Level Curves of the Hyperbolic Paraboloid

Starring - The Hyperbolic Paraboloid

hyp = 
      Plot3D[x^2 - y^2, {x, -2, 2}, {y, -2, 2}, 
        BoxRatios -> {1, 1, 2}]

Implicit curves

s2 = Plot3D[2, {x, -2, 2}, {y, -2, 2}]

<< Graphics`ImplicitPlot`

ls2 = ImplicitPlot[x^2 - y^2 == 2, {x, -2, 2}, {y, -2, 2}]

s175 = Plot3D[1.75, {x, -2, 2}, {y, -2, 2}]

ls175 = 
      ImplicitPlot[x^2 - y^2 == 1.75, {x, -2, 2}, {y, -2, 2}]

s15 = Plot3D[1.5, {x, -2, 2}, {y, -2, 2}]

ls15 = 
      ImplicitPlot[x^2 - y^2 == 1.5, {x, -2, 2}, {y, -2, 2}]

s125 = Plot3D[1.25, {x, -2, 2}, {y, -2, 2}]

ls125 = 
      ImplicitPlot[x^2 - y^2 == 1.25, {x, -2, 2}, {y, -2, 2}]

ls1 = ImplicitPlot[x^2 - y^2 == 1, {x, -2, 2}, {y, -2, 2}]

s075 = Plot3D[0.75, {x, -2, 2}, {y, -2, 2}]

ls075 = 
      ImplicitPlot[x^2 - y^2 == 0.75, {x, -2, 2}, {y, -2, 2}]

s05 = Plot3D[0.5, {x, -2, 2}, {y, -2, 2}]

ls05 = 
      ImplicitPlot[x^2 - y^2 == 0.5, {x, -2, 2}, {y, -2, 2}]

s025 = Plot3D[0.25, {x, -2, 2}, {y, -2, 2}]

ls025 = 
      ImplicitPlot[x^2 - y^2 == 0.25, {x, -2, 2}, {y, -2, 2}]

s0 = Plot3D[0, {x, -2, 2}, {y, -2, 2}]

ls0 = ImplicitPlot[x^2 - y^2 == 0, {x, -2, 2}, {y, -2, 2}]

t025 = Plot3D[-0.25, {x, -2, 2}, {y, -2, 2}]

lt025 = 
      ImplicitPlot[x^2 - y^2 == -0.25, {x, -2, 2}, {y, -2, 2}]

t05 = Plot3D[-0.5, {x, -2, 2}, {y, -2, 2}]

lt05 = 
      ImplicitPlot[x^2 - y^2 == -0.5, {x, -2, 2}, {y, -2, 2}]

t075 = Plot3D[-0.75, {x, -2, 2}, {y, -2, 2}]

lt075 = 
      ImplicitPlot[x^2 - y^2 == -0.75, {x, -2, 2}, {y, -2, 2}]

t1 = Plot3D[-1, {x, -2, 2}, {y, -2, 2}]

lt1 = 
      ImplicitPlot[x^2 - y^2 == -1, {x, -2, 2}, {y, -2, 2}]

t125 = Plot3D[-1.25, {x, -2, 2}, {y, -2, 2}]

lt125 = 
      ImplicitPlot[x^2 - y^2 == -1.25, {x, -2, 2}, {y, -2, 2}]

t15 = Plot3D[-1.5, {x, -2, 2}, {y, -2, 2}]

lt15 = 
      ImplicitPlot[x^2 - y^2 == -1.5, {x, -2, 2}, {y, -2, 2}]

t175 = Plot3D[-1.75, {x, -2, 2}, {y, -2, 2}]

lt175 = 
      ImplicitPlot[x^2 - y^2 == -1.75, {x, -2, 2}, {y, -2, 2}]

t2 = Plot3D[-2, {x, -2, 2}, {y, -2, 2}]

lt2 = 
      ImplicitPlot[x^2 - y^2 == -2, {x, -2, 2}, {y, -2, 2}]

Explicit sections

hyp2 = Show[hyp, s2, PlotRange -> {-4, 4}]

hyp175 = Show[hyp, s175, PlotRange -> {-4, 4}]

hyp15 = Show[hyp, s15, PlotRange -> {-4, 4}]

hyp125 = Show[hyp, s125, PlotRange -> {-4, 4}]

hyp1 = Show[hyp, s1, PlotRange -> {-4, 4}]

hyp075 = Show[hyp, s075, PlotRange -> {-4, 4}]

hyp05 = Show[hyp, s05, PlotRange -> {-4, 4}]

hyp025 = Show[hyp, s025, PlotRange -> {-4, 4}]

hyp0 = Show[hyp, s0, PlotRange -> {-4, 4}]

hypt025 = Show[hyp, t025, PlotRange -> {-4, 4}]

hypt05 = Show[hyp, t05, PlotRange -> {-4, 4}]

hypt075 = Show[hyp, t075, PlotRange -> {-4, 4}]

hypt1 = Show[hyp, t1, PlotRange -> {-4, 4}]

hypt125 = Show[hyp, t125, PlotRange -> {-4, 4}]

hypt15 = Show[hyp, t15, PlotRange -> {-4, 4}]

hypt175 = Show[hyp, t175, PlotRange -> {-4, 4}]

hypt2 = Show[hyp, t2, PlotRange -> {-4, 4}]

Cast - in order of appearance

f1 = Show[GraphicsArray[{hyp2, ls2}]]

f2 = Show[GraphicsArray[{hyp175, ls175}]]

f3 = Show[GraphicsArray[{hyp15, ls15}]]

f4 = Show[GraphicsArray[{hyp125, ls125}]]

f5 = Show[GraphicsArray[{hyp1, ls1}]]

f6 = Show[GraphicsArray[{hyp075, ls075}]]

f7 = Show[GraphicsArray[{hyp05, ls05}]]

f8 = Show[GraphicsArray[{hyp025, ls025}]]

f9 = Show[GraphicsArray[{hyp0, ls0}]]

f10 = Show[GraphicsArray[{hypt025, lt025}]]

f11 = Show[GraphicsArray[{hypt05, lt05}]]

f12 = Show[GraphicsArray[{hypt075, lt075}]]

f13 = Show[GraphicsArray[{hypt1, lt1}]]

f14 = Show[GraphicsArray[{hypt125, lt125}]]

f15 = Show[GraphicsArray[{hypt15, lt15}]]

f16 = Show[GraphicsArray[{hypt175, lt175}]]

f17 = Show[GraphicsArray[{hypt2, lt2}]]

The unabridged version

<< Graphics`Animation`

ShowAnimation[{f1, f2, f3, f4, f5, f6, f7, f8, f9, f10, f11, f12, f13, 
        f14, f15, f16, f17}]