If a = b≠c, then theellipsoid is called a spheroid; it can be obtained by rotating the ellipse x2/a2+ z2/c2 =1, y =0lying in the plane OXZ about the axis OZ see Fig.. The section of a hy
Trang 1The distance between them can be calculated by the formula
d= [(r1– r2)R1R2]
|R1×R2| =
x1l – x2 y1m – y1 2 z1n – z1 2
l m1
l m2
2+m1 n1
m2 n2
2+n1 l
n2 l
2
(4.6.3.30)
The condition that the determinant in the numerator in (4.6.3.30) is zero is the condition for
the two lines in space to meet.
Remark 1 The numerator of the fraction in (4.6.3.30) is the volume of the parallelepiped spanned by the
vectors r1– r2, R1, and R2, while the denominator of the fraction is the area of its base Hence the fraction
itself is the altitude d of this parallelepiped.
Remark 2. If the lines are parallel (i.e., l1= l2 = l, m1= m2 = m, and n1= n2= n, or R1 = R2= R), then the distance between them should be calculated by formula (4.6.3.29) with r0replaced by r2
4.7 Quadric Surfaces (Quadrics)
4.7.1 Quadrics (Canonical Equations)
4.7.1-1 Central surfaces
A segment joining two points of a surface is called a chord If there exists a point in space,
not necessarily lying on the surface, that bisects all chords passing through it, then the
surface is said to be central and the point is called the center of the surface.
The equations listed below in Paragraphs 4.7.1-2 to 4.7.1-4 for central surfaces are given
in canonical form; i.e., the center of a surface is at the origin, and the surface symmetry
axes are the coordinate axes Moreover, the coordinate planes are symmetry planes
4.7.1-2 Ellipsoid
An ellipsoid is a surface defined by the equation
x2
a2 +
y2
b2 +
z2
c2 =1, (4.7.1.1)
where the numbers a, b, and c are the lengths of the segments called the semiaxes of the ellipsoid (see Fig 4.52a).
a
a
c
c
Figure 4.52 Triaxial ellipsoid (a) and spheroid (b).
Trang 2If a≠ b≠c , then the ellipsoid is said to be triaxial, or scalene If a = b≠c, then the
ellipsoid is called a spheroid; it can be obtained by rotating the ellipse x2/a2+ z2/c2 =1,
y =0lying in the plane OXZ about the axis OZ (see Fig 4.52b) If a = b > c, then the ellipsoid is an oblate spheroid, and if a = b < c, then the ellipsoid is a prolate spheroid If
a = b = c, then the ellipsoid is the sphere of radius a given by the equation x2+ y2+ z2= a2
An arbitrary plane section of an ellipsoid is an ellipse (in a special case, a circle) The
volume of an ellipsoid is equal to V = 43πabc
Remark About the sphere, see also Paragraph 3.2.3-3.
4.7.1-3 Hyperboloids
A one-sheeted hyperboloid is a surface defined by the equation
x2
a2 +
y2
b2 –
z2
c2 =1, (4.7.1.2)
where a and b are the real semiaxes and c is the imaginary semiaxis (see Fig 4.53a).
A two-sheeted hyperboloid is a surface defined by the equation
x2
a2 +
y2
b2 –
z2
c2 = –1, (4.7.1.3)
where c is the real semiaxis and a and b are the imaginary semiaxes (see Fig 4.53b).
Y
Y
X
X a
c b
Figure 4.53 One-sheeted (a) and two-sheeted (b) hyperboloids.
A hyperboloid approaches the surface
x2
a2 +
y2
b2 –
z2
c2 =0,
which is called an asymptotic cone, infinitely closely.
A plane passing through the axis OZ intersects each of the hyperboloids (4.7.1.2) and
(4.7.1.3) in two hyperbolas and the asymptotic cone in two straight lines, which are the
Trang 3Y Z
X a
c
b O
Figure 4.54 A cone.
asymptotes of these hyperbolas The section of a hyperboloid by a plane parallel to OXY
is an ellipse The section of a one-sheeted hyperboloid by the plane z = 0is an ellipse,
which is called the gorge or throat ellipse.
For a = b, we deal with the hyperboloid of revolution obtained by rotating a hyperbola with semiaxes a and c about its focal axis2c(which is an imaginary axis for a one-sheeted
hyperboloid and a real axis for a two-sheeted hyperboloid) If a = b = c, then the hyperboloid
of revolution is said to be right, and its sections by the planes OXZ and OY Z are equilateral
hyperbolas
A one-sheeted hyperboloid is a ruled surface (see Paragraph 4.7.1-6)
4.7.1-4 Cone
A cone is a surface defined by the equation
x2
a2 +
y2
b2 –
z2
c2 =0 (4.7.1.4) The cone (see Fig 4.54) defined by (4.7.1.4) has vertex at the origin, and for its base we
can take the ellipse with semiaxes a and b in the plane perpendicular to the axis OZ at the distance c from the origin This cone is the asymptotic cone for the hyperboloids (4.7.1.2) and (4.7.1.3) For a = b, we obtain a right circular cone.
A cone is a ruled surface (see Paragraph 4.7.1-6)
Remark About the cone, see also Paragraph 3.2.3-2.
4.7.1-5 Paraboloids
In contrast to the surfaces considered above, paraboloids are not central surfaces For the
equations listed below, the vertex of a paraboloid lies at the origin, the axis OZ is the symmetry axis, and the planes OXZ and OY Z are symmetry planes.
An elliptic paraboloid (see Fig 4.55a) is a surface defined by the equation
x2
p + y
2
q =2z, (4.7.1.5)
Trang 4where p >0and q >0are parameters.
Y
Y
O
Figure 4.55 Elliptic (a) and hyperbolic (b) paraboloids.
The sections of an elliptic paraboloid by planes parallel to the axis OZ are parabolas, and the sections by planes parallel to the plane OXY are ellipses For example, let the parabola x2 =2pz, y = 0, obtained by the section of an elliptic paraboloid by the plane
OXZ be fixed and used as the directrix, and let the parabola x2=2qz, x =0, obtained by the
section of the elliptic paraboloid by the plane OY Z be movable and used as the generator.
Then the paraboloid can be obtained by parallel translation of the movable parabola (the generator) in a given direction along the fixed parabola (the directrix)
If p = q, then we have a paraboloid of revolution, which is obtained by rotating the
parabola2pz= x2lying in the plane OXZ about its axis.
The volume of the part of an elliptic paraboloid cut by the plane perpendicular to its
axis at a height h is equal to V = 12πabh, i.e., half the volume of the elliptic cylinder with the same base and altitude
A hyperbolic paraboloid (see Fig 4.55b) is a surface defined by the equation
x2
p – y
2
q =2z, (4.7.1.6)
where p >0and q >0are parameters
The sections of a hyperbolic paraboloid by planes parallel to the axis OZ are parabolas, and the sections by planes parallel to the plane OXY are hyperbolas For example, let the parabola x2=2pz, y =0, obtained by the section of the hyperbolic paraboloid by the plane
OXZ be fixed and used as the directrix, and let the parabola x2 = –2qz, x = 0, obtained
by the section of the hyperbolic paraboloid by the plane OY Z be movable and used as
the generator Then the paraboloid can be obtained by parallel translation of the movable parabola (the generator) in a given direction along the fixed parabola (the directrix)
A hyperbolic paraboloid is a ruled surface (see Paragraph 4.7.1-6)
4.7.1-6 Rulings of ruled surfaces
A ruled surface is a surface swept out by a moving line in space The straight lines forming a ruled surface are called rulings Examples of ruled surfaces include the cone
(see Paragraph 3.2.3-2 and 4.7.1-4), the cylinder (see Paragraph 3.2.3-1), the one-sheeted hyperboloid (see Paragraph 4.7.1-3), and the hyperbolic paraboloid (see Paragraph 4.7.1-5)
Trang 5The cone (4.7.1.4) has one family of rulings,
αx = βy,
α2a2+ β2b2
ab x= β
c z
Properties of rulings of the cone:
1 There is a unique ruling through each point of the cone
2 Two arbitrary distinct rulings of the cone meet at the point O(0,0,0)
3 Three pairwise distinct rulings of the cone are not parallel to any plane
The one-sheeted hyperboloid (4.7.1.2) has two families of rulings:
α
x
a + z
c
= β
1+ y
b
x
a – z
c
= α
1– y
b
;
γ
x
a + z
c
= δ
1– y
b
x
a – z
c
= γ
1+ y
b
(4.7.1.7)
One of these families is shown in Fig 4.56a.
X
X
Figure 4.56 Families of rulings for one-sheeted hyperboloid (a) and for hyperbolic paraboloid (b).
Properties of rulings of the one-sheeted hyperboloid:
1 In either family, there is a unique ruling through each point of the one-sheeted hyper-boloid
2 Any two rulings in different families lie in a single plane
3 Any two distinct rulings in the same family are skew
4 Three distinct rulings in the same family are not parallel to any plane
The hyperbolic paraboloid (4.7.1.6) has two families of rulings:
α
x
√ p + √ y q
=2β, β
x
√ p – √ y q
= αz;
γ
x
√ p + √ y q
= δz, δ
x
√ p – √ y q
=2γ
(4.7.1.8)
One of these families is shown in Fig 4.56b.
Trang 6Properties of rulings of a hyperbolic paraboloid:
1 In either family, there is a unique ruling through each point of the hyperbolic paraboloid
2 Any two rulings in different families lie in a single plane and meet
3 Any two distinct rulings in the same family are skew
4 All rulings in either family are parallel to a single plane
4.7.2 Quadrics (General Theory)
4.7.2-1 General equation of quadric
A quadric is a set of points in three-dimensional space whose coordinates in the rectangular
Cartesian coordinate system satisfy a second-order algebraic equation
a11x2+ a
22y2+ a
33z2+2a12xy+2a13xz+2a23yz
+2a14x+2a24y+2a34z + a44=0, (4.7.2.1) or
(a11x + a12y + a13z + a14)x + (a21x + a22y + a23z + a24)y
+ (a31x + a32y + a33z + a34)z + a41x + a42y + a43z + a44=0,
where a ij = a ji (i, j = 1,2,3,4) If equation (4.7.2.1) does not define a real geometric
object, then one says that this equation defines an imaginary quadric Equation (4.7.2.1) in
vector form reads
(Ar)⋅r +2a⋅r + a44=0, (4.7.2.2)
where A is the affinor with coordinates A i j = a ij and a is the vector with coordinates a i = a i4
4.7.2-2 Classification of quadrics
There exists a rectangular Cartesian coordinate system in which equation (4.7.2.1),
depend-ing on the coefficients, has 1 of 17 canonical forms, each of which is associated with a
certain class of quadrics (see Table 4.3)
4.7.2-3 Invariants of quadrics
The shape of a quadric can be determined by using four invariants and two semi-invariants
without reducing equation (4.7.2.1) to canonical form
The main invariants are the quantities
S = a11+ a22+ a33, (4.7.2.3)
T =a11 a12
a21 a22
+a11 a13
a31 a33
+a22 a23
a23 a33
, (4.7.2.4)
δ =
a11 a12 a13
a12 a22 a23
a13 a23 a33
Δ =
a11 a12 a13 a14
a12 a22 a23 a24
a13 a23 a33 a34
a14 a24 a34 a44
whose values are preserved under parallel translations and rotations of the coordinate axes
Trang 7TABLE 4.3 Canonical equations and classes of quadrics
Irreducible surfaces
a2 +y
2
b2 +z
2
c2 = 1
Elliptic
a2 +y
2
b2 +z
2
c2 = – 1
3 One-sheeted hyperboloid x2
a2 +y
2
b2 –z
2
c2 = 1
Hyperbolic Nondegenerate
4 Two-sheeted hyperboloid x2
a2 + y
2
b2 –z
2
c2 = – 1
p +y
2
q = 2z
Parabolic
(p >0, q >0 )
6 Hyperbolic paraboloid x2
p –y
2
q = 2z
a2 +y
2
b2 = 1
8 Imaginary elliptic cylinder x2
a2 + y
2
b2 = – 1
Cylindrical
a2 – y
2
b2 = 1
Degenerate
a2 +y
2
b2 –z
2
c2 = 0
Conic
12 Imaginary cone with real vertex x2
a2 +y
2
b2 +z
2
c2 = 0
Reducible surfaces
13 Pair of real intersecting planes x2
a2 – y
2
b2 = 0
14 intersecting in a real straight linePair of imaginary planes x2
a2 +y
2
b2 = 0
Pairs of planes Degenerate
15 Pair of real parallel planes x2= a2
16 Pair of imaginary parallel planes x2= –a2
17 Pair of real coinciding planes x2 = 0
The semi-invariants are the quantities
σ=Δ11+Δ22+Δ33, (4.7.2.7)
Σ =a11 a14
a41 a44
+a22 a24
a42 a44
+a33 a34
a44 a44
, (4.7.2.8)
whose values are preserved only under rotations of the coordinate axes HereΔij is the
cofactor of the entry a ij inΔ
The classification of quadrics based on the invariants S, T , δ, and Δ and the
semi-invariants σ andΣ is given in Tables 4.4 and 4.5
... coefficients, has of 17 canonical forms, each of which is associated with acertain class of quadrics (see Table 4.3)
4.7.2-3 Invariants of quadrics
The shape of a quadric...
One of these families is shown in Fig 4.56a.
X
X
Figure 4.56 Families of rulings for one-sheeted hyperboloid (a) and for hyperbolic... z
Properties of rulings of the cone:
1 There is a unique ruling through each point of the cone
2 Two arbitrary distinct rulings of the cone meet at the point