1. The Plane: Recall that our derivation for an equation involved (x,y,z) as any point in space. Then, if Q:(x,y,z) is any point in space that is on a plane with Normal vector v = (a,b,c) and if P: is a particular point in that plane, the equation for the plane is the result of the dot product of a vector between P and Q and the Normal vector. We saw that Ax + By + Cz + D = 0 follows from that equation. I remind you of this because we can use a similar argument for our second surface in space, namely, the sphere.

2. The Sphere:If C: is the center of a sphere of radius R and Q:(x,y,z) is any point in space that is on the sphere then the equation for the sphere is the result of the computation of the norm of the vector between C and Q set equal to R.

This example should help: The center of a sphere is at ( -1,2,4) and the radius is 5 units. An equation for the sphere is

the norm of the vector ( x+1, y-2, z-4 ) set equal to 5. Then by squaring we get 

which results in 

And one more example:

3. The Cylindrical Surface or simply a Cylinder:

above. You can image that this cylinder is generated by the vertical line revolving around the circle in the xy-plane. This circle is the generating curve for the cylinder as indicated in the following definition.

Let C be a curve in a plane and let L be a line not in a parallel plane. The set of all lines parallel to L and intersecting C is called a cylinder. C is called the generating curve ( or directrix ) of the cylinder , and the parallel lines are called rulings.

To find an equation for the cylinder, note that you can generate any one of the rulings by fixing the values of x and y and then allowing z to take on all real values.( An example would be the vertical line in the picture above. ) In this sense the value of z is arbitrary and is , therefore, not included in the equation. In other words, the equation of this cylinder is simply the equation of the generating curve.( Points on the cylinder pictured above must stay on a circle but may be at any height, so to speak.)

4. Quadric Surfaces:

Look at the following solved problems in the Exercises at the end of 11.6 :1-6 ( list traces in the coordinate planes by observing the graphs; use algebra to get the same traces from the equations; use the info to do the matches ) ; 7-15 odd ( in this set you are not required to use a computer system to graph--instead, do paper and pencil sketches) ; 19-25 odd ; Then do the following unsolved as homework : 8-16 even( in this set you are not required to use a computer system to graph--instead, do paper and pencil sketches) ; 20-24 even