Paraboloids Available at www-math.bgsu.edu/~zirbel/calc3

> with(plots): Needed for contour plots.

Elliptic Paraboloid

> f:=(x,y)->(x^2)/3 + y^2; This defines a function f of two variables, x and y . It is said to be parabolic because of the and terms. Like a parabola, the whole surface opens up.

> plot3d(f,-10..10,-10..10,axes=BOXED,labels=["x axis", "y axis", "z axis"]);

> contourplot(f,-10..10,-10..10,labels=["x axis", "y axis"]); The contours are ellipses in the xy plane; that's why it's called an elliptic paraboloid.

Hyperbolic Paraboloid

> f:=(x,y)->-(x^2)/3 + y^2;

> plot3d(f,-10..10,-10..10,axes=BOXED,labels=["x axis", "y axis", "z axis"]); This surface curves both up and down, like a saddle or Pringles potato chip.

> contourplot(f,-10..10,-10..10,labels=["x axis", "y axis"]); This paraboloid is called hyperbolic because the contours are hyperbolae.

Rotated Hyperbolic Paraboloid

> f:=(x,y)->x^2 + 4*x*y + y^2; This paraboloid has an xy term. It isn't clear right away whether it is elliptic or hyperbolic, but the graph makes it clear.

> plot3d(f,-10..10,-10..10,axes=BOXED,style=PATCHCONTOUR,labels=["x axis", "y axis", "z axis"]); The PATCHCONTOUR style draws a surface plot with contour lines on it. This can be helpful, but isn't always good.

> contourplot(f,-10..10,-10..10,labels=["x axis", "y axis"], contours=20); The hyperbolae do not line up with the x and y axes; they are rotated by the xy term.

Parabolic cylinder

> f:=(x,y) -> (2*x-3*y)^2;

> plot3d(f(x,y),x=-10..10,y=-10..10,axes=boxed,style=patchcontour,labels=["x axis", "y axis", "z axis"]); It's hard to see unless you rotate this graph, but this surface only opens up, never down, but there is one direction in which the height of the surface stays constant. This paraboloid is balanced perfectly between the elliptic and hyperbolic cases.

> contourplot(f,-10..10,-10..10,labels=["x axis", "y axis"], contours=20); The contours are neither ellipses nor hyperbolae, but just straight lines.