|
        
4. Working with Graphics3D Packages
Standard
Mathematica provides built-in commands to render
function graphs, parametrized surfaces and curves in 3D.
Function Graph
| Graph of a scalar
function on a rectangular domain. |
In[1]:= fg = Plot3D[y/6 Sin[x], {x, -Pi, Pi}, {y, 0.,
6}]
In[2]:=
JavaView[fg]
|
Compare with Mathematica |
Parametrized Surfaces
| Sphere in standard
parametrization with meridians and parallels. |
In[1]:= gp = ParametricPlot3D[{Cos[u] Cos[v], Sin[u] Cos[v],
Sin[v]}, {u, 0, 2Pi}, {v, -Pi/2, Pi/2}, PlotPoints -> {40, 10}];
In[2]:=
JavaView[gp]
|
Compare with Mathematica |
3D Curves
| A closed curve in
space. |
In[1]:= cv = ParametricPlot3D[{Cos[5t], Sin[3t], Sin[t]}, {t,
0, 2 Pi}];
In[2]:=
JavaView[cv]
|
Compare with Mathematica |
Graphics`ContourPlot3D`
Compute implicit surfaces as zero set of a function or
of a scalar values given at vertices of a ZxZxZ grid.
| Zero level of a
function.
|
In[1]:= <<Graphics`ContourPlot3D`
In[2]:= gc = ContourPlot3D[ Cos[Sqrt[x^2 + y^2 + z^2]], {x,
-2, 2}, {y, 0, 2}, {z, -2, 2}];
In[3]:=
JavaView[gc]
|
Compare with Mathematica |
| Zero level of a
regular scalar field in 3D. |
In[1]:= <<Graphics`ContourPlot3D`
In[2]:= data = Table[ x^2 + 2*y^2 + 3*z^2, {z, -1, 1, .25},
{y, -1, 1, .25}, {x, -1, 1, .25}];
In[3]:= gl = ListContourPlot3D[data, MeshRange -> {{-1,
1}, {-1, 1}, {-1, 1}}, Lighting -> False, Contours -> {1.5, 3.},
Axes -> True, ContourStyle -> {{RGBColor[0, 1, 0]}, {RGBColor[1, 0,
0]}}]
In[4]:=
JavaView[gl]
|
Compare with Mathematica |
Graphics`Graphics3D`
Package supplies different representations of 3D data
like bar chart and shadow plots.
Bar Charts
Display scalar values on a uv-mesh as vertical columns
whose height represent the value.
| This gives a bar
chart made from a two-by-three array of bars with integer height. |
In[1]:= <<Graphics`Graphics3D`
In[2]:= gb = BarChart3D[{{1, 2, 3}, {4, 5, 6}}, BoxRatios
-> Automatic];
In[3]:=
JavaView[gb]
|
Compare with Mathematica |
ScatterPlot3D
ScatterPlot3D[] shows a set of 3D points and thereby
extends ListPlot to 3D. Optionally, the points may be connected by a line.
| Here is a list of
points in three dimensions. The scatter plot of points lies on a conical
helix. |
In[1]:= <<Graphics`Graphics3D`
In[2]:= lpts = Table[{t Cos[t], t Sin[t], t}, {t, 0, 4Pi,
Pi/20}];
In[3]:= sc = ScatterPlot3D[lpts]
In[4]:=
JavaView[sc]
|
Compare with Mathematica |
| Join the points of
the same list with a thicker line. |
In[1]:= <<Graphics`Graphics3D`
In[2]:= lpts = Table[{t Cos[t], t Sin[t], t}, {t, 0, 4Pi,
Pi/20}];
In[3]:= scl = ScatterPlot3D[lpts, PlotJoined -> True,
PlotStyle -> Thickness[0.01]]
In[4]:=
JavaView[scl]
|
Compare with Mathematica |
ListSurfacePlot3D uses an array of points and generates
the faces of a surface.
| Take an array of
points in three dimensions.
The array of points is used to generate vertices
of a polygonal mesh. It creates a piece of a sphere. |
In[1]:= <<Graphics`Graphics3D`
In[2]:= apts = Table[{Cos[t] Cos[u], Sin[t] Cos[u], Sin[u]},
{t, 0, Pi, Pi/5}, {u, 0, Pi/2, Pi/10}];
In[3]:= lsp = ListSurfacePlot3D[apts]
In[4]:=
JavaView[lsp]
|
Compare with Mathematica |
Shadow Plot
Draw shadows of a surface at its bounding box.
ShadowPlot3D and ListShadowPlot3D work exactly like the builtin Plot3D and
ListPlot3D, except shadows are drawn.
| Plot a 2D shadow
below a 3D surface. |
In[1]:= <<Graphics`Graphics3D`
In[2]:= sp = ShadowPlot3D[Sin[x y], {x, 0, 3}, {y, 0,
3}]
In[3]:=
JavaView[sp]
|
Compare with Mathematica |
| ShadowPosition
determines whether shadow is above or below. |
In[1]:= <<Graphics`Graphics3D`
In[2]:= st = ShadowPlot3D[Exp[-(x^2 + y^2)], {x, -2, 2}, {y,
-2, 2}, ShadowPosition -> 1]
In[3]:=
JavaView[st]
|
Compare with Mathematica |
| Plot shadows at
the bounding box of a 3D surface. |
In[1]:= <<Graphics`Graphics3D`
In[2]:= dbell = ParametricPlot3D[{Sin[t], Sin[2t] Sin[u],
Sin[2t] Cos[u]}, {t, -Pi/2, Pi/2}, {u, 0, 2Pi}, Ticks -> None];
In[3]:= sa := Shadow[dbell, ZShadow -> False];
In[4]:=
JavaView[sa]
|
Compare with Mathematica |
Graphics`ParametricPlot3D`
The command ParametricPlot3D[] in this package expands
the built-in command by allowing du and dv increments to be specified.
| Determine number
of lines through du and dv instead of PlotPoints. |
In[1]:= <<Graphics`ParametricPlot3D`
In[2]:= gps = ParametricPlot3D[{Cos[u] Cos[v], Sin[u] Cos[v],
Sin[v]}, {u, 0, 2Pi, Pi/20}, {v, -Pi/2, Pi/2, Pi/10}];
In[3]:=
JavaView[gps]
|
Compare with Mathematica |
| Only a collection
of points is shown when you use PointParametricPlot3D. |
In[1]:= <<Graphics`ParametricPlot3D`
In[2]:= gpp = PointParametricPlot3D[{Cos[u] Cos[v], Sin[u]
Cos[v], Sin[v]}, {u, 0, 2Pi}, {v, -Pi/2, Pi/2}];
In[3]:=
JavaView[gpp]
|
Compare with
Mathematica |
Graphics`PlotField3D`
Creates and renders vector fields in 3D.
| A 3D vortex vector
field. |
In[1]:= <<Graphics`PlotField3D`
In[2]:= gv3 = PlotVectorField3D[{y, -x, 0}/z, {x, -1, 1}, {y,
-1, 1}, {z, 1, 3}]
In[3]:=
JavaView[gv3]
|
Compare with Mathematica |
| The same vector
field with VectorHeads. |
In[1]:= <<Graphics`PlotField3D`
In[2]:= gv3h = PlotVectorField3D[{y, -x, 0}/z, {x, -1, 1},
{y, -1, 1}, {z, 1, 3}, VectorHeads -> True]
In[3]:=
JavaView[gv3h]
|
Compare with Mathematica |
Graphics`Polyhedra`
This package contains a collection of Platonic solids
and commands for their modification like Stellate[] and Truncate[].
| Load package
Graphics`Polyhedra and display a dodecahedron in a JavaView display. |
In[1]:= <<Graphics`Polyhedra`
In[2]:= dode = Graphics3D[Dodecahedron[]];
In[3]:=
JavaView[dode]
|
Compare with Mathematica |
Graphics`Shapes`
A collection of primitive surfaces including sphere,
torus, Möbius band etc. is provided by this package.
| Combine two shapes
and position them in a scene.
JavaView accepts a list of geometries. |
In[1]:= << Graphics`Shapes`
In[2]:= g1 =
TranslateShape[Graphics3D[Torus[]], {1., 0., 0.}];
In[3]:= g2 = RotateShape[Graphics3D[MoebiusStrip[1., 0.2,
30]], Pi/2., Pi/2., 0.];
In[4]:=
JavaView[{g1,g2}]
|
Compare with Mathematica |
Graphics`SurfaceOfRevolution`
This package simplifies the task to generate a surface
of revolution from a planar (or non-planar) curve by rotation around the default
z-axis, or about a user specified rotation axis.
| The curve Sin[x]
is rotated around the z-axis with an offset. |
In[1]:= << Graphics`SurfaceOfRevolution`
In[2]:= g = SurfaceOfRevolution[{1.2 + Sin[x], x}, {x, 0,
2Pi}];
In[3]:=
JavaView[g]
|
Compare with
Mathematica |
The Pseudosphere is obtained by rotating a tractix
around the z-axis.
| Rotate the
tractrix {f[x],g[x]} around the z-axis. |
In[1]:= << Graphics`SurfaceOfRevolution`
In[2]:= gPsd = SurfaceOfRevolution[{ Exp[-Abs[x]],
-Sign[x](Sqrt[1 - Exp[-2 Abs[x]]] - Log[(1 + Sqrt[1 - Exp[-2
Abs[x]]])/Exp[-Abs[x]]])}, {x, -2., 2.}]
In[3]:=
JavaView[gPsd]
|
Compare with Mathematica | |
