Rotate a conic section in 2 dimensions to create a quadratic surface in 3 dimensions. For instance, rotate the parabola about its axis to get a paraboloid. Its equation is x2+y2 = z. In other words, the distance from the z axis, when squared, gives the height (z coordinate). This is the paraboloid, the shape of the parabolic mirror in your flashlight, and in the Space Telescope. If a light ray approaches the mirror running parallel to the z axis, restrict attention to the plane containing the z axis and the light ray. This plane intersects the paraboloid in a parabola, which reflects the light ray towards the focus as described earlier. Thus the paraboloid has the same optical properties as the parabola, but in 3 dimensions.
Similarly, an ellipse, rotated about its major axis, produces a prolate spheroid, and light emanating from one focus is reflected to the other.
So - what are the quadratic surfaces in 3 dimensions anyways?
If one of the variables is missing, e.g. there is no term associated with z, then we have a conic section in the xy plane as before, and the same curve appears for every z coordinate. A circle extends to an infinite cylinder, and an ellipse becomes an elliptical cylinder. The degenerate conics extend as well. Parallel lines become parallel planes, and intersecting lines become intersecting planes. With this case behind us, let's assume all variables are present, in squared or linear form.
If two of the variables are linear then the third must be squared, else we wouldn't have a quadratic form. Consider the following example.
x2 + 5y + 7z = 0
Let u = 5y+7z and let v = 7y-5z. Our equation becomes x2+u = 0, and v does not participate. This is an extended parabola, as described above. whenever there are two linear variables, apply a change of basis to produce an extended parabola.
Next assume there is one linear variable and two squares, as in:
ax2 + by2 = z
If there were a constant c, we may as well fold it into the variable z.
We now have a conic section for each level z, and a degenerate conic when z = 0. Consider a = b = 1. This produces a point at the origin, and a circle for each positive z, with radius sqrt(z). This is the paraboloid discussed earlier. When a and b are not equal, the resulting surface is an elliptical paraboloid. Each cross section is an ellipse.
If a and b have opposite sign, then each cross section is a hyperbola, except z = 0, which yields two intersecting lines. As we move down towards z = 0, the branches of the hyperbola squeeze closer and closer to their asymptotes. At z = 0 they reach their asymptotes. As z goes negative the branches move to the other sides of their asymptotes, pulling farther away as z approaches -infinity. This is called a hyperbolic paraboloid. You can see the hyperbolas, but you have to turn your head a bit to see the parabolas. Restrict attention to the plane y = cx for some constant c. Now some multiple of x2 equals z, and that's a parabola.
This shape is also called a saddle, since it curves upward in front and back, and down on the sides - just like a real saddle. You may have seen this shape in another form, namely z = xy.
Finally assume all three variables are squared. We can't simply fold the constant into z any more, so there are actually four coefficients to consider.
ax2 + by2 + cz2 = d
Start with d = 0, the degenerate surfaces. At least two coefficients have the same sign, so assume a and b are positive. If z is positive, only the origin will do, but if z is negative, the result is an elliptical cone. Set a = b to produce a traditional cone, circular in cross section. The distance from the z axis, squared, is proportional to the z coordinate squared, hence the radius is proportional to the height, and that's a cone.
If d is nonzero, divide through so that d = 1. If a b and c are negative we are out of luck. If they are positive we have an ellipsoid. If they are all equal the surface is a sphere. If two of the three coefficients are equal, the surface is a spheroid.
If a and b are positive and c is negative, move cz2 to the other side. As z increases or decreases, the ellipse in the xy plane expands according to 1+cz2. If the ellipse is a circle, i.e. a = b, the surface is a hyperboloid of one sheet. It has a double cone as its asymptote. This is a surface of revolution. Place the branches of your favorite hyperbola to the left and right of the y axis, with a giant letter X acting as asymptotes, and spin the whole thing around the y axis. The X becomes the double cone, and the branches form the hyperboloid. If the base conic is an ellipse, rather than a circle, the surface is not a surface of revolution any more, since each cross section is an ellipse. Yet the surface is still called a hyperboloid, or perhaps an elliptical hyperboloid if you want to be precise. It has an elliptical cone as asymptote.
Last but not least, two of the coefficients could be negative. Rewrite the equation in the following form, using all positive coefficients.
ax2 + by2 = cz2 - 1
When z is close to 0, there is no solution. When z = sqrt(1/c), x = y = 0, a single point. As z moves farther away from 0 the ellipse grows larger. This is a hyperboloid, or elliptical hyperboloid, of two sheets. The term "two sheets" indicates two surfaces that are not connected. One lies above the xy plane and the other lies below the xy plane. When the cross section is a circle we have a surface of revolution. Draw the capital X as before, but put the branches of the hyperbola above and below the X. Now rotate about the y axis. The X creates the double cone, the upper branch becomes an upper sheet inside the top cone, and the lower branch becomes a lower sheet inside the bottom cone.
That completes the quadratic surfaces in 3 space.