If a curve is given parametrically 9.1.1.2, then the positive sense is defined on this curve, i.e., the direction in which the point M xt, yt of the curve moves as the parameter t increa
Trang 1Differential Geometry
9.1 Theory of Curves
9.1.1 Plane Curves
9.1.1-1 Regular points of plane curve
A plane curve Γ in a Cartesian coordinate system can be defined by equations in the
following form:
Explicitly,
Implicitly,
F (x, y) =0
Parametrically,
In vector form,
r = r(t), where r(t) = x(t)i + y(t)j
In a polar coordinate system, the curve is usually given by the equation
r = r(ϕ),
where the relationship between Cartesian and polar coordinates is given by formulas
x = r cos ϕ and y = r sin ϕ.
Remark The explicit equation (9.1.1.1) can be obtained from the parametric equations (9.1.1.2) if the
abscissa is taken for the parameter: x = t, y = f (t).
A point M (x(t), y(t)) is said to be regular if the functions x(t) and y(t) have continuous
first derivatives not simultaneously equal to zero in a sufficiently small neighborhood
of this point For implicitly defined functions, a point M (x, y) is said to be regular if grad F =∇F ≠ 0at this point
If a curve is given parametrically (9.1.1.2), then the positive sense is defined on this curve, i.e., the direction in which the point M (x(t), y(t)) of the curve moves as the parameter t
increases If the curve is given explicitly by (9.1.1.1), then the positive sense corresponds
to the direction in which the abscissa increases (i.e., moves from left to right) In a polar
coordinate system, the positive sense corresponds to the direction in which the angle ϕ
increases (i.e., the positive sense is counterclockwise)
If s is the curve length from some constant point M0to M , then the infinitesimal length increment of the arc M0M is approximately expressed by the formula for the arc length
367
Trang 2differential ds; i.e., the following formulas hold:
1+ (y x )2dx, if the curve is given explicitly,
(x t 2+ (y t 2dt, if the curve is given parametrically,
r2+ (r
ϕ)2dϕ, for a curve in the polar coordinate system.
Example 1 The arc length differential of the curve y = cos x has the form ds = √
1+ sin x dx.
Example 2 For the semicubical parabola x = t2, y = t3, the arc length differential is equal to ds =
t √
4 + 9t 2dt.
Example 3 For the hyperbolic spiral r = a/ϕ for r >0, the arc length differential is equal to ds =
a
1+ ϕ2/ϕ2dϕ.
9.1.1-2 Tangent and normal
The tangent to a curve Γ at a regular point M0is defined to be the straight line that is the
limit position of the secant M0M1as the point M1approaches the point M0; the normal is
defined to be the straight line passing through M1and perpendicular to the tangent (Fig 9.1)
normal
tangent
M
Figure 9.1 Tangent and normal.
At each regular point M (x0, y0) = M (x(t0), y(t0)), the curveΓ has a unique tangent
given by one of the equations (depending on how the curve is defined)
y – y0= y x (x – x0), if the curve is given explicitly,
F x (x – x0) + F y (y – y0) =0, if the curve is given implicitly,
y – y0
y
t
= x – x0
x t
, if the curve is given parametrically,
r = r0+ λr t, if the curve is given in vector form,
where r0is the position vector of the point M0, λ is an arbitrary parameter, and all derivatives are evaluated at x = x0, y = y0, and t = t0
The slope of the tangent is determined by the angle α between the positive direction of the OX-axis and the positive direction of the tangent (Fig 9.2a) The slope of the tangent (and the angle α) is determined by the formulas
tan α = y x= –F x
F y =
y t
x
t.
Trang 3M O
α
θ
X Y
Figure 9.2 Slope of the tangent.
If a curve is given in the polar coordinate system, then the slope of the tangent is
determined by the angle θ between the direction of the position vector r = OM and the positive direction of the tangent (Fig 9.2b) The angle θ is determined by the formula
tan θ = r
ϕ r
The normal at each regular point M (x0, y0) = M (x(t0), y(t0)) is given, depending on the method for defining the curve, by the equations
y – y0 = –x – x0
y
x , if the curve is given explicitly,
x – x0
F x =
y – y0
F y , if the curve is given implicitly,
(y – y0)y t + (x – x0)x t=0, if the curve is given parametrically,
where all the derivatives are evaluated at x = x0, y = y0, and t = t0
The positive sense of the tangent coincides with the positive sense of the curve at the point of tangency; and the positive sense of the normal can in some way be made consistent
with the positive sense of the tangent; for example, it can be obtained from the positive
sense of the tangent by counterclockwise rotation around M by an angle of90◦ (Fig 9.3).
The point M divides the tangent and the normal into positive and negative half-lines.
M
Figure 9.3 Positive sense of tangent.
Example 4 Let us find the equations of the tangent and the normal to the parametrically given semicubical
parabola x = t2, y = t3at the point M0 (1, 1), t = 1.
The equation of the tangent,
y – t3
3t 2 = x – t
2
2t or y=23tx–12t3,
at the point M0 (1, 1) is
y= 3
2x– 1
2.
Trang 4The equation of the normal,
2t(x – t 2 ) + 3t 2(y – t3) = 0 or 2x + 3ty= t2(2 + 3t 2 ),
at the point M0 (1, 1) is (Fig 9.4a)
2x + 3y = 5.
M
y=32x 12
X
1 1
1
2
2
3
3
Y
2 +3
=5
x y
2 2
2 4
Y
x y
+
=8
y=x
0
Figure 9.4 Tangents and normals to the semicubical parabola (a) and to the circle (b).
Example 5 Let us find the equation of the tangent and the normal to the circle x2+ y2= 8 at the point
M0(2, 2).
We write the equation of the circle as F (x, y) =0:
x2+ y2– 8 = 0,
i.e., F (x, y) = x2+ y2– 8 Obviously, we obtain
F x= 2x, F y= 2y.
The equation of the tangent is
2x 0(x – x0) + 2y 0(y – y0) = 0,
or, taking into account the original equation of the circle,
xx0+ yy0= 8.
At the point M0 (2, 2), we have
x + y =4.
The equation of the normal is
x – x0
2x 0 = y2y– y00, or
y= y0
x0x.
At the point M0 (2, 2) (Fig 9.4b), we have
y = x.
Trang 5M O
1
1
X π Y
0
y=x
y= x
y=cosx
π
2
Figure 9.5 The tangent and the normal to the curve y = cos x.
Example 6 Let us find the equations of the tangent and the normal to the curve y = cos x at the point
M0(π/2,0).
The equation of the tangent is
y – cos x0= – sin x0(x – x0) or y = cos x0– sin x0(x – x0).
At the point M0(π/2,0), we have
y= π2 – x.
The equation of the normal is
y – cos x0= – x – x0
– sin x0 or y = cos x0+x – x0
sin x0
At the point M0(π/2,0) (Fig 9.5), we have
y = x – π
2.
9.1.1-3 Singular points
A point is said to be singular if it is not regular.
Implicit equations of the form F (x, y) =0are used as a rule to find singular points of a
curve and analyze their character At any singular point M0(x0, y0), both partial derivatives
of the function F (x, y) are zero:
F x (x0, y0) =0 and F y (x0, y0) =0
If both first partial derivatives are zero at M0and simultaneously at least one of the second
derivatives F xx , F xy , and F yy is nonzero, then M0is called a double point This is the most
widely known case of singular points If both first partial derivatives and simultaneously all
second partial derivatives are zero at M0but not all third partial derivatives are zero at M0,
then the point M0 is said to be triple In general, if all partial derivatives of F (x, y) up to order n –1inclusive are zero at M0but at least one of the nth derivatives is nonzero at M0,
then the point M0 is called an n-fold singular point At an n-fold singular point M0, the
curve has n tangents, some of which may coincide or be imaginary For example, for a double singular point M0, the slopes λ = y x of the two tangents at this point are the roots
of the quadratic equation
F yy (x0, y0)λ2+2F xy (x0, y0)λ + F xx (x0, y0) =0 (9.1.1.3) The roots of equation (9.1.1.3) depend on the sign of the expression
Δ =F xx F xy
F yx F yy
= F xx F yy – F xy2 ,
where the second derivatives are evaluated at the point M0(x0, y0)
this case, a sufficiently small neighborhood of M0(x0, y0) does not contain any other points
of the curve except for M0itself Such a point is called an isolated point.
Trang 6Example 7 The curve
y2+ 4x 2– x4= 0 has the isolated point (0, 0) (Fig 9.6a).
1 2
5
5
2
Y
Figure 9.6 Examples of the isolated point (a) and the node (b).
IfΔ <0, then the quadratic equation (9.1.1.3) has two distinct real roots In this case,
there are two branches of the curve passing through the point M0(x0, y0); these branches have distinct tangents whose directions are just determined by equation (9.1.1.3) Such a
point is called a node (a point of self-intersection).
Example 8 The point (0, 0) of the curve
y2– x2= 0
is the node (0, 0) (Fig 9.6b).
IfΔ =0, then the roots of the quadratic equation (9.1.1.3) coincide In this case, the singular point of the curve is either isolated or characterized by the fact that all branches
approaching the singular point M0have a common tangent at this point:
1 Cusps of the first kind are points approached by two branches of the curve that have a
common tangent at this point and lie on the same side of the common normal and on opposite sides of the common tangent
2 Cusps of the second kind are points approached by two branches of the curve that have
a common tangent at this point and lie on the same side of the common normal and on the same side of the common tangent
3 Points of osculation are points at which the curve is tangential to itself.
Example 9 For the curve (cissoid of Diocles)
(2a– x)y2– x3= 0,
shown in Fig 9.7a, the origin is a cusp of the first kind, which is clear from the explicit equation
y= x3 2a– x
of the curve For x <0, no y satisfy the equation, and for x > 0the values y lie on opposite sides of the tangent
x= 0 at the origin.
1
1 1 1
1
2
Figure 9.7 Examples of a cusp of the first kind (a), a cusp of the second kind (b), and an osculation point (c).
Trang 7Example 10 For the curve (Fig 9.7b)
(y2– x2)2– x5= 0, the origin is a cusp of the second kind, which easily follows from equation
y = x2 x5/2
of the curve It is also obvious that the curve consists of two branches tangent to the axis OX at the origin, and
for 0< x <1the value of y is positive for both branches.
Example 11 The curve (Fig 9.7c)
y2– x4= 0 has an osculation point at the origin.
Remark 1. If all the second partial derivatives are zero at the point M0, i.e., F xx = F xy = F yy= 0, then
more than two branches of the curve can pass through this point For example, for the trefoil
(x2+ y2)2– ax(x2– y2) = 0,
three branches with tangents x =0and x y= 0 pass through the origin (Fig 9.8).
Y
Figure 9.8 The trefoil.
Remark 2. If the equation F (x, y) =0 does not contain constant terms and terms of degree 1, then the origin is a double point The equation of the tangent at a double point can readily be obtained by equating all terms of degree 2 with zero For example, for the cissoid of Diocles (Example 9), the equation of the tangent
x= 0follows from the equation –xy2– x3= 0 If the equation F (x, y) = 0 does not contain constant terms and terms of degrees 1 and 2, then the origin is a triple point, etc.
Along with the singular points listed above, there are many other singular points with specific names:
1 Break points are points at which the curve changes its direction by a “jump” and, in contrast to cusps, the tangents to both parts of the curve are distinct (Fig 9.9a).
2 Termination points are points at which the curve terminates (Fig 9.9b).
3 Asymptotic points are points around which the curve winds infinitely many times while infinitely approaching them (Fig 9.9c).
Figure 9.9 The break point (a), the termination point (b), and the asymptotic point (c).
Remark. A break point corresponds to a jump discontinuity of the derivative dy/dx Termination points correspond to either a jump discontinuity or termination of the function y = f (x) Asymptotic points can be
found most easily in curves given in the polar coordinate system.
... y0, and t = t0The positive sense of the tangent coincides with the positive sense of the curve at the point of tangency; and the positive sense of the normal... Cusps of the first kind are points approached by two branches of the curve that have a
common tangent at this point and lie on the same side of the common normal and on opposite sides of. .. Cusps of the second kind are points approached by two branches of the curve that have
a common tangent at this point and lie on the same side of the common normal and on the same side of