Asymptotes.A straight line is called an asymptote of a curve Γ if the distance from a point Mx, y of the curve to this straight line tends to zero as x2+ y2→ ∞.. The center of this circl
Trang 19.1.1-4 Asymptotes.
A straight line is called an asymptote of a curve Γ if the distance from a point M(x, y) of
the curve to this straight line tends to zero as x2+ y2→ ∞ The limit position of the tangent
to a regular point of the curve is an asymptote; the converse assertion is generally not true
For a curve given explicitly as y = f (x), vertical asymptotes are determined as points
of discontinuity of the function y = f (x), while horizontal and skew asymptotes have the form y = kx + b, where
k= lim
t→∞
f (x)
x , b= lim
t→∞ [f (x) – kx];
both limits must be finite
Example 12 Let us find the asymptotes of the curve y = x3/(x2+ 1) Since the limits
k= lim
t→∞
f(x)
x = lim
t→∞
x3 x(x2+ 1) =1,
b= lim
t→∞ [f (x) – kx] = lim
t→∞
x3
x2+ 1– x
= lim
t→∞
–x
x2+ 1
= 0
exist, the asymptote is given by the equation y = x.
To find an asymptote of a parametrically defined curve x = x(t), y = y(t), one should find the values t = t i for which x(t) → ∞ or y(t) → ∞.
If
x (t i) =∞ but y(t i ) = c≠∞,
then the straight line y = c is a horizontal asymptote If
y (t i) =∞ but x(t i ) = a≠∞,
then the straight line x = a is a vertical asymptote If
x (t i) =∞ and y(t i) =∞,
then one should calculate the following two limits:
k= lim
t→t i
y (t)
x (t) and b= limt→t i [y(t) – kx(t)]. (9.1.1.4)
If both limits exist, then the curve has the asymptote y = kx + b.
Example 13 Let us find the asymptote of the Folium of Descartes
x= 3at
t3+ 1, y=
3at 2
t3+ 1 (–∞≤t≤∞).
Since x(–1) = ∞ and y(–1) = ∞, one should use formulas (9.1.1.4),
k= lim
t→–1
y(t) x(t) = limt→–1
3at 2
t3+ 1 3at
t3+ 1
= lim
t→–1t= –1,
b= lim
t→–1[y(t) – kx(t)] = lim
t→–1
3at 2
t3+ 1+t33at+ 1
= lim
t→–1
3at(t + 1)
(t +1)(t 2– t +1) = –a,
which imply that the asymptote is given by the equation y = –x – a.
Trang 2Suppose that the function F (x, y) in the equation F (x, y) = 0 is a polynomial in the
variables x and y We choose the terms of the highest order in F (x, y) By Φ(x, y) we
denote the set of highest-order terms and solve the equation for the variables x and y:
x = ϕ(y), y = ψ(x).
The values y i = c for which x = ∞ give the horizontal asymptotes y = c; the values x i = a for which y = ∞ give the horizontal asymptotes x = a.
To find skew asymptotes, one should substitute the expression y = kx + b into F (x, y).
We write the resulting polynomial F (x, kx + b) as
F (x, kx + b) = f1(k, b)x n + f2(k, b)x n–1 +
If the system of equations
f1(k, b) =0, f2(k, b) =0
is consistent, then its solutions k, b are the parameters of the asymptotes y = kx + b.
9.1.1-5 Osculating circle
The osculating circle (circle of curvature) of a curve Γ at a point M0 is defined to be the
limit position of the circle passing through M0and two neighboring points M1and M2of
the curve as M1→ M0and M2→ M0(Fig 9.10)
C
M
M
M
2
1
0
Figure 9.10 The osculating circle.
The center of this circle (the center of curvature of the curve Γ at the point M1) is called the center of the osculating circle and lies on the normal to this curve (Fig 9.10) The coordinates of the center of curvature can be found by the following formulas:
for a curve defined explicitly,
x c = x0– y
x
1+ (y x)2
y
xx
, y c = y0+ 1+ (y x )2
y
xx
;
for a curve defined implicitly,
x c = x0– F x F x
2+ F
y2
2F xy F x F y – F x2F yy – F y2F xx, y c = y0+
F y F x2+ F y2
2F xy F x F y – F x2F yy – F y2F xx;
Trang 3for a curve defined parametrically,
x c = x0– y
t
(x t 2+ (y t 2
x
t y tt – y t x tt , y c = y0+
x t
(x t 2+ (y t2
x
t y tt – y t x tt ;
for a curve in polar coordinates,
x c = r0cos ϕ0–
r2
0+ r ϕ 2
r0cos ϕ0+ r ϕ sin ϕ0
r2
0+2 r
ϕ 2
– r0r
ϕϕ
,
y c = r0sin ϕ0–
r2
0+ r ϕ 2
r0sin ϕ0– r ϕ cos ϕ0
r2
0+2 r
ϕ 2
– r0r
ϕϕ
,
x0= r0cos ϕ0, y0= r0sin ϕ0,
where all derivatives are evaluated at x = x0, y = y0, t = t0, and ϕ = ϕ0
The radius of the osculating circle is called the radius of curvature of the curve at the point M0(x0, y0); its length in a Cartesian coordinate system is
ρ=
1+ (y x )23 2
y xx = F x
2+ F
y2 3 2
2F xy F x F y – F2
x F yy – F2F xx = x t 2
+ y
t 23 2
x
t y tt – y t x tt ,
and in the polar coordinate system it is
ρ=
r2
0+ r ϕ
23 2
r2
0+2 r ϕ
2
– r0r
ϕϕ ,
where all derivatives are evaluated at x = x0, y = y0, t = t0, and ϕ = ϕ0
9.1.1-6 Curvature of plane curves
The limit ratio of the tangent rotation angleΔϕ to the corresponding arc length Δs of the
curveΓ as Δs →0(Fig 9.11),
k= lim
Δs→0
Δϕ Δs,
is called the curvature of Γ at the point M1
M1
d = α Δφ
Δs
X Y
Figure 9.11 The curvature of the curve.
Trang 4The curvature and the radius of curvature are reciprocal quantities,
k= 1
ρ
The more bent a curve is near a point, the larger k is and the smaller ρ is at this point For a circle of radius a, the radius of curvature is ρ = a and the curvature is k =1/a(they
are constant at all points of the circle); for a straight line, ρ = ∞ and k =0; for all other curves, the curvature varies from point to point
9.1.1-7 Fr´enet formulas
To each point M of a plane curve, one can naturally assign a local coordinate system The role of the origin O is played by point M itself, and the role of the axes OX and OY are
played by the tangent and normal at this point The unit tangent and normal vectors to the
curve are usually denoted by t and n, respectively (Fig 9.12).
n
Figure 9.12 The unit tangent t and normal n vectors to the curve.
Suppose that the arc length is taken as a (natural) parameter on the curve:
r = r(s);
then the Fr´enet formulas
t s = kn, n s = –kt
hold, where k is the curvature of the curve.
With first-order accuracy, the Fr´enet formulas determine the rotation of the vectors t
and n when translated along the curve to a close point, s → s + Δs.
9.1.1-8 Envelope of a family of curves
A one-parameter family of curves is the set of curves defined by the equation
which is called the equation of the family Here C is a parameter varying in a certain range,
C1≤C ≤C2; in particular, the range can be –∞≤C+∞.
A curve that is tangent at each point to some curve of a one-parameter family of
curves (9.1.1.5) is called the envelope of the family The point of tangency of the envelope
to a curve of the family is called a characteristic point of the curve of the family (Fig 9.13).
Figure 9.13 The envelope of the family of curves.
Trang 5The equation of the envelope of a one-parameter family is obtained by elimination of
the parameter C from the system of equations
F (x, y, C) =0,
∂F (x, y, C)
which determines the discriminant curve of this family.
The discriminant curve (9.1.1.6) of a one-parameter family is an envelope if it does not consist of singular points of the curves
X Y
Figure 9.14 The envelope of the family of semicubical parabolas.
Example 14 Consider the family of semicubical parabolas (Fig 9.14)
3(y – C) 2 – 2(x – C) 3 = 0.
Differentiating with respect to the parameter C, we obtain
y – C – (x – C)2= 0.
Solving the system
3(y – C) 2 – 2(x – C) 3 = 0,
y – C – (x – C)2= 0,
we obtain
x – C =0, y – C =0;
x – C = 2
3, y – C =
4
9.
Eliminating the parameter C, we see that the discriminant curve splits into the pair of straight lines x = y and
x – y =2/9 Only the second of these two straight lines is an envelope, because the first straight line is the locus of singular points.
9.1.1-9 Evolute and evolvent
The locus of centers of curvature of a curve is called its evolute If a curve is defined via a
natural parameter s, then the position vector of a point of the evolute of a plane curve p can
be expressed in terms of the radius vector r of a point of this curve, the normal vector n,
and its radius of curvature ρ as follows:
p = r + ρn.
To obtain the equation of the evolute, it also suffices to treat x c and y c in the equations for the coordinates of the center of curvature as the current coordinates of the evolute
Trang 6Geometric properties of the evolute:
1 The normals to the original curve coincide with the tangents to the evolute at the corresponding points
2 If the radius of curvature ρ varies monotonically on a given part of the curve, then its
increment is equal to the distance passed by the center of curvature along the evolute
3 At the points of extremum of the radius of curvature ρ, the evolute has a cusp of the first
kind
4 Since the radius of curvature ρ is always positive, any point of the evolute lies on a
normal to the curve on the concave side
5 The evolute is the envelope of the family of normals to the original curve
A curve that intersects all curves of a family at the right angle is called an orthogonal
trajectory of a one-parameter family of curves A trajectory orthogonal to tangents to a
given curve is called an evolvent of this curve.
If a curve is defined via its natural parameter s, then the vector equation of its evolvent
has the form
p = r + (s0– s)t,
where s0is an arbitrary constant
Basic properties of the evolvent:
1 The tangent to the original curve at each point is the normal to the evolvent at the corresponding point
2 The distance between the corresponding points of two evolvents of a given curve is constant
3 For s = s0, the evolvent has cusps of the first kind
The evolute and the evolvent are related to each other The original curve is the evolvent
of its evolute The converse assertion is also true; i.e., the original curve is the evolute of its evolvent The normal to the evolvent is tangent to the evolute
9.1.2 Space Curves
9.1.2-1 Regular points of space curve
A space curveΓ is in general determined parametrically or in vector form by the equations
x = x(t), y = y(t), z = z(t) or r = r(t) = x(t)i + y(t)j + z(t)k,
where i, j, and k are the unit vectors (see (4.5.2.2)), t is an arbitrary parameter (t[t1, t2]),
and t1and t2can be –∞ and +∞, respectively.
A point M (x(t), y(t), z(t)) is said to be regular if the functions x(t), y(t), and z(t) have
continuous first derivatives in a sufficiently small neighborhood of this point and these
derivatives are not simultaneously zero, i.e., if dr/dt≠ 0
If the functions x(t), y(t), and z(t) have continuous derivatives with respect to t and
dr/dt≠ 0for all t[t1, t2], thenΓ is a regular arc.
For the parameter t it is convenient to take the arc length s, that is, the length of the arc from a point M0(x(t0), y(t0), z(t0)) to M1(x(t1), y(t1), z(t1)),
s=
Γ ds=
Γ
√
dr⋅dr =
t1
t0
(x t 2+ (y t2+ (z t 2dt (9.1.2.1)
Trang 7The sign of ds is chosen arbitrarily, and it determines the positive sense of the curve and
the tangent
A space curve can also be defined as the intersection of two surfaces
F1(x, y, z) =0, F2(x, y, z) =0 (9.1.2.2)
For a curve determined as the intersection of two planes, a point M0 is regular if the vectors∇F1and∇F2are not linearly dependent at this point.
9.1.2-2 Tangents and normals
A straight line is called the tangent to a curve Γ at a regular point M0if it is the limit position
of the secant passing through M0and a point M1infinitely approaching the point M0
At a regular point M0, the equation of the tangent has the form
r = r0+ λr t (t0), (9.1.2.3)
where λ is a variable parameter.
Eliminating the parameter λ from (9.1.2.3), we obtain the canonical equation
x – x0
x
t (t0) =
y – y0
y
t (t0) =
z – z0
z
t (t0)
of the tangent
The equation of the tangent at a point M0 of the curve Γ obtained as the
intersec-tion (9.1.2.2) of two planes is
x – x0
(F1)y (F2)z – (F1)z (F2)y =
y – y0 (F1)z (F2)x – (F1)x (F2)z =
z – z0 (F1)x (F2)y – (F1)y (F2)x,
where all derivatives are evaluated at x = x0and y = y0
A perpendicular to the tangent at the point of tangency is called a normal to a space
curve Obviously, at each point of the curve, there are infinitely many normals that form the plane perpendicular to the tangent
The plane passing through the point of tangency and perpendicular to the tangent is
called the normal plane (Fig 9.15).
rt
Figure 9.15 The normal plane.