TABLE 4.1 Classification of second-order curves Δ I > 0 Real ellipse for I2=4δ or a11= a22 and a12 = 0, this is a circle Elliptic δ> 0 Δ Pair of imaginary straight lines intersecting ata
Trang 1whose values do not change under parallel translation and rotation of the coordinate axes,
and the semi-invariant
σ =a22 a23
a32 a33
+a11 a13
a31 a33
, (4.4.5.3) whose value does not change under rotation of the coordinate axes
The invariantΔ is called the large discriminant of equation (4.4.2.1) The invariant δ
is called the small discriminant.
Table 1 presents the classification of second-order curves based on invariants
TABLE 4.1 Classification of second-order curves
Δ
I > 0 Real ellipse (for I2=4δ or a11= a22
and a12 = 0, this is a circle) Elliptic
δ> 0 Δ Pair of imaginary straight lines intersecting ata real point (ellipse degenerating into a point)
I < 0 Imaginary ellipse (no real points)
Hyperbolic
δ< 0 Hyperbola Pair of real intersecting straight lines(degenerate hyperbola)
σ> 0 Pair of imaginary parallel straight lines
σ< 0 Pair of real parallel straight lines Parabolic
σ= 0 Pair of coinciding straight lines(a single straight line)
4.4.5-4 Characteristic equation of second-order curves
The properties of second-order curves can be studied with the use of the characteristic equation
a11a – λ a12
21 a22– λ
=0 or λ2– Iλ + δ =0 (4.4.5.4)
The roots λ1and λ2of the characteristic equation (4.4.5.4) are the eigenvalues of the
real symmetric matrix [a ij] and, as a consequence, are real
Obviously, the invariants I and δ of second-order curves are expressed as follows in terms of the roots λ1and λ2of the characteristic equation (4.4.5.4):
I = λ1+ λ2, δ = λ1λ2. (4.4.5.5)
4.4.5-5 Centers and diameters of second-order curves
A straight line passing through the midpoints of parallel chords of a second-order curve is
called a diameter of this curve A diameter is said to be conjugate to the chords (or to the
direction of chords) which it divides into two parts The diameter conjugate to chords forms
an angle ϕ with the positive direction of the OX-axis and is determined by the equation
(a11x + a12y + a13) cos ϕ + (a21x + a22y + a23) sin ϕ =0 (4.4.5.6)
Trang 2All diameters of a second-order curve with δ≠ 0meet at a single point called the center
of the curve, and in this case the curve is said to be central The center coordinates (x0, y0) satisfy the system of equations
a11x0+ a12y0+ a13=0,
a21x0+ a22y0+ a23=0, (4.4.5.7) which implies that
x0= –1
δ
a a13 a12
23 a22
, y0= –1
δ
a a11 a13
21 a23
(4.4.5.8)
All diameters of a second-order curve with δ =0are parallel or coincide A second-order
curve does not have a center if and only if δ =0and Δ≠ 0 A second-order curve has a
center line if and only if δ =0andΔ =0
Example 1 The centers of nine canonical second-order curves are as follows:
1 Ellipse, δ =1 and Δ = –1: the single center O(0, 0);
2 Hyperbola, δ = –1and Δ = 1: the single center O(0, 0);
3 Parabola, δ =0 and Δ = –1: no centers;
4 Imaginary ellipse, δ =1 and Δ = –1: the single center O(0, 0);
5 Pair of intersecting straight lines, δ = –1and Δ = 0: the single center O(0, 0);
6 Pair of imaginary intersecting straight lines, δ =1 and Δ = 0: the single center O(0, 0);
7 Pair of imaginary parallel straight lines, δ =0 and Δ = 0: the center line y = 0;
8 Pair of coinciding straight lines, δ =0 and Δ = 0: the center line y = 0.
Each of the two conjugate diameters of a central second-order curve bisects the chords parallel to the other diameter
4.4.5-6 Principal axes
A diameter perpendicular to the chords conjugate to it is called a principal axis A principal axis is a symmetry axis of a second-order curve For each central second-order curve (δ≠ 0), either there are two perpendicular principal axes or each of its diameters is a principal axis
(the circle) A second-order curve with δ = 0has a unique principal axis The points of
intersection of a second-order curve with its principal axes are called its vertices.
The directions of principal axes coincide with the directions of the eigenvectors of the
symmetric matrix [a ij ] (i, j =1,2); i.e., the direction cosines cos θ, sin θ of the normals to
the principal axes are determined from the system of equations
(a11– λ) cos θ + a12sin θ =0,
a21cos θ + (a22– λ) sin θ =0, (4.4.5.9)
where λ is a nonzero root of characteristic equation (4.4.5.4).
The directions of principal axes and of their conjugate chords are called the principal directions of a second-order curve The angle between the positive direction of the axis OX
and each of the two principal directions of a second-order curve is given by the formula
tan2ϕ= tan2θ= 2a12
a11– a22 . (4.4.5.10) Remark The circle has undetermined principal directions.
Trang 34.4.5-7 Reduction of central second-order curves to canonical form.
A second-order curve with δ≠ 0has a center By shifting the origin to the center O1(x0, y0) whose coordinates are determined by formula (4.4.5.8), we can reduce equation (4.4.5.1)
to the form
a11x2
1+2a12x1y1+ a22y2
1+ Δ
where x1and y1are the new coordinates
By rotating the axes O1X1 and O1Y1 by the angle θ determined by (4.4.5.10), we
transform equation (4.4.5.11) as follows:
A11ˆx2+ A22ˆy + Δ
The coefficients A11and A22are the roots of the characteristic equation (4.4.5.4)
We note the following formulas for the ellipse:
a2= – 1
λ2
Δ
λ1λ2 2
, b2= – 1
λ1
Δ
λ2
1λ2
, (4.4.5.13)
where a and b are the parameters of the canonical equation, δ andΔ are the invariants, and
λ1and λ2are the roots of the characteristic equation (4.4.5.4)
Similarly, for the hyperbola one has
a2= – 1
λ1
Δ
λ2
1λ2 , b
2= 1
λ1
Δ
λ2
1λ2 . (4.4.5.14)
4.4.5-8 Reduction of noncentral second-order curves to canonical form
If δ =0, then the curve does not have any center or does not have a definite center, and its equation can be written as
(αx + βy)2+2a13x+2a23y + a33=0 (4.4.5.15)
If the coefficients a13and a23are proportional to the coefficients α and β, i.e., a13= kα and a23= kβ, then equation (4.4.5.15) becomes (αx + βy)2+2k (αx + βy) + a33 =0, and hence
is a pair of real parallel straight lines
If the coefficients a13 and a23 are not proportional to the coefficients α and β, then
equation (4.4.5.15) can be written as
(αx + βy + γ)2+2k (βx – αx + q) =0 (4.4.5.17)
The parameters k, γ, and q can be determined by comparing the coefficients in equa-tions (4.4.5.15) and (4.4.5.17) For the axis O1X one should take the line αx + βy + γ =0,
and for the axis O1Y , the line βx – αx + q =0 We denote
ˆx = βx – αx + q
α2+ β2 , ˆy =
αx + βy + γ
α2+ β2 ; (4.4.5.18)
Trang 4then equation (4.4.5.17) acquires the form
where p = |k|/
α2+ β2 The axis O1X points into the half-plane where the sign of
βx – αx + q is opposite to that of k.
If one only needs to find the canonical equation of a parabola and it is not necessary to
construct the graph of the parabola in the coordinate system OXY , then the parameter p
is determined via the invariants I, δ, and Δ and the roots λ1 and λ2 (λ1 ≥ λ2) of the characteristic equation (4.4.5.4) by the formulas
p= 1
I = 1
λ1 –
Δ
λ1 >0, λ2=0 (4.4.5.20) 4.4.5-9 Geometric definition of nondegenerate second-order curve
There exists a coordinate system in which equation (4.4.5.1) has the form
y2 =2px– (1– e2)x2, (4.4.5.21)
where p >0is a parameter and e is the eccentricity Obviously, the curve (4.4.5.21) passes through the origin of the new coordinate system The axis OX is a symmetry axis of the
curve
The equation of the directrix of the curve (4.4.5.21) is
The coordinates of the focus are
1+ e , y=0 (4.4.5.23)
The distance from the focus to the directrix is equal to p/e For a central second-order
curve, the line
1– e2 = a (4.4.5.24)
is a symmetry axis
Remark All types of second-order curves can be obtained as plane sections of a right circular cone for various positions of the secant plane with respect to the cone.
4.4.5-10 Tangents and normals to second-order curves
The equation of the tangent to a second-order curve at a point M0(x0, y0) has the form
(a11x0+ a12y0+ a13)x + (a21x0+ a22y0+ a23)y + a31x0+ a32y0+ a33=0 (4.4.5.25)
The equation of the normal to a second-order curve at the point M0(x0, y0) has the form
x – x0
a11x0+ a12y0+ a13 =
y – y0
a21x0+ a22y0+ a23 . (4.4.5.26)
Trang 5TABLE 4.2 Ellipse, hyperbola, parabola Main formulas
Canonical
2
a2 +y
2
a2 –y
2
Equation
in polar coordinates ρ=
p
p
1 e cos ϕ ρ=
p
1– cos ϕ
Eccentricity e= c
a = 1
Focal radii
(distance from the
foci to an arbitrary
point (x, y) of curve)
r1= a + ex
r2= a – ex
r1=
a + ex for x >0
–a – ex for x <0
r2 =
–a + ex for x >0
a – ex for x <0
r = x + p
2 Focal
b2
2
a
p
Equation
a
Equation of
diameter conjugate
to chords with slope k y= –
b2
2
k
Area of segment
bounded by an arc
convex to the left
and the chord
joining points
(x0, y0) and (x0, –y0 )
πab
2 +
b a
x0
a2– x2 + a2arcsinx0
a
x0y0– ab ln
x0
a +y0
b
= x0y0– ab arccosh x0
a
4
3x0y0
Curvature radius
at point (x, y) a2b2
x2
a4 +y
2
b4
3/2
=
(r1r2 )3
2b2
x2
a4 +y
2
b4
3/2
=
(r1r2 )3
ab
(p +2x)3/2
√ p
Equations of
tangents to a curve
which pass through
an arbitrary
point (x0, y0 )
y – y0
x – x0
=–x0y0
a2y2+b2x2–a2b2
a2– x2
y – y0
x – x0
=–x0y0
a2y2–b2x2+a2b2
a2– x2
y – y0
x – x0
= y0
y2– 2px0
2x0
Equation of tangent
at point (x0, y0)
x0x
a2 + y0y
a2 – y0y
b2 = 1 yy0= p(x + x0) Equation of
tangent with slope k y = kx
√
k2a2– b2 y = kx + p
2k
Equation of normal
at point (x0, y0 )
y – y0
x – x0 =
a2y0
b2x0
y – y0
x – x0 = –
a2y0
b2x0
y – y0
x – x0 = –y0
p
Coordinates of pole
(x0, y0 ) of straight line
Ax + By + C =0
w.r.t a curve
x0= –a
2A
C , y0= –b
2B
2A
2B
C , y0= –pB
C
The polar is the set of points Q that are harmonically conjugate to a point P , called the pole of a second-order curve, with respect to points R1 and R2 of intersection of the
second-order curve with secants passing through P ; i.e.,
R1P
P R2 = –
R1Q
QR2 .
Trang 6If the point P lies outside the second-order curve (one can draw two tangents through P ),
then the polar line passes through the points at which this curve is tangent to straight lines
drawn through point P If the point lies on a second-order curve, then the polar line is a straight line tangent to this curve at this point If the polar line of a point P passes through
a point Q, then the polar line of Q passes through P
4.5 Coordinates, Vectors, Curves,
and Surfaces in Space
4.5.1 Vectors Cartesian Coordinate System
4.5.1-1 Notion of vector
A directed segment with initial point A and endpoint B (see Fig 4.26) is called the vector
−−→
AB A nonnegative number equal to the length of the segment AB joining the points A and B is called the length|AB −−→|of the vector −− AB → The vector − BA −→ is said to be opposite to the vector − AB −→.
A
B
Figure 4.26 Vector.
Two directed segments − AB −→ and −−→ CDof the same length and the same direction determine
the same vector a; i.e., a = −− AB → = −−→ CD.
Two vectors are said to be collinear (parallel) if they lie on the same straight line or
on parallel lines Three vectors are said to be coplanar if they lie in the same plane or
in parallel planes A vector0whose initial point and endpoint coincide is called the zero vector The length of the zero vector is equal to zero (|0|=0), and the direction of the zero
vector is assumed to be arbitrary A vector e of unit length is called a unit vector.
The sum a + b of vectors a and b is defined as the vector directed from the initial point
of a to the endpoint of b under the condition that b is applied at the endpoint of a The rule
for addition of vectors, which is contained in this definition, is called the triangle rule or the
rule of closing a chain of vectors (see Fig 4.27a) The sum a + b can also be found using the parallelogram rule (see Fig 4.27b) The difference a – b of vectors a and b is defined
as follows: b + (a – b) = a (see Fig 4.27c).
a
b
Figure 4.27 The sum of vectors: triangle rule (a) and parallelogram rule (b) The difference of vectors (c).
The product λa of a vector a by a number λ is defined as the vector whose length is
equal to|λa| =|λ||a|and whose direction coincides with that of the vector a if λ >0or is
opposite to the direction of the vector a if λ <0
Trang 7Remark. If a =0or λ =0, then the absolute value of the product is zero, i.e., it is the zero vector In this
case, the direction of the product λa is undetermined.
Main properties of operations with vectors:
1 a + b = b + a (commutativity).
2 a + (b + c) = a + (b + c) (associativity of addition).
3 a +0= a (existence of the zero vector).
4 a + (–a) =0(existence of the opposite vector)
5 λ(a + b) = λa + λb (distributivity with respect to addition of vectors).
6 (λ + μ)a = λa + μa (distributivity with respect to addition of constants).
7 λ(μa) = (λμ)a (associativity of product).
8 1a = a (multiplication by unity)
4.5.1-2 Projection of vector onto axis
A straight line with a unit vector e lying on it determining the positive sense of the line is
called an axis The projection pre a of a vector a onto the axis (see Fig 4.28) is defined as
the directed segment on the axis whose signed length is equal to the scalar product of a by the unit vector e, i.e., is determined by the formula
where ϕ is the angle between the vectors a and e.
a e
φ
Figure 4.28 Projection of a vector onto the axes.
Properties of projections:
1 pre (a + b) = pr e a + pr e b (additivity).
2 pre(λa) = λ pre a (homogeneity).
4.5.2 Coordinate Systems
4.5.2-1 Cartesian coordinate system
If a one-to-one correspondence between points in space and numbers (triples of numbers)
is given, then one says that a coordinate system is introduced in space.
A rectangular Cartesian coordinate system is determined by a scale segment for mea-suring lengths and three pairwise perpendicular directed straight lines OX, OY , and OZ (the coordinate axes) concurrent at a single point O (the origin) The three coordinate axes divide the space into eight parts called octants.
We choose an arbitrary point M in space and project it onto the coordinate axes, i.e., draw the perpendiculars to the axes OX, OY , and OZ through M We denote the points
... sign of< /sub>βx – αx + q is opposite to that of k.
If one only needs to find the canonical equation of a parabola and it is not necessary to
construct the graph of. .. invariants I, δ, and Δ and the roots λ1 and λ2 (λ1 ≥ λ2) of the characteristic equation (4.4.5.4) by the formulas
p=... coordinate system The axis OX is a symmetry axis of the
curve
The equation of the directrix of the curve (4.4.5 .21) is
The coordinates of the focus are
1+ e ,