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Cylinders and Quadric Surfaces: A Mathematical Analysis, Study notes of Advanced Calculus
An introduction to cylinders and quadric surfaces, discussing their definitions, properties, and specific examples. Topics covered include cylinders as the intersection of lines parallel to a given line and passing through a plane curve, and quadric surfaces as the graph of a second-degree equation in three variables. The document also explores various types of quadric surfaces, such as ellipsoids, paraboloids, hyperboloids, and cones.
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Cylinders and quadric surfaces
Math 2163
Cylinders
A cylinder is a surface that consists of all lines (rulings) thatare parallel to a given line and pass through a given planecurve.
e.g. x 2
+
y 2
= 1
Cylinders
e.g. y 2
+
z 2
= 1
Cylinders
e.g. z
=
x 2
Quadric surfaces
A quadric surface is the graph of a second-degree equationin three variables
x
,
y
and
z
. e.g.
Ax 2
+
By 2
+
Cz 2
+
Dxy
+
Eyz
+
F xz
+
Gx
+
Hy
+
Iz
+
J
= 0
Ax 2
+
By 2
+
Cz 2
+
J
= 0
Ax 2
+
By 2
+
Iz
= 0
We will study the following types of quadric surfaces:
ellipsoid
paraboloid
hyperboloid
cone
ellipsoid
x 2 a 2
+
y 2 b 2
+
z 2 c 2
= 1
Paraboloid
Elliptic Paraboloid z^ c
=
x 2 a 2
+
y 2 b 2 Hyperbolic Paraboloid z^ c
=
x 2 a 2
−
y 2 b 2
Hyperboloid
Hyperboloid of one sheet x 2 a 2
+
y 2 b 2
−
z 2 c 2
= 1
Hyperboloid of two sheets
−
x 2 a 2
−
y 2 b 2
+
z 2 c 2
= 1
Cone
Cone 1 z 2 c 2
=
x 2 a 2
+
y 2 b 2 Cone 2 y 2 b 2
=
x 2 a 2
+
z 2 c 2