Quadric Surfaces

Hyperboloid of One Sheet  x^2 + y^2 -z^2 = 1

Define Function

f[x_,y_]:=Sqrt[x^2+y^2-1];

3D Surface Plot - Standard Style

P1 = Plot3D[f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1",BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic];

P2 = Plot3D[-f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"x","y","z"}, PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1", BoxRatios->Automatic];

Show[{P1,P2}, PlotRange->All]

Contour Plot of z = f(x,y) Level Curves

ContourPlot[f[x,y],{x,-2,2},{y,-2,2}, FrameLabel->{"X","Y"}, PlotLabel->"Level Curves For (x^2) + (y^2) - (z^2) = 1", ColorFunction->Hue, Contours -> 40,BaseStyle->Directive[FontSize->16, FontFamily->"Times"]]

3D Surface Plot - Hue Color Function Option

P1 = Plot3D[f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1", ColorFunction->Function[{x,y,z},Hue[z]],BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic];

P2 = Plot3D[-f[x,y],{x,-2,2},{y,-2,2}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1", ColorFunction->Function[{x,y,z},Hue[z]],BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic];

Show[{P1,P2}, PlotRange->All]

3D Surface Plot  RegionFunction restriction to circular domain

P1 = Plot3D[f[x,y],{x,-3,3},{y,-3,3}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1", ColorFunction->Function[{x,y,z},Hue[z]], RegionFunction->Function[{x,y,z},x^2+y^2≤ 9],BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic];

P2 = Plot3D[-f[x,y],{x,-3,3},{y,-3,3}, AxesLabel->{"X","Y","Z"}, PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1", ColorFunction->Function[{x,y,z},Hue[z]], RegionFunction->Function[{x,y,z},x^2+y^2≤ 9],BaseStyle->Directive[FontSize->16, FontFamily->"Times"], BoxRatios->Automatic];

Show[{P1,P2}, PlotRange->All]

Plot plane x = 1.5 and show that it intersects the hyperboloid of one sheet as a hyperbola possessing the z-axis as a symmetry axis.

IntersectPlanePlot = ParametricPlot3D[{1.5, s,t},{s,-3,3},{t,-3,3},PlotStyle->Cyan];

Show[P1,P2,IntersectPlanePlot, PlotRange->All,PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1 w/ Intersecting Plane x = 1.5"]

Plot plane x = 0.5 and show that it intersects the hyperboloid of one sheet as a hyperbola possessing the z-axis as a symmetry axis.

IntersectPlanePlot = ParametricPlot3D[{0.5, s,t},{s,-3,3},{t,-3,3},PlotStyle->Cyan];

Show[P1,P2,IntersectPlanePlot, PlotRange->All,PlotLabel->"Plot of Surface (x^2) + (y^2) - (z^2) = 1 w/ Intersecting Plane x = 0.5"]