios -> {1, 1, 1}] Out[1]= [image] ``` --- Give three sets of exclusions: ```wl In[1]:= SphericalPlot3D[θ Sin[ϕ] / Abs[Sin[ϕ]], {θ, 0, Pi}, {ϕ, 0, 2Pi}, Exclusions -> {ϕ == 0, ϕ == Pi, ϕ == 2Pi}] Out[1]= [image] ``` --- Use both automatically computed and explicit exclusions: ```wl In[1]:= SphericalPlot3D[Clip[Tan[θ]Tan[ϕ], {-5, 5}], {θ, 0, Pi}, {ϕ, 0, Pi}, Exclusions -> {Automatic, Cos[ϕ] == 0, Cos[θ] == 0}, PlotRange -> All] Out[1]= [image] ``` #### ExclusionsStyle (2) Style the boundary with a red line: ```wl In[1]:= SphericalPlot3D[Floor[ϕ], {θ, 0, Pi}, {ϕ, 0, 3Pi}, Mesh -> None, ExclusionsStyle -> {None, Red}] Out[1]= [image] ``` --- Style the boundary with a red line and the surface in between with yellow: ```wl In[1]:= SphericalPlot3D[Floor[ϕ], {θ, 0, Pi}, {ϕ, 0, 3Pi}, Mesh -> None, ExclusionsStyle -> {Yellow, Red}] Out[1]= [image] ``` #### ImageSize (7) Use named sizes such as ``Tiny``, ``Small``, ``Medium`` and ``Large`` : ```wl In[1]:= {SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> Tiny], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> Small]} Out[1]= [image] ``` --- Specify the width of the plot: ```wl In[1]:= {SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> 150], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AspectRatio -> 1.5, ImageSize -> 150]} Out[1]= {[image], [image]} ``` Specify the height of the plot: ```wl In[2]:= {SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> {Automatic, 150}], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AspectRatio -> 2, ImageSize -> {Automatic, 150}]} Out[2]= [image] ``` --- Allow the width and height to be up to a certain size: ```wl In[1]:= {SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> UpTo[200]], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AspectRatio -> 2, ImageSize -> UpTo[200]]} Out[1]= {[image], [image]} ``` --- Specify the width and height for a graphic, padding with space if necessary: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> {200, 300}, Background -> LightBlue] Out[1]= [image] ``` Setting ``AspectRatio -> Full`` will fill the available space: ```wl In[2]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AspectRatio -> Full, ImageSize -> {200, 300}, Background -> LightBlue] Out[2]= [image] ``` --- Use maximum sizes for the width and height: ```wl In[1]:= {SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> {UpTo[150], UpTo[100]}], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AspectRatio -> 2, ImageSize -> {UpTo[150], UpTo[100]}]} Out[1]= [image] ``` --- Use ``ImageSize -> Full`` to fill the available space in an object: ```wl In[1]:= Framed[Pane[SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ImageSize -> Full, Background -> LightBlue], {200, 200}]] Out[1]= [image] ``` --- Specify the image size as a fraction of the available space: ```wl In[1]:= Framed[Pane[SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}, AspectRatio -> Full, ImageSize -> {Scaled[0.5], Scaled[0.5]}, Background -> LightBlue], {200, 200}]] Out[1]= [image] ``` #### MaxRecursion (1) Refine the surface where it changes quickly: ```wl In[1]:= Table[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> All, MaxRecursion -> r], {r, {0, 2}}] Out[1]= [image] ``` #### Mesh (5) Show the initial and final sampling meshes: ```wl In[1]:= {SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> Full], SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> All]} Out[1]= [image] ``` --- Use 10 mesh levels evenly spaced in the parameter directions: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 10] Out[1]= [image] ``` --- Use a different number of mesh lines in different directions: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> {5, 20}] Out[1]= [image] ``` --- Use an explicit list of values for the mesh in the $\theta$ parameter and no mesh in the $\phi$ parameter: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> {Range[0, Pi, Pi / 12], 0}] Out[1]= [image] ``` --- Use explicit value and style for the $\theta$ mesh: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, MeshFunctions -> {#4&}, Mesh -> {{{Pi / 3, Thick}, {2Pi / 3, Directive[Thick, Red]}}}] Out[1]= [image] ``` #### MeshFunctions (2) Use a mesh evenly spaced in the $x$, $y$, $z$, $\theta$, $\phi$, and $r$ directions: ```wl In[1]:= Table[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, BoundaryStyle -> None, Mesh -> 5, MeshFunctions -> {Function[{x, y, z, θ, ϕ, r}, Evaluate[f]]}, PlotLabel -> f, Axes -> None], {f, {x, y, z, θ, ϕ, r}}] Out[1]= [image] ``` --- Show five mesh levels in the $\theta$ direction (red) and ten in the $\phi$ direction (blue): ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> {5, 10}, MeshFunctions -> {#4&, #5&}, MeshStyle -> {Red, Blue}] Out[1]= [image] ``` #### MeshShading (7) Alternate red and blue arcs in the $u$ direction: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 15, MeshFunctions -> {#5&}, MeshShading -> {Red, Blue}] Out[1]= [image] ``` --- Use ``None`` to remove segments: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 15, MeshFunctions -> {#5&}, MeshShading -> {Red, None}] Out[1]= [image] ``` --- ``MeshShading`` has higher priority than ``PlotStyle`` for styling: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 15, MeshFunctions -> {#5&}, PlotStyle -> Green, MeshShading -> {Red, Blue}] Out[1]= [image] ``` --- Use the ``PlotStyle`` for some segments by setting ``MeshShading`` to ``Automatic`` : ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 15, MeshFunctions -> {#5&}, PlotStyle -> Directive[Opacity[0.5], Yellow], MeshShading -> {Red, Automatic}] Out[1]= [image] ``` --- ``MeshShading`` can be used with ``ColorFunction`` : ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> 15, MeshFunctions -> {#5&}, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, Hue[θ]], MeshShading -> {Black, Automatic}] Out[1]= [image] ``` --- Fill between regions defined by multiple mesh functions: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, Hue[θ]], Mesh -> 15, MeshShading -> {{Black, Automatic}, {None, Orange}}] Out[1]= [image] ``` --- Use ``FaceForm`` to use different styles for different sides of a surface: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, MeshShading -> {{FaceForm[Red, None], FaceForm[None, Blue]}, {FaceForm[None, Cyan], FaceForm[Orange, None]}}] Out[1]= [image] ``` #### MeshStyle (2) Use a red mesh in the $x$ direction: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, MeshFunctions -> {#1&}, MeshStyle -> Red, BoundaryStyle -> None] Out[1]= [image] ``` --- Use a red mesh in the $x$ direction and a blue mesh in the $y$ direction: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, MeshFunctions -> {#1&, #2&}, MeshStyle -> {Red, Blue}] Out[1]= [image] ``` #### NormalsFunction (3) Normals are automatically calculated: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None] Out[1]= [image] ``` --- Use ``None`` to get flat shading for all the polygons: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, NormalsFunction -> None, Mesh -> None] Out[1]= [image] ``` --- Vary the effective normals used on the surface: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, NormalsFunction -> Function[{x, y, z, θ, ϕ, r}, RandomReal[{0, 2}, {3}]]] Out[1]= [image] ``` #### PerformanceGoal (2) Generate a higher-quality plot: ```wl In[1]:= Timing[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, PerformanceGoal -> "Quality"]] Out[1]= [image] ``` --- Emphasize performance, possibly at the cost of quality: ```wl In[1]:= Timing[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, PerformanceGoal -> "Speed"]] Out[1]= {0.020963, [image]} ``` #### PlotLegends (3) Use placeholders to identify plot styles: ```wl In[1]:= SphericalPlot3D[{Sin[θ], 2Sin[θ], 3Sin[θ]}, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}, PlotLegends -> Automatic] Out[1]= [image] ``` Use specific labels: ```wl In[2]:= SphericalPlot3D[{Sin[θ], 2Sin[θ], 3Sin[θ]}, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}, PlotLegends -> {"one", "two", "three"}] Out[2]= [image] ``` Use the expressions as legends: ```wl In[3]:= SphericalPlot3D[{Sin[θ], 2Sin[θ], 3Sin[θ]}, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}, PlotLegends -> "Expressions"] Out[3]= [image] ``` --- Use ``Placed`` to control legend position: ```wl In[1]:= SphericalPlot3D[{Sin[θ], 2Sin[θ], 3Sin[θ]}, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}, PlotLegends -> Placed[{"one", "two", "three"}, Below]] Out[1]= [image] ``` --- ```wl In[1]:= SphericalPlot3D[{Sin[θ], 2Sin[θ], 3Sin[θ]}, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}, PlotLegends -> SwatchLegend[{"one", "two", "three"}, LegendFunction -> "Frame"]] Out[1]= [image] ``` #### PlotPoints (1) Use more initial points to get a smoother plot: ```wl In[1]:= Table[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotPoints -> i, MaxRecursion -> 0, Mesh -> None, Axes -> None], {i, {5, 10, 15, 25}}] Out[1]= {[image], [image], [image], [image]} ``` #### PlotStyle (3) Use different style directives: ```wl In[1]:= Table[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotStyle -> ps], {ps, {Red, Directive[Orange, Specularity[White, 50]], Opacity[0.3], Directive[Purple, Opacity[0.5]]}}] Out[1]= [image] ``` --- Explicitly specify the style for different surfaces: ```wl In[1]:= SphericalPlot3D[{1, 2 + Sin[5ϕ] / 10}, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}, PlotStyle -> {Yellow, Red}] Out[1]= [image] ``` --- Use a different style inside the surface: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 3Pi / 2}, PlotStyle -> FaceForm[Red, Blue]] Out[1]= [image] ``` #### PlotTheme (3) Use a theme with detailed ticks, grid lines, and legends: ```wl In[1]:= SphericalPlot3D[1 + Sin[5 ϕ] Sin[10 θ] / 10, {θ, 0, π}, {ϕ, 0, 2π}, PlotTheme -> "Detailed"] Out[1]= [image] ``` --- Turn off the mesh lines: ```wl In[1]:= SphericalPlot3D[1 + Sin[5 ϕ] Sin[10 θ] / 10, {θ, 0, π}, {ϕ, 0, 2π}, PlotTheme -> "Detailed", Mesh -> None] Out[1]= [image] ``` --- Create a thick surface for 3D printing: ```wl In[1]:= SphericalPlot3D[ϕ, {θ, 0, Pi}, {ϕ, 0, 3Pi}, PlotTheme -> "ThickSurface"] Out[1]= [image] ``` #### RegionFunction (2) Select a region in $x$, $y$, $z$, $\theta$, $\phi$, and $r$: ```wl In[1]:= Table[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, RegionFunction -> Function[{x, y, z, θ, ϕ, r}, Evaluate[f]], PlotLabel -> f, Mesh -> None, BoundaryStyle -> Black, PlotRange -> 1.1, Axes -> None], {f, {0 < x < 1, 0 < y < 1, 0 < z < 1, 0 < θ < 1, 0 < ϕ < Pi / 2, 1 / 2 < r < 1}}] Out[1]= [image] ``` --- Select a region in parameter space: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, RegionFunction -> (Sin[5(#4 + #5)] > 0&), Mesh -> None, BoundaryStyle -> Black] Out[1]= [image] ``` #### ScalingFunctions (3) By default, plots have linear scales in all directions: ```wl In[1]:= SphericalPlot3D[θ ϕ, {θ, 0, Pi}, {ϕ, 0, 3Pi}] Out[1]= [image] ``` --- Use a shifted log scale to show a function with negative values: ```wl In[1]:= SphericalPlot3D[θ ϕ, {θ, 0, Pi}, {ϕ, 0, 3Pi}, ScalingFunctions -> {None, None, "SignedLog"}, BoxRatios -> {1, 1, 1}] Out[1]= [image] ``` --- Use ``ScalingFunctions`` to reverse the coordinate direction in the $z$ direction: ```wl In[1]:= SphericalPlot3D[θ ϕ, {θ, 0, Pi}, {ϕ, 0, 3Pi}, ScalingFunctions -> {None, None, "Reverse"}] Out[1]= [image] ``` #### TextureCoordinateFunction (4) Textures use scaled $\theta$ and $\phi$ parameters by default: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, PlotStyle -> Texture[ExampleData[{"ColorTexture", "MultiSpiralsPattern"}]]] Out[1]= [image] ``` --- Use the $x$ and $y$ coordinates: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, TextureCoordinateFunction -> ({#1, #2}&), PlotStyle -> Texture[ExampleData[{"ColorTexture", "MultiSpiralsPattern"}]]] Out[1]= [image] ``` --- Use unscaled coordinates: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, TextureCoordinateScaling -> False, PlotStyle -> Texture[ExampleData[{"ColorTexture", "MultiSpiralsPattern"}]]] Out[1]= [image] ``` --- Use textures to highlight how parameters map onto a surface: ```wl In[1]:= texture = ArrayPlot[{{1, 1, 1, 2, 1, 1}, {2, 0, 0, 2, 0, 0}, {2, 0, 0, 2, 0, 0}, {2, 1, 1, 1, 1, 1}, {2, 0, 0, 2, 0, 0}, {2, 0, 0, 2, 0, 0}}, ColorRules -> {1 -> Red, 2 -> Blue, 0 -> White}, Frame -> False, PlotRangePadding -> None, ImagePadding -> None, ImageSize -> 100] Out[1]= [image] In[2]:= SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, TextureCoordinateScaling -> False, PlotStyle -> Texture[texture]] Out[2]= [image] ``` #### TextureCoordinateScaling (1) Use scaled or unscaled coordinates for textures: ```wl In[1]:= texture = ArrayPlot[{{1, 1, 1, 2, 1, 1}, {2, 0, 0, 2, 0, 0}, {2, 0, 0, 2, 0, 0}, {2, 1, 1, 1, 1, 1}, {2, 0, 0, 2, 0, 0}, {2, 0, 0, 2, 0, 0}}, ColorRules -> {1 -> Red, 2 -> Blue, 0 -> White}, Frame -> False, PlotRangePadding -> None, ImagePadding -> None, ImageSize -> 100] Out[1]= [image] In[2]:= Table[SphericalPlot3D[1 + Sin[5ϕ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, TextureCoordinateScaling -> s, PlotStyle -> Texture[texture], PlotLabel -> s], {s, {True, False}}] Out[2]= [image] ``` #### Ticks (6) Ticks are placed automatically in each plot: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}] Out[1]= [image] ``` --- Use ``Ticks -> None`` to not draw any tick marks: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, Ticks -> None] Out[1]= [image] ``` --- Place tick marks at specific positions: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, Ticks -> {{-2.5, 0, 2.5}, {-2.5, 0, 2.5}, {-2.5, 0, 2.5}}] Out[1]= [image] ``` --- Draw tick marks at the specified positions with the specified labels: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, Ticks -> {{{-2.5, -a}, {0, 0}, {2.5, a}}, {{-2.5, -a}, {0, 0}, {2.5, a}}, {{-2.5, -a}, {0, 0}, {2.5, a}}}] Out[1]= [image] ``` --- Specify tick marks with scaled lengths: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, Ticks -> {{{-2.5, -a, .18}, {0, 0, .2}, {2.5, a, .05}}, {{-2.5, -a, .05}, {0, 0, .05}, {2.5, a, .05}}, {{-2.5, -a, .1}, {0, 0, .15}, {2.5, a, .15}}}] Out[1]= [image] ``` --- Customize each tick with position, length, labeling and styling: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, Ticks -> {{{-2.5, -a, .18, Directive[Red, Dashed, Thick]}, {0, 0, .2, Directive[Red]}, {2.5, a, .05, Directive[Red, Thick]}}, {{-2.5, -a, .05, Directive[Blue, Thick]}, {0, 0, .05, Directive[Blue, Thick]}, {2.5, a, .05, Directive[Blue, Thick]}}, {{-2.5, -a, .1, Directive[Darker@Green, Thick]}, {0, 0, .15, Directive[Darker@Green, Dashed, Thick]}, {2.5, a, .15, Directive[Darker@Green, Dashed]}}}] Out[1]= [image] ``` #### TicksStyle (3) By default, the ticks and tick labels use the same styles as the axis: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, AxesStyle -> Directive[Red, Thick]] Out[1]= [image] ``` --- Specify the overall tick style, including the tick labels: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, TicksStyle -> Directive[Red, Thick]] Out[1]= [image] ``` --- Specify the overall tick style for each of the axes: ```wl In[1]:= SphericalPlot3D[{1, 2, 3}, {θ, 0, Pi}, {ϕ, 0, 3 Pi / 2}, TicksStyle -> {Directive[Blue, Thick], Directive[Red, Thick], Directive[Brown, Thick]}] Out[1]= [image] ``` #### WorkingPrecision (2) Evaluate functions using machine-precision arithmetic: ```wl In[1]:= SphericalPlot3D[1 + Sin[5(θ + 10 ^ 16)], {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Evaluate functions using arbitrary-precision arithmetic: ```wl In[1]:= SphericalPlot3D[1 + Sin[5(θ + 10 ^ 16)], {θ, 0, Pi}, {ϕ, 0, 2Pi}, WorkingPrecision -> 20] Out[1]= [image] ``` ### Applications (5) Plot a sphere: ```wl In[1]:= SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- A spiraling shell: ```wl In[1]:= SphericalPlot3D[ϕ, {θ, 0, Pi}, {ϕ, 0, 3Pi}] Out[1]= [image] ``` --- An oscillation around a sphere: ```wl In[1]:= SphericalPlot3D[1 + Sin[5ϕ]Sin[10θ] / 10, {θ, 0, Pi}, {ϕ, 0, 2Pi}] Out[1]= [image] ``` --- Plot an eigenfunction to the Laplace equation in spherical coordinates: ```wl In[1]:= GraphicsGrid@Table[SphericalPlot3D[Evaluate@Abs@SphericalHarmonicY[l, m, θ, ϕ], {θ, 0, Pi}, {ϕ, 0, 2Pi}, PlotRange -> 0.6, Mesh -> None, Boxed -> False, Axes -> None], {l, 0, 3}, {m, 0, l}] Out[1]= [image] ``` --- Plot the absolute value and color by phase: ```wl In[1]:= SphericalPlot3D[Evaluate@Abs@SphericalHarmonicY[5, 2, θ, ϕ], {θ, 0, π}, {ϕ, 0, 2π}, Mesh -> None, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, Evaluate[Hue@Rescale[Arg@SphericalHarmonicY[5, 2, θ, ϕ], {-π, π}]]], ColorFunctionScaling -> False, PlotPoints -> 35] Out[1]= [image] ``` ### Properties & Relations (8) ``SphericalPlot3D`` is a special case of ``ParametricPlot3D`` : ```wl In[1]:= SphericalPlot3D[θ ^ 2, {θ, 0, Pi}, {ϕ, 0, 5Pi / 3}] Out[1]= [image] In[2]:= ParametricPlot3D[{θ ^ 2 Sin[θ]Cos[ϕ], θ ^ 2 Sin[θ]Sin[ϕ], θ ^ 2 Cos[θ]}, {θ, 0, Pi}, {ϕ, 0, 5Pi / 3}] Out[2]= [image] ``` --- Use ``RevolutionPlot3D`` for revolved surfaces and cylindrical coordinates: ```wl In[1]:= RevolutionPlot3D[t ^ 2, {t, 0, 2}] Out[1]= [image] In[2]:= RevolutionPlot3D[Sin[t] Sin[5θ], {t, 0, Pi}, {θ, 0, 2Pi}] Out[2]= [image] ``` --- Use ``ParametricPlot3D`` for arbitrary curves and surfaces in three dimensions: ```wl In[1]:= ParametricPlot3D[Evaluate[{Cos[u], Sin[u], -u / 5} / Sqrt[1 + (u / 5) ^ 2]], {u, -20, 20}] Out[1]= [image] In[2]:= ParametricPlot3D[{u - u ^ 3 / 3 + u v ^ 2, -v - u ^ 2v + v ^ 3 / 3, u ^ 2 - v ^ 2}, {u, -1.5, 1.5}, {v, -1.5, 1.5}] Out[2]= [image] ``` --- Use ``PolarPlot`` for curves in polar coordinates: ```wl In[1]:= PolarPlot[Sin[4t], {t, 0, 2Pi}] Out[1]= [image] ``` --- Use ``ParametricPlot`` for curves and regions in two dimensions: ```wl In[1]:= {ParametricPlot[{u Cos[u], u Sin[u]}, {u, 0, 4Pi}], ParametricPlot[{(u + v) Cos[u], (u + v) Sin[u]}, {u, 0, 4Pi}, {v, 0, 3}]} Out[1]= [image] ``` --- Use ``ContourPlot3D`` and ``RegionPlot3D`` for implicitly defined surfaces and regions: ```wl In[1]:= ContourPlot3D[x ^ 2 + y ^ 2 == z ^ 2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] Out[1]= [image] In[2]:= RegionPlot3D[x ^ 2 + y ^ 2 < z ^ 2, {x, -2, 2}, {y, -2, 2}, {z, -2, 2}] Out[2]= [image] ``` --- Use ``ListPlot3D`` and ``ListSurfacePlot3D`` for data: ```wl In[1]:= ListPlot3D[Table[Sin[x y], {x, 0, 3, 0.1}, {y, 0, 3, 0.1}]] Out[1]= [image] In[2]:= ListSurfacePlot3D[Table[{x = RandomReal[{-1, 1}], y = RandomReal[{-1, 1}], RandomChoice[{-1, 1}]Sqrt[1 - x ^ 2 - y ^ 2]}, {1000}]] Out[2]= [image] ``` --- Use ``Sphere`` for generating spheres: ```wl In[1]:= Graphics3D[Table[{RGBColor[i, j, k], Sphere[{i, j, k}, 0.1]}, {i, 0, 1, 0.2}, {j, 0, 1, 0.2}, {k, 0, 1, 0.2}], Lighting -> "Neutral"] Out[1]= [image] ``` ### Possible Issues (1) Surfaces that have multiple coverings may exhibit unusual behavior: ```wl In[1]:= {SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, -Pi, 7Pi / 4}, PlotStyle -> Opacity[0.5]], SphericalPlot3D[1, {θ, 0, Pi}, {ϕ, -Pi, 7Pi / 4}, ColorFunction -> Function[{x, y, z, θ, ϕ, r}, Hue[ϕ]]]} Out[1]= [image] ``` ### Neat Examples (2) An oscillating spherical surface: ```wl In[1]:= SphericalPlot3D[1 + Sin[5 ϕ] Sin[10 θ] / 10, {θ, 0, π}, {ϕ, 0, 2π}, ColorFunction -> (ColorData["Rainbow"][#6]&), Mesh -> None, PlotPoints -> 25, Boxed -> False, Axes -> False] Out[1]= [image] ``` --- An oscillating piecewise spherical surface: ```wl In[1]:= SphericalPlot3D[1 + FractionalPart[10ϕ / (2Pi)] / 5, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, ExclusionsStyle -> {None, Red}, PlotStyle -> Opacity[0.6]] Out[1]= [image] In[2]:= SphericalPlot3D[1 + FractionalPart[10θ / Pi] / 5, {θ, 0, Pi}, {ϕ, 0, 2Pi}, Mesh -> None, ExclusionsStyle -> {None, Red}, PlotStyle -> Opacity[0.6]] Out[2]= [image] ``` ## See Also * [`RevolutionPlot3D`](https://reference.wolfram.com/language/ref/RevolutionPlot3D.en.md) * [`ParametricPlot3D`](https://reference.wolfram.com/language/ref/ParametricPlot3D.en.md) * [`PolarPlot`](https://reference.wolfram.com/language/ref/PolarPlot.en.md) * [`Sphere`](https://reference.wolfram.com/language/ref/Sphere.en.md) * [`RotationMatrix`](https://reference.wolfram.com/language/ref/RotationMatrix.en.md) ## Tech Notes * [Some Special Plots](https://reference.wolfram.com/language/tutorial/GraphicsAndSound.en.md#31896) ## Related Guides * [Function Visualization](https://reference.wolfram.com/language/guide/FunctionVisualization.en.md) * [Angles and Polar Coordinates](https://reference.wolfram.com/language/guide/AnglesAndPolarCoordinates.en.md) ## History * [Introduced in 2007 (6.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn60.en.md) \| [Updated in 2008 (7.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn70.en.md) ▪ [2010 (8.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn80.en.md) ▪ [2014 (10.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn100.en.md) ▪ [2016 (11.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn110.en.md) ▪ [2022 (13.1)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn131.en.md)