Quadric surfaces

formulario a de geometria analitica de espacio

1. Point coordinates of the quadric surfaces: x,
y, z, x1, y1, z1, …
Real numbers: A, B, C, …, a, b, c, k1, k2, k3
Invariants: e, E, Δ
Radius of a sphere: R
Center of a sphere: (a,b,c)
FormulasFormulas
Analytic Geometry
Quadric Surfaces
1General equation of a quadric surface
Ax
2
+By
2
+Cz
2
+2Fyz+2Gzx +2Hxy+2Px +2Qy+2Rz +D=0,
where x, y, z are the Cartesian coordinates of the points of the surface, A, B, C,… are real numbers.
2Classification of quadric surfaces
This classification is based on invariants of the quadric surfaces. Invariants are special expressions composed of the coefficients
of the general equation which do not change under parallel translation or rotation of the coordinate system. In total, there are 17
different (canonical) classes of the quadric surfaces.
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2. Here the invariants are the ranks of the matrices e and E, the determinant Δ of the matrix E, and signs of the
eigenvalues k of the matrix e. The matrices e and E are given by
e=
⎡
⎢
⎣
AHG
HBF
GFC
⎤
⎥
⎦
, E=
⎡
⎢
⎢
⎢
⎢
⎣
AHQP
HBFQ
GFCR
PQRD
⎤
⎥
⎥
⎥
⎥
⎦
, Δ=det(E),
The roots k1, k2, k3 are obtained from the solution of the equation
∣
∣
∣
∣
A−kH G
H B−kF
G F C−k
∣
∣
∣
∣
=0.
3Real Ellipsoid (#1)
++ =1
4Imaginary Ellipsoid (#2)
++ =−1
5Hyperboloid of One Sheet (#3)
+− =1
x
2
a
2
y
2
b
2
z
2
c
2
x
2
a
2
y
2
b
2
z
2
c
2
x
2
a
2
y
2
b
2
z
2
c
2
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3. 6Hyperboloid of Two Sheets (#4)
+− =−1
7Real Quadric Cone (#5)
+− =0
8Imaginary Quadric Cone (#6)
++ =0
9Elliptic Paraboloid (#7)
+−z =0
x
2
a
2
y
2
b
2
z
2
c
2
x
2
a
2
y
2
b
2
z
2
c
2
x
2
a
2
y
2
b
2
z
2
c
2
x
2
a
2
y
2
b
2
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4. 10Hyperbolic Paraboloid (#8)
−−z =0
11Real Elliptic Cylinder (#9)
+ =1
12Imaginary Elliptic Cylinder (#10)
+ =−1
13Hyperbolic Cylinder (#11)
− =1
x
2
a
2
y
2
b
2
x
2
a
2
y
2
b
2
x
2
a
2
y
2
b
2
x
2
a
2
y
2
b
2
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5. 14Real Intersecting Planes (#12)
− =0
15Imaginary Intersecting Planes (#13)
+ =0
16Parabolic Cylinder (#14)
−y=0
17Real Parallel Planes (#15)
=1
18Imaginary Parallel Planes (#16)
=−1
19Coincident Planes (#17)
x
2
=0
20Equation of a sphere centered at the origin
A sphere is a special case of an ellipsoid when the three semi-axes are the same and equal to the radius of the sphere. The
equation of a sphere of radius R centered at the origin is given by
x
2
+y
2
+z
2
=R
2
.
x
2
a
2
y
2
b
2
x
2
a
2
y
2
b
2
x
2
a
2
x
2
a
2
x
2
a
2
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6. 21Equation of a sphere centered at any point
(x−a)
2
+(y−b)
2
+(z−c)
2
=R
2
,
where (a,b,c) are the coordinates of the center of the sphere.
22Diameter form of the equation of a sphere
(x−x1)(x−x2) +(y−y1)(y−y2) +(z−z1)(z−z2) =0,
where P1(x1,y1,z1), P2(x2,y2,z2) are the endpoints of a diameter.
23Four points form of the equation of a sphere
∣
∣
∣
∣
∣
∣
∣
∣
x
2
+y
2
+z
2
xyz1
x
2
1
+y
2
1
+z
2
1
x1y1z11
x
2
2
+y
2
2
+z
2
2
x2y2z21
x
2
3
+y
2
3
+z
2
3
x3y3z31
x
2
4
+y
2
4
+z
2
4
x4y4z41
∣
∣
∣
∣
∣
∣
∣
∣
=0.
The points P1(x1,y1,z1), P2(x2,y2,z2), P3(x3,y3,z3), P4(x4,y4,z4) belong to the given sphere.
24General equation of a sphere
Ax
2
+Ay
2
+Az
2
+Dx+Ey +Fz+M =0, (A≠0)
The center of the sphere has the coordinates (a,b,c) where
a=−, b=−, c=−.
The radius of the sphere is given by
R= .
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