Example 1: The paraboloid example shows that any function with three variables can be visually represented using the Plot3D function. Also, the ColorFunction option allows you to easily see the changes of the function.

Plot3D[4*x^2 + y^2, {x, 0, 50}, {y, 0, 50}, Axes -> True, AxesLabel -> {y, x, z}, Boxed -> False, LabelStyle -> Italic, Mesh -> False, Filling -> Bottom, FillingStyle -> Opacity[.8], ColorFunction -> Hue, AspectRatio -> Automatic]

Example 2: The VectorPlot3D function is used to plot vector fields. Vector fields are used to represent equations that model many physical quantities such as the flow of liquid, strength of a force, and velocity of an object.

VectorPlot3D[{2*x^2, 3*y^2, -4*z^2}, {x, -20, 20}, {y, -10, 10}, {z, -10, 10}, BoxRatios -> {1, 1, 1}, VectorStyle -> Magenta]

\( \mbox{viviani}[a](t) = a (1+\cos t , \sin t, 2\sin (t/2)) \qquad (-2\pi \le t \le 2\pi ) \)

To plot the viviani curve, we first define its equation and then plot it

Vincenzo Viviani	Intersection of a cylinder with sphere	Viviani curve

Now we randomly cut out some balls, using the RandomPoint function, to obtain a piece of cheese. This code takes a while to generate an output due to the randomness. However, each time you run the code, a new figre will appear. The following code creates a random number of balls of radius 0.65 within the three dimensional region.

RegionPlot3D[ Fold[RegionDifference, ImplicitRegion[ x^2 + y^2 <= 6^2 && 0 < z < 4 && 0 < y <= x Sin[Pi/6], {x, y, z}], Ball[#, 0.65] & /@ RandomPoint[ ImplicitRegion[ x^2 + y^2 <= 6^2 && 0 < z < 4 && 0 < y <= x Sin[Pi/6], {x, y, z}], 10]], PlotPoints -> 100]

RegionPlot3D[ Fold[RegionDifference, ImplicitRegion[ x^2 + y^2 <= 6^2 && 0 <= z <= 4 && 0 <= y <= x Sin[Pi/6], {x, y, z}], Ball[#, 0.35] & /@ RandomPoint[ImplicitRegion[x^2 + y^2 <= 6^2 && 0 < z < 4 && 0 < y <= x Sin[Pi/6], {x, y, z}], 10]], PlotPoints -> 100]

RegionPlot3D[x^2 + y^2 <= 16, {x, -4, 4}, {y, -4, 4}, {z, 0, 2}, BoxRatios -> Automatic, AxesLabel -> Automatic, PlotStyle -> Gray, Mesh -> 8]

RegionPlot3D[x^2 + y^2 <= 4, {x, 0, 2}, {y, 0, 2}, {z, 0, 3}, BoxRatios -> Automatic, AxesLabel -> Automatic, ColorFunction -> "BlueGreenYellow"]

RegionPlot3D[ x^2 + y^2 + z^2 <= 9 && x^2 + y^2 + z^2 >= 4, {x, -3, 3}, {y, -3, 0}, {z, -3, 3}, BoxRatios -> Automatic, AxesLabel -> Automatic, ColorFunction -> "DarkRainbow", MeshShading -> Automatic]

RegionPlot3D[x^2 + y^2 + z^2 <= 1, {x, 0, 1}, {y, 0, 1}, {z, -1, 0}, BoxRatios -> Automatic, AxesLabel -> Automatic, ColorFunction -> Hue, MeshShading -> Automatic]

Example 5: Logarithmic functions are used to represent real world situations like population growth, decay of a substance, and financial interest rates. Most times these functions are shown in 2 dimensional spaces, but can still be represented in 3D.

G[x_, y_] := (Log[ Min[1, (1 + x)/(1 + y)]* Min[1, (1 + y)/(1 + x)]])/(Log[(Min[x/y, y/x])^2]) Plot3D[G[x, y], {x, 0.1, 15}, {y, 0.1, 15}, ColorFunction -> Function[{x, y, z}, Hue[0.65*(1 - z)]]]

Example 6: Now we plot surface of revolution with respect to x-axis. Using the RevolutionPlot3D function, you can create a simple 2D function into a cool 3D shape:

RevolutionPlot3D[{t^2, t}, {t, 0, 4}, {\[Theta], 0, 2 Pi},  RevolutionAxis -> "X"]

Revolution of parabola	Revolution of sine

Example 7: A catenoid is a type of surface, arising by rotating a catenary curve about an axis. It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally described in 1744 by the mathematician Leonhard Euler. The catenoid may be defined by the following parametric equations:

A helicoid minimal surface formed by a soap film on a helical frame	A catenoid	A catenoid obtained from the rotation of a catenary	Deformation of a helicoid into a catenoid

Example 8:

a = SphericalPlot3D[{1}, {\[Theta], 0, 2 Pi}, {\[Phi], 0, 4 Pi/2}, PlotStyle -> Directive[Blue, Opacity[0.5], Specularity[White, 20]], Mesh -> None, PlotPoints -> 450];
b = SphericalPlot3D[{2}, {\[Theta], 0, Pi}, {\[Phi], 0, 5 Pi/3}, PlotStyle -> Directive[Blue, Opacity[0.5], Specularity[White, 20]], Mesh -> {{0}, {0}, {0}}, PlotPoints -> 50];
c = Graphics3D[Arrow[Tube[{{2, 2, 2}, {1, 0, 0}}, .03]]];
d = Graphics3D[Arrow[Tube[{{0, -3, 1}, {0, -2, 1}}, .03]]];
e = Graphics3D[Arrow[{{0, 0, 0}, {3, 0, 0}}]];
f = Graphics3D[Arrow[{{0, 0, 0}, {0, -3, 0}}]];
g = Graphics3D[Arrow[{{0, 0, 0}, {0, 0, 3}}]];
h = Graphics3D[Arrow[{{0, 0, 0}, {-3, 0, 0}}]];
i = Graphics3D[Arrow[{{0, 0, 0}, {0, 3, 0}}]];
j = Graphics3D[Arrow[{{0, 0, 0}, {0, 0, -3}}]];
k = Graphics3D[Text[P == 1, {2, 2, 2}]];
l = Graphics3D[Text[P == 2, {0, -3, 1}]];
m = Graphics3D[Text[x, {0, -3.1, 0}]];
n = Graphics3D[Text[y, {3.1, 0, 0}]];
o = Graphics3D[Text[z, {0, 0, 3}]];
az = Show[a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, PlotRange -> {{-3, 3}, {-3, 3}, {-3, 3}}, Axes -> {False, False, False}, Boxed -> False] We can plot sphere with different colors.

Two spheres	Two spheres with different colors

c = SphericalPlot3D[4, {\[Theta], 0, \[Pi]}, {\[Phi], 1, 2 \[Pi]}]
d = SphericalPlot3D[2, {\[Theta], 0, Pi}, {\[Phi], 0, 2 Pi}]
Show[c, d]

Two spheres	Two spheres with different colors

a = Graphics3D[Arrow[Tube[{{0, 0, 1}, {0, 0, 2}}, .025]]];
c = Graphics3D[Arrow[Tube[{{0, 0, 1}, {0, 0, 0}}, .025]]];
o = Graphics3D[Text[height, {-0.5, -0.5, 1}]];
p = Graphics3D[Arrow[Tube[{{1, 0, 1}, {0, 0, 1}}, .025]]];
q = Graphics3D[Arrow[Tube[{{1, 0, 1}, {1.75, 0, 1}}, .025]]];
r = Graphics3D[Text[radius, {1, 0, 0.8}]];
d = Graphics3D[Cylinder[{{0, 0, 0}, {0, 0, .0001}}, 2]];
b = SphericalPlot3D[{2}, {\[Theta], 0, Pi}, {\[Phi], 0, 3 Pi/2}, PlotStyle -> Directive[Red, Opacity[0.7], Specularity[White, 20]], Mesh -> {{0}, {0}, {0}}, PlotPoints -> 40];
g = Graphics3D[Arrow[{{0, 0, 0}, {0, 0, 3}}]];
h = Graphics3D[Arrow[{{0, 0, 0}, {0, -3, 0}}]];
i = Graphics3D[Arrow[{{0, 0, 0}, {3, 0, 0}}]];
j = Graphics3D[Text[z, {0, 0, 3.2}]];
k = Graphics3D[Text[x, {3.2, 0, 0}]];
l = Graphics3D[Text[y, {0, -3.2, 0}]];
m = Graphics3D[Arrow[{{0, 0, 0}, {-3, 0, 0}}]];
n = Graphics3D[Arrow[{{0, 0, 0}, {0, 3, 0}}]];
b = Show[a, b, c, d, g, h, i, j, k, l, m, n, o, p, q, r, PlotRange -> {{-4, 4}, {-4, 4}, {0, 4}}]

a = Graphics3D[Arrow[Tube[{{0, 0, 0}, {0, 0, Sqrt[2]}}, .025]]];
c = Graphics3D[Arrow[Tube[{{0, 0, 0}, {1, 0, 1}}, .025]]];
d = Graphics3D[Cylinder[{{0, 0, 0}, {0, 0, .0001}}, 2]];
b = SphericalPlot3D[{2}, {\[Theta], 0, Pi}, {\[Phi], 0, 5 Pi/4}, PlotStyle -> Directive[Red, Opacity[0.7], Specularity[White, 20]], Mesh -> {{0}, {0}, {0}}, PlotPoints -> 40];
g = Graphics3D[Arrow[{{0, 0, 0}, {0, 0, 3}}]];
h = Graphics3D[Arrow[{{0, 0, 0}, {0, -3, 0}}]]; i = Graphics3D[Arrow[{{0, 0, 0}, {3, 0, 0}}]];
j = Graphics3D[Text[z, {0, 0, 3.2}]];
k = Graphics3D[Text[y, {3.2, 0, 0}]];
l = Graphics3D[Text[x, {0, -3.2, 0}]];
m = Graphics3D[Arrow[{{0, 0, 0}, {-3, 0, 0}}]];
n = Graphics3D[Arrow[{{0, 0, 0}, {0, 3, 0}}]];
b = Show[a, b, c, d, g, h, i, j, k, l, m, n, PlotRange -> {{-4, 4}, {-4, 4}, {0, 4}}]

RegionPlot3D[ x^2 + y^2 <= 6^2 && 0 < z < 4 && 0 < y <= .1 x Sin[Pi/6], {x, 0, 6}, {y, 0, Sin[Pi/6]}, {z, 0, 4}]
a = SphericalPlot3D[4, {\[Theta], 0, \[Pi]/2}, {\[Phi], 1, 2 \[Pi]}]
b = SphericalPlot3D[4, {\[Theta], .8, \[Pi]/2}, {\[Phi], 0, 2 \[Pi]}]
Show[a, b]

One can put a box inside a sphere:

e = SphericalPlot3D[4, {\[Theta], 0, \[Pi]/2}, {\[Phi], 3, 2 \[Pi]}]
f = Graphics3D[Cuboid[{-2, -1, 2}]]
Show[e, f]

Spherical shell with a box	Meshed spherical shell with a box

Example 9: Arrow plots can be used to show the direction of a line by adding arrows to the functions. A typical arrow plot:

Arrow[Tube[ Table[{Cos[phi], Sin[phi], .2 phi}, {phi, 0, 10, 0.1}]]] // Graphics3D

Typical arrow	Colored arrow	Colored line arrow

With[{pt = {-1, 2, 3}, nrm = {3, -2, 1}}, Show[ParametricPlot3D[ pt + RotationTransform[{{0, 0, 1}, nrm}][ {t Cos[2 t], t Sin[2 t], 0}] // Evaluate, {t, 0, 2 π}], Graphics3D[{{Red, AbsolutePointSize[8], Point[pt]}, {LightBlue, InfinitePlane[pt, Rest[Orthogonalize[{nrm, {1, 0, 0}, {0, 1, 0}}]]]}}], Lighting -> "Neutral"]]

p = {-1, 2, 3}; point = ListPointPlot3D[{p}, PlotStyle -> {Blue, PointSize[Large]}]

eqPlane = 3 (x + 1) - 2 (y - 2) + (z - 3) == 0;
f[x_, y_] := z /. First @ Solve[eqPlane, z]
f[x, y]
plane = ParametricPlot3D[{x, y, f[x, y]}, {x, -10, 10}, {y, -10, 10}, Mesh -> None, BoxRatios -> 1

{a[t_], b[t_], c[t_]} := {t Cos[2 t], t Sin[2 t], f[a[t], b[t]]}

s0 = {a[0], b[0], c[0]}

spiral = ParametricPlot3D[{a[t], b[t], c[t]} - s0 + p, {t, 0, 2 π}, PlotStyle -> Red, BoxRatios -> 1]
view = Coefficient[Simplify[First@eqPlane], #] & /@ {x, y, z}
Show[spiral, plane, point, ViewPoint -> view]

{x[t_], y[t_]} = {t Cos[2 t], t Sin[2 t]};
p = ({x, y, z} /. First@Solve[(x + 1) - 2 (y - 2) + (z - 3) == 0, z])
q = p /. {x -> x[t], y -> y[t]};
Show[Plot3D[Evaluate[p[[3]]], {x, -10, 10}, {y, -10, 10}, Mesh -> None, PlotStyle -> {LightBlue, Opacity[0.5]}, ViewPoint -> {5, 5, 5}], ParametricPlot3D[Evaluate[q], {t, 0, 2 Pi}, PlotStyle -> Red]]

View point (5,5,5)	View point (-5,5,5)	View point (5,-5,d105)

2. Visualization and Graphics by John Boccio, Wolfram Mathematica Tutorial Collection.