Handbook of mathematics for engineers and scienteists part 60 potx

9.1. THEORY OF CURVES 381
At a regular point M
0
, the equation of the normal plane has the form
(x – x
0
)x

t
(t
0
)+(y – y
0
)y

t
(t
0
)+(z – z
0
)z

t
(t
0
)=0 or (r – r
0
) ⋅ r

t
(t

0
)=0,(9.1.2.4)
and for the curve Γ obtained as the intersection (9.1.2.2) of two planes, we have





x – x
0
y – y
0
z – z
0
(F
1
)
x
(F
1
)
y
(F
1
)
z
(F
2
)
x

(F
2
)
y
(F
2
)
z





= 0,(9.1.2.5)
where all derivatives are evaluated at x = x
0
and y = y
0
.
1
◦
. Consider a curve defined via its natural parameter s, r = r(s). Unlike the derivatives
with respect to an arbitrary parameter, the derivatives with respect to s will be denoted by
primes. Consider the vectors r

s
and r

ss
. It follows from (9.1.2.1) that |r


s
| = 1. Thus the
first derivative with respect to the natural parameter s of the position vector of a point on a
curve is the unit vector tangent to the curve.
2
◦
. The second derivative with respect to the natural parameter s of the position vector of
a point of a curve is equal to the first derivative of the unit vector r

s
, i.e., of a vector of
constant length, and hence it is perpendicular to this vector. But since the vector of the first
derivative is tangent to the curve, the vector of the second derivative with respect to the
natural parameter s is normal to the curve. This normal is called the principal normal to
the curve.
At a regular point M
0
, the equation of the principal normal has the form
x – x
0
y

t
n – z

t
m
=
y – y

0
z

t
l – x

t
n
=
z – z
0
x

t
m – y

t
l
or r = r
0
+ λr

t
× (r

t
× r

tt
),

l = y

t
z

tt
– y

tt
z

t
, m = z

t
x

tt
– z

tt
x

t
, n = x

t
y

tt

– x

tt
y

t
,
(9.1.2.6)
where all derivatives are evaluated at t = t
0
.
9.1.2-3. Osculating plane.
Any plane passing through the tangent line to a curve is called a tangent plane. A tangent
plane passing through a principal normal to the curve is called the osculating plane.
A curve has exactly one osculating plane at each of its points (assuming that the vectors
r

t
and r

tt
are linearly independent). This plane passes through the vectors r

t
and r

tt
drawn from the point of tangency. The osculating plane is independent of the choice of the
parameter t on the curve.
Remark. If t is time and r = r(t) is an equation of motion, then the vector r


tt
is called the acceleration
vector of a moving point. The acceleration vector always lies in the osculating plane of the trajectory of a
moving point.
Since the family of all planes in space depends on three parameters and the position of
a plane is determined by three noncollinear points, the osculating plane can be determined
as the limit position of the plane passing through three points of the curve that infinitely
approach one another (Fig. 9.16).
Figure 9.16. The osculating plane.
382 DIFFERENTIAL GEOMETRY
The equation of the osculating plane at the point M
0
has the form





x – x
0
y – y
0
z – z
0
x

t
(t
0

) y

t
(t
0
) z

t
(t
0
)
x

tt
(t
0
) y

tt
(t
0
) z

tt
(t
0
)






= 0 or

(r – r
0
)r

t
(t
0
)r

tt
(t
0
)

= 0.
This equation becomes meaningless for the points of the curve at which
r

t
(t
0
) × r

tt
(t
0

)=0 .(9.1.2.7)
The points satisfying (9.1.2.7) are called points of rectification. The osculating plane at
these points is undefined and, in what follows, we exclude them from consideration together
with singular points of the curve.
Remark. This is inadmissible for a curve entirely consisting of points of rectification, because a curve
consisting of points of rectification is a straight line.
9.1.2-4. Moving trihedral of curve.
The normal perpendicular to the osculating plane is called the binormal.
At each point of a curve, the tangent, the principal normal, and the binormal determine
a trihedral with three right angles at the vertex, which lies on the curve. This trihedral is
called the moving or natural trihedral of a curve. The faces of the moving trihedral are
three mutually perpendicular planes:
1. The normal plane is the plane containing the principal normal and the binormal.
2. The osculating plane is the plane containing the tangent and the principal normal.
3. The rectifying plane is the plane containing the tangent and the binormal.
The direction vector of the tangent is equal to the fi rst derivative
T = r

t
.
The direction vector of the binormal is equal to the vector product of the vectors of the first
and second derivatives (see Fig. 9.17):
B = r

t
× r

tt
.
r

Nr= B
r
Brr=
t
t
t
t
t
t
t
Figure 9.17. Direction vectors of the tangent, of the binormal, and of the principal normal.
9.1. THEORY OF CURVES 383
The direction vector of the principal normal is perpendicular to the tangent and binormal
vectors. Hence it can be set equal to their vector product
N = r

t
× B.
The moving trihedral permits one to assign a rectangular coordinate system to each
point of the curve; the axes of this coordinate system coincide with the tangent, the principal
normal, and the binormal. To determine the sense of these axes, one introduces positive
unit vectors of these axes.
Consider a curve defi ned via the natural parameter s.
1. The unit tangent vector is defined to coincide with the fi rst derivative of the position
vector with respect to the natural parameter,
t = r

s
.(9.1.2.8)
2. The unit principal normal vector n is defined in such a way that its sense coincides with

that of the vector of the second derivative with respect to the parameter,
n =
r

ss
|r

ss
|
=
1
k
r

ss
.
The vector kn = r

ss
is called the curvature vector;herek is the curvature, studied in
Paragraph 9.1.2-6.
3. The unit binormal vector b is chosen from the condition that t, n,andb is a right triple
of vectors,
b = t × n.
If the direction of arc length increase is changed, then the tangent and binormal vectors
also change their sense, but the sense of the principal normal vector remains the same.
9.1.2-5. Equations for elements of trihedral.
At a regular point M
0
, the equation of the tangent has the form [see also (9.1.2.3)]

r = r
0
+ λt,
where λ is a variable parameter.
At a regular point M
0
, the equation of the principal normal has the form [see also
(9.1.2.5)]
r = r
0
+ λn,
where λ is a variable parameter.
At a regular point M
0
, the equation of the binormal has the form
r = r
0
+ λb,
where λ is a variable parameter.
The vector equation of the osculating plane at a regular point M
0
is
(r – r
1
) ⋅ b = 0
[see also (9.1.2.6)].
The vector equation of the normal plane at a regular point M
0
is
(r – r

0
) ⋅ t = 0
[see also (9.1.2.4)].
The vector equation of the rectifying plane at a regular point M
0
is
(r – r
0
) ⋅ n = 0.
384 DIFFERENTIAL GEOMETRY
9.1.2-6. Curvature of space curves.
The curvature of a space curve at a point M
0
is determined by analogy with the curvature of
a plane curve (see Paragraph 9.1.1-7); i.e., the curvature of a curve at a point M
0
is the limit
ratio of the tangent rotation angle along an arc shrinking to M
0
to the arc length (Fig. 9.18),
k = lim
Δs→0



Δt
Δs




=



dt
ds



.(9.1.2.9)
M
M
1
0
t
t
ttΔ+
ttΔ+
Δt
Δφ
Δs
Figure 9.18. The curvature of a space curve.
Remark. The curvature k can be determined as the angular velocity of the vector t (or, which is the same,
the angular velocity of the tangent) at a given point of the curve with respect to the distance s passed along the
curve.
It follows from (9.1.2.8) and (9.1.2.9) that
k = |r

ss
| =


(x

ss
)
2
+(y

ss
)
2
+(z

ss
)
2
.(9.1.2.10)
For an arbitrary choice of the parameter, the curvature k is calculated by the formulas
k =
|r

t
× r

tt
|
|r

t
|

3
,
k =


(x

t
)
2
+(y

t
)
2
+(z

t
)
2

(x

tt
)
2
+(y

tt
)

2
+(z

tt
)
2

–

x

t
x

tt
+ y

t
y

tt
+ z

t
z

tt

2


(x

t
)
2
+(y

t
)
2
+(z

t
)
2

3/2
.
The radius of curvature and the curvature are reciprocal quantities; i.e., ρ = 1/k.For
space curves, ρ and k are always positive.
Example 1. Let us find the curvature of the helix x = a cos t, y = a sin t, z = bt.
Expressing the parameter t in terms of the arc length s as
s =
√
a
2
+ b
2
t,
we obtain

x = a cos
s
√
a
2
+ b
2
, y = a sin
s
√
a
2
+ b
2
, z =
bs
√
a
2
+ b
2
.
Formula (9.1.2.10) implies
k =
a
a
2
+ b
2
, ρ =

a
2
+ b
2
a
.
9.1. THEORY OF CURVES 385
M
M
1
0
bbΔ+
bbΔ+
Δb
b
b
β
Δs
Figure 9.19. The torsion of a space curve.
9.1.2-7. Torsion of space curves.
In addition to curvature, space curves are also characterized by torsion. The torsion of
a space curve is defined to be a quantity whose absolute value is equal to the limit ratio
of the rotation angle of the binormal along an arc, shrinking to a given point to the arc
length (Fig. 9.19):
|τ| = lim
Δs→0
β
Δs
= lim
Δs→0




Δb
Δs



=



db
ds



.
Remark 1. The torsion τ does not change (including the sign) if the direction of increase of the parameter s
is changed to the opposite.
Remark 2. The torsion τ can be defined as the angular velocity of the binormal b at the corresponding
point with respect to the distance s passed along the curve.
In the special case of a plane curve, the torsion is zero at all points; conversely, if the
torsion is zero at all points of a curve, then the entire curve lies in a single plane, which is
the common osculating plane for all of its points.
For an arbitrary choice of the parameter t, the torsion τ is calculated by the formulas
τ = ρ
2
r


t
r

tt
r

ttt
|r

t
|
3
=






x

t
y

t
z

t
x


tt
y

tt
z

tt
x

ttt
y

ttt
z

ttt







(x

t
)
2
+(y


t
)
2
+(z

t
)
2

(x

tt
)
2
+(y

tt
)
2
+(z

tt
)
2

–(x

t
x


tt
+ y

t
y

tt
+ z

t
z

tt
)
2
.
(9.1.2.11)
But if the arc length is taken as the parameter, then τ is calculated by the formulas
τ = ρ
2
(r

s
r

ss
r

sss
)

|r

s
|
3
=






x

s
y

s
z

s
x

ss
y

ss
z

ss

x

sss
y

sss
z

sss







(x

ss
)
2
+(y

ss
)
2
+(z

ss
)

2

2
.
The quantity ρ
τ
= 1/τ inverse to the torsion is called the radius of torsion.
Example 2. Let us find the torsion of the helix x = a cos t, y = a sint, z = bt. Using formula (9.1.2.11),
we obtain
τ =





–a sin tacos tb
–a cos t –a sin t 0
a sin t –a cos t 0






a
2
sin
2
t + a
2

cos
2
t + b
2

a
2
cos
2
t + a
2
sin
2
t

–

a
2
sin t cos t – a
2
cos t sin t

2
=
b
a
2
+ b
2

.
386 DIFFERENTIAL GEOMETRY
9.1.2-8. Serret–Fr
´
enet formulas.
The derivatives of the vectors t, n,andb with respect to the parameter s at each point of the
curve satisfy the Serret–Fr
´
enet formulas
t

s
= kn, n

s
=–kt + τ b, b

s
=–τn,
where k is the curvature of the curve and τ is the torsion of the curve.
As the parameter s increases, the point M moves along Γ; in this case
1. The tangent rotates around the instantaneous position of the binormal b at a positive
angular velocity equal to the curvature k of Γ at M.
2. The binormal b rotates around the instantaneous position of the tangent at an angular
velocity equal to the torsion τ of Γ at M; here the torsion is positive if the shape of the
curve resembles a right helix.
3. The trihedral rotates as a solid around an instantaneous axis whose direction is deter-
mined by the Darboux vector Ω = τ t + kb at a positive angular velocity equal to the
total curvature of Γ M, i.e., to |Ω| =
√

τ
2
+ k
2
.
The Serret–Fr
´
enet formulas permit one to find the coefficients in the decomposition of
the derivatives of the position vector in the vectors of the moving trihedral. The following
recursion formulas hold for the decomposition of the derivatives of the position vector of a
point on the curve with respect to the arc length:
r
(n)
s
= A
n
t + B
n
n + C
n
b,
where
A
n+1
=(A
n
)

s
– B

n
k, B
n+1
=(B
n
)

s
+ A
n
k – C
n
τ, C
n+1
=(C
n
)

s
+ B
n
τ.
For example, using the relation r

ss
= kn, we can decompose r

sss
as
r


sss
=–k
2
t + k

s
n + kτb.
9.2. Theory of Surfaces
9.2.1. Elementary Notions in Theory of Surfaces
9.2.1-1. Equation of surface.
Any surface can be determined by equations in one of the following forms:
Explicitly,
z = f (x, y).
Implicitly,
F (x, y, z)=0.(9.2.1.1)
Parametrically,
x = x(u, v), y = y(u, v), z = z(u, v).
In vector form,
r = r(u, v), or r = x(u, v)i + y(u, v)j + z(u, v)k.
9.2. THEORY OF SURFACES 387
Varying the parameters u and v arbitrarily, we obtain the position vector and the coordi-
nates of various points on the surface. Eliminating the parameters u and v from a parametric
equation of a surface, we obtain an implicit equation of the surface. The explicit equation
is a special case (u = x and v = y) of the parametric equation.
It is assumed that the vectors r
u
≡ ∂r/∂u and r
v
≡ ∂r/∂v are nonparallel, i.e.,

r
u
× r
v
≠ 0.(9.2.1.2)
The points at which condition (9.2.1.2) holds are said to be regular. This condition
satisfied at a point M on the surface guarantees that the equation of the surface near this
point can be solved for one of the coordinates. Condition (9.2.1.2) also guarantees a one-
to-one correspondence between points on the surface and pairs of values of u and v in the
corresponding range of u and v.
Condition (9.2.1.2) for a surface defined implicitly becomes
grad F ≠ 0 .
For a surface defined parametrically, all functions have continuous first partial derivatives,
and the rank of the matrix

x
u
y
u
z
u
x
v
y
v
z
v

is equal to 2.
Example 3. The sphere defined by the implicit equation

x
2
+ y
2
+ z
2
– a
2
= 0
is described in parametric form as
x = a cos u sin v, y = a sin u sin v, z = a cos v
and in vector form as
r = a cos u sin vi + a sinu sin vj + a cos vk.
9.2.1-2. Curvilinear coordinates on surface.
We consider a surface or part of the surface such that it can be topologically (i.e., bijectively
and continuously) mapped onto a plane domain and assume that a point M of this surface
is taken to a point M
0
with rectangular coordinates u and v on the plane (see Fig. 9.20). If
such a mapping is given, then the surface is said to be parametrized,andu and v are called
curvilinear (Gaussian) coordinates of the point M on the surface. Since the mapping is
continuous, each curve on the plane gives rise to a curve on the surface. In particular, the
straight lines u = const and v = const are associated with the curves on the surface, which
are called parametric or curvilinear lines of the surface. Since the mapping is one-to-one,
there is a single curve of the family u = const and a single curve of the family v = const that
pass through each point of the parametrized surface. Both families together form a regular
net, which is called the coordinate net.
In the case of rectangular Cartesian coordinates, the coordinate net on the plane is formed
by all possible straight lines parallel to coordinate axes; in the case of polar coordinates, the
coordinate net is formed by circles centered at the pole and half-lines issuing from the pole.

Example 4. In the parametric equations
x = a cos u sin v, y = a sin u sin v, z = a cos v
of the sphere, u is the longitude and v the polar distance of a point.