1 國立交通大學應用數學系 莊重教授
§12.6 Cylinders and Quadrate Surface
Definition：
1. Cylinders：A surface consisting of all lines (called rulings) that parallel to a given line and pass through a given plane curve.
Example：
z = x2 (Parabolic cylinder)
i. The rulings of the cylinder are parallel to the y-axis.
ii. The rulings of the cylinder pass through a given plane curve z = x2 and y = 0.
iii. If one of the variables x, y or z is missing from the equation of a surface, then the surface is a cylinder.
2. Quadratic Surface vs. Quadratic Curves
2nd – degree equation in 3 (resp. z) variable x, y and z (resp. x and y) 高中：Quadratic Curves
0 0
2 2 2
) (
Hy Ax
J By
矩陣 Ax
平移和轉軸
大學：Quadratic Surfaces
0 0
2 2
2 2 2
) (
Iz By Ax
J Cz By
矩陣 Ax
平移和轉軸
3. Standard Quadratic Surfaces
3 國立交通大學應用數學系 莊重教授
Example 1：
Match the equation with its graph (labeled I - VIII). Give reason for your choice.
2 2
2 2 2
2 2 2
2 2 2
2 2
2 2
2 2 2
2 2 2
. 8
2 .
6
1 .
4
1 4
9 . 2
1 2 . 7
2 . 5
1 .
3
1 9 4 . 1
z x y
z x y
z y x
z y x
z x
z x y
z y x
z y x
Example 2：
Identify the trace cross section of the surface x = 2y2 + 3z2 in the plane x = 1.
(1) Ellipse but not circle (2) Parabola
(3) Hyperbola (4) Circle
(5) Two Parallel straight lines (6) Two intersection straight lines
(7) Point (8) Straight Line
5 國立交通大學應用數學系 莊重教授
Example 3：
Given the following graph：
Which equation in the following matches its graph as above?
A) 9x2 + 4y2 + z2 = 1 B) y = x2 – z2 C) x2 – y2 + z2 = 1 D) –x2 + y2 – z2 = 1 Example 4：
The graph
is the equation of
A) x2 + 4y2 + 9z2 = 1 B) x2 – y2 + z2 = 1 C) –x2 + y2 – z2 = 1 D) y2 = x2 + 2z2
