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Cylinders and quadric surfaces
Math 2163
. – p.1/9
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. x2+ y2
= 1
. – p.2/9
Cylinders
e.g. y2+ z2
= 1
. – p.3/9
Cylinders
e.g. z = x2
. – p.4/9
Quadric surfaces
A quadric surface is the graph of a second-degree equationin three variables x, y and z. e.g.
Ax2+ By2
+ Cz2+ Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0
Ax2+ By2
+ Cz2+ J = 0
Ax2+ By2
+ Iz = 0
We will study the following types of quadric surfaces:
ellipsoid paraboloid hyperboloid cone
. – p.5/9
ellipsoid
x2
a2+
y2
b2+
z2
c2= 1
. – p.6/9
Paraboloid
Elliptic Paraboloidz
c=
x2
a2+
y2
b2
Hyperbolic Paraboloidz
c=
x2
a2−
y2
b2
. – p.7/9
Hyperboloid
Hyperboloid of one sheetx2
a2+
y2
b2−
z2
c2= 1
Hyperboloid of two sheets −
x2
a2−
y2
b2+
z2
c2= 1
. – p.8/9
Cone
Cone 1z2
c2=
x2
a2+
y2
b2
Cone 2y2
b2=
x2
a2+
z2
c2
. – p.9/9