1
QUADRATIC SURFACES
Sphere
S = (m, n, p) = (0, 0, 0)
Ellipsoid
(x −
(y − n)2 (z − p)2
+
+
=1
a2
b2
c2
S = (m, n, p) = (0, 0, 0)
r=3
a = 3, b = 2, c = 1
Sphere
Ellipsoid
(upper part)
p
z = p + c r2 − (x − m)2 − (y − n)2
S = (m, n, p) = (0, 0, 0)
(lower part)
r
(x − m)2 (y − n)2
z =p−c 1−
−
a2
b2
S = (m, n, p) = (0, 0, 0)
r=3
a = 3, b = 2, c = 1
(x − m)2 + (y − n)2 + (z − p)2 = r2
m)2
c 2014 doc. Ing. Ivana Linkeová, Ph.D.
2
QUADRATIC SURFACES
Hyperboloid of revolution of one sheet, o k z
(x − m)2 (y − n)2 (z − p)2
+
−
=1
a2
b2
c2
S = (m, n, p) = (0, 0, 0)
Elliptic hyperboloid of one sheet, o k z
(x − m)2 (y − n)2 (z − p)2
+
−
=1
a2
b2
c2
S = (m, n, p) = (0, 0, 0)
a = b = 2, c = 3
a = 1, b = 2, c = 3
Cone of revolution, o k z
(x − m)2 (y − n)2 (z − p)2
+
−
=0
a2
b2
c2
V = (m, n, p) = (0, 0, 0)
Eliptic cone, o k z
(x −
(y − n)2 (z − p)2
+
−
=0
a2
b2
c2
V = (m, n, p) = (0, 0, 0)
a = b = 2, c = 3
a = 1, b = 2, c = 3
m)2
c 2014 doc. Ing. Ivana Linkeová, Ph.D.
3
QUADRATIC SURFACES
Hyperboloid of revolution of two sheets, o k z
(x − m)2 (y − n)2 (z − p)2
−
−
+
=1
a2
b2
c2
S = (m, n, p) = (0, 0, 0)
Elliptic hyperboloid of two sheets, o k z
(x − m)2 (y − n)2 (z − p)2
−
−
+
=1
a2
b2
c2
S = (m, n, p) = (0, 0, 0)
a = b = 3, c = 4
a = 2, b = 3, c = 4
Paraboloid of revolution, o k +z
(x − m)2 (y − n)2
z−p
+
=
2
2
a
b
c
V = (m, n, p) = (0, 0, 0)
Elliptic paraboloid, o k +z
(x − m)2 (y − n)2
z−p
+
=
2
2
a
b
c
V = (m, n, p) = (0, 0, 0)
a = b = 2, c = 5
a = 2, b = 3, c = 5
c 2014 doc. Ing. Ivana Linkeová, Ph.D.
4
QUADRATIC SURFACES
Hyperbolic paraboloid, o k +z
(x − m)2 (y − n)2
z−p
−
=
2
2
a
b
c
V = (m, n, p) = (0, 0, 0)
Elliptic cylinder, o k z
(x − m)2 (y − n)2
+
=1
a2
b2
S = (m, n, z) = (0, 0, z)
a = 3, b = 2, c = 1
a = 2, b = 3
Parabolic cylinder, o k z
(x − m)2
= 2p(z − p)
a2
V = (m, n, p) = (0, 0, 0)
Hyperbolic cylinder, o k z
(x − m)2 (y − n)2
−
=1
a2
b2
S = (m, n, p) = (0, 0, 0)
p=
1
2
a = 2, b = 3
c 2014 doc. Ing. Ivana Linkeová, Ph.D.