Math 207
Cylinders and Quadratic Surfaces
Mariam Kreydem Office Room: 152-C
Building No. 07 2016, second semester.
This is a very short summary of the basic ideas we are facing in this course.
Note that this summary is NOT a replacement of the textbook provided for this level. Kindly, DO NOT try
to rely on it totally in order to make a feasible excuse
for textbook replacement.
• The word surface means the outside part or uppermost layer of some- thing. There are three types of surfaces:
- plane (flat) surface, such as books, tables, etc,..
- curved surface like apples, balls, earth, etc,
- surfaces which have the two of above characterization, that is, they are both curved and flat in the same time such as a clock, drums, etc,..
• To draw surfaces in 3D-space, we need to review some curves in 2D- plane.
-Parabolas. The simplest general form of this kind of curve is given by:
y=ax2. (1) Ifa >0, then the graph is open upward.
(2) Ifa <0, then the parabola is open downward.
-Ellipses. The general form is given by x2
a2 +y2 b2 = 1, where a and b are positive numbers.
-Hyperbolas. The general form is given by x2
a2 − y2 b2 = 1.
•If we ”flicked” the above curves about their axes, we will get 3D-shapes which are originated from them. To know which curve is used to create those shapes we need to add another axis so that we have 3 axes in total. Also, we need to study the traces; a shape makes with each axis:
- To findxy-trace, we put z = 0.
- To findxz-trace, we put y= 0.
- To findyz-trace, we put x= 0.
A Quadratic Surface
1. Ellipsoid.
x2 a2 + y2
b2 +z2 c2 = 1 Characteristic:
- All three variables present and have degree 2.
- All three terms of the equation are positive when equals to 1.
- All three traces are ellipses.
2. Hyperboloid of One Sheet.
x2 a2 + y2
b2 − z2 c2 = 1 Characteristic:
- All three variables present and have degree 2.
- Two terms of the equation are positive and one is negative when equals to 1.
- One trace is an ellipses.
- Two traces are hyperbolas.
- The axis of symmetry corresponds to the variable of negative sign.
3. Hyperboloid of Two Sheets.
x2 a2 − y2
b2 − z2 c2 = 1 Characteristic:
- All three variables present and have degree 2.
- Two terms of the equation are negative and one is positive when equals to 1.
- One trace is an ellipses.
- Two traces are hyperbolas.
- The axis of symmetry corresponds to the variable of positive sign.
Note that if we putx= 0, then we get
−y2 b2 −z2
c2 = 1
and this equation has no solution in R. To handle this problem, we setx=k such that
1− k2
a2 =−1.
For example,
x2 16− y2
9 − z2 4 = 1 Put x=√
32, and bring it to the other side of the equation as follows
−y2 9 − z2
4 = 1−
√32
16 =−1.
Now, this form has a solution in R, since in this case we have y2
9 + z2 4 = 1.
So, for all values less than √
32, the output is nothing, and this is the reason why we see a split in the shape into two parts with no connection in between.
4. Nothing!
−x2 a2 −y2
b2 − z2 c2 = 1.
This equation has no solution in all planes, and so the outcome is nothing!.
5. Original Point.
x2 a2 +y2
b2 + z2 c2 = 0.
The only possible solution for this equation is (0,0,0).
6. Elliptical Cone.
x2 a2 + y2
b2 − z2 c2 = 0 Characteristic:
- All three variables present and have degree 2.
- Two terms of the equation are positive and one is negative when equals to 0.
- Two traces are hyperbolas.
- One trace is the original point or an ellipse parallel to the third plane.
- The axis of symmetry corresponds to the variable of negative sign.
7. Elliptical Paraboloid.
x2 a2 + y2
b2 = z c2 Characteristic:
- All three variables present such that two terms have degree 2 and one term has degree 1.
- Two terms of the equation are positive and one is negative when equals to 0.
- Two traces are parabolas.
- One trace is an ellipse.
- The axis of symmetry corresponds to the variable of degree one.
8. Hyperbolic Paraboloid.
x2 a2 − y2
b2 = z c2 Characteristic:
- All three variables present such that two terms have degree 2 and one term has degree 1.
- Two terms of the equation are negative and one is positive when equals to 0.
- Two traces are parabolas.
- One trace is a hyperbola.
- The axis of symmetry corresponds to the variable of degree one.