#### You've reached the end of your free preview.

Want to read all 13 pages?

**Unformatted text preview: **Lecture 6 - Friday, April 11 CYLINDERS AND QUADRIC SURFACES (§12.6)
We have seen that a linear equation like
ax + by + cz = d
gives a plane in R3.
What about surfaces defined by quadratic equations
(quadratic in x, y, z)?
The situation in R2
In R2 a quadratic equation gives one of the following curves:
• y = ax 2 → parabola
a>0
y a<0
y x x 41 • ( ax )2 + ( yb )2 = 1 → ellipse y
x • hyperbola y y
x x ( yb )2 − ( ax )2 = 1 ( ax )2 − ( yb )2 = 1 Remark: All the curves can be shifted 42 Scaling and translating
• Scaling:
y
2 2 1
−2 −1
−1 y 1 x
1 2 −2
x 2 + y2 = 1 1 x −2 −1
−1 2 1 y −2 −1
−1 2 −2
( x2 )2 + y2 = 1 x
1 −2
(2x)2 + y2 = 1 Dividing x by a means stretching the object by a factor
a along the x axis
• Translating:
2 y 2 1
−2 −1
−1 1 2 −2
x 2 + y2 = 1 y 1 x −2 −1
−1 x
1 2 −2
(x − 1)2 + y2 = 1 Subtracting a from x means moving the object by a in
direction of the x-axis. 43 2 Def: Given a plane curve in R3, a cylinder is a set of all
parallel lines through this curve.
Example: ( x4 )2 + z 2 = 1 describes an elliptic cylinder
z
1
x
4 z y x 44 Example: z = y2 describes an parabolic cylinder
z
y z y x
Observe: Intersecting any parallel translation of the yzplane with the cylinder gives the same plane curve. 45 Recall: x 2 + y2 + z 2 = r 2 defines a sphere of radius r.
z y x 46 An ellipsoid is of the form ( ax )2 + ( yb )2 + ( zc )2 = 1.
z y x
Intersection with any coordinate planes (= trace) gives an
ellipse 47 An cone is of the form ( zc )2 = ( ax )2 + ( yb )2.
z y x Intersections with shifted xy-plane are ellipses.
For a = b = c, the intersection with z = z0 is a sphere of
radius |z0|. z0
radius 48 An elliptic paraboloid is of the form z
c = ( ax )2 + ( yb )2. z y x
Intersections with shifted xy-plane are ellipses.
For a = b = c = 1, the intersection with z = z0 is a sphere
√
of radius z0.
z0 radius 49 An hyperboloid of one sheet is of the form
y
z
x
( )2 + ( )2 − ( )2 = 1.
a
b
c
z y x
Intersections with shifted xy-plane are ellipses
For a =
qb = c = 1, intersection with z = z0 is sphere of
radius 1 + z02
z0 radius
50 An hyperboloid of two sheets is of the form
( ax )2 + ( yb )2 = ( zc )2 − 1.
z y x Intersections with xy-plane are ellipses
For a = b = c = 1, intersection with z = z0 is sphere of
√
radius z 2 − 1
z0
radius 51 Example 1: What type of surface is z = x 2 + y2?
Solution: Intersections with shifted xy-plane is a sphere of
√
radius z ⇒ elliptic paraboloid. Example 2: What type of surface is 36x 2 − 9y2 + 4z 2 = 36?
Solution: After scaling/shrinking along x, y and z-axis,
the surface is of the same type as x 2 − y2 + z 2 = 1.
Hence this is a hyperboloid of one sheet. 52 Example: Classify the surface given by
x 2 + 2z 2 − 6x − y + 4z + 11 = 0.
Idea: Bring eq. into form (x−?)2 − y + (z−?)2 =?. We rewrite the equation (x 2 − 6x) + 2(z 2 + 2z) = y − 11 ⇔ (x 2 − 6x + 32) + 2(z 2 + 2z + 1) = y − 11 + 2 + 9
⇔ (x − 3)2 + 2(z + 1)2 = y This is an elliptic paraboloid, shifted by (3, 0, −1). The central axis is (3, 0, −1) + t(0, 1, 0) for t ≥ 0
z y x
(3, 0, −1) 53 ...

View
Full Document

- Spring '18
- ANNDREW