Differential Geometry
Differential Geometry was initially developed in the 18th and 19th centuries as a method
of using calculus to study problems in geometry. It is the language in which Einstein’s
general theory of relativity is expressed and has applications to econometrics and
computer-aided design. The treatment in FP3 gives an introduction to the theory of plane
and space curves in 3-dimensional space and is a good choice for students who are
studying the chapter on multivariable calculus.
Arc length
If a curve is ‘sufficiently smooth’ the arc length between two points can be defined as:
2
⎛ dx ⎞ ⎛ dy ⎞
ds
= ⎜ ⎟ +⎜ ⎟
dp
⎝ dp ⎠ ⎝ dp ⎠
2
From this, s can be found by integrating with respect to p.
δy
δs
δx
2
The special cases:
ds
⎛ dy ⎞
= 1+ ⎜ ⎟
dx
⎝ dx ⎠
and
ds
⎛ dr ⎞
2
= ⎜
⎟ +r
dθ
⎝ dθ ⎠
when y = f(x)
2
when r and θ are polar coordinates
are worth remembering.
1
Example: Find the length of the astroid x = a cos3 θ , y = sin 3 θ
ds
= (−3a cos 2 θ sin θ ) 2 + (3a sin 2 θ cos θ ) 2
dθ
= 3a cos θ sin θ
So the length of arc in the first quadrant is:
∫
π
2
0
3a cos θ sin θ dθ
3a
2
3a
and the length of the complete astroid is 4 x
= 6a
2
=
NB the integrand must be positive (if you are not careful you could end up with ‘arc
length = 0’since the expression 3a cos θ sin θ is negative in the second and fourth
quadrants).
The above use of the calculus only works if the curve is sufficiently smooth
arc PQ
ie.
→ 1 as P → Q
chord PQ
One example where this method does not work is Von Koch’s ‘snowflake’.
2
Intrinsic Equations
With cartesian or polar coordinates an equation of a curve may take many forms. For
example, x 2 − y 2 = 2 and xy = 1 represent the same equation but with respect to different
axes. From the middle of the 19th century, under the influence of Riemann, curves were
studied as free-standing objects.
The intrinsic equation of a curve connects its arc length s with the angle ψ which its
tangent makes with a fixed direction (normally s = 0 when ψ = 0). We also need to
specify the sense along the curve in which s increases.
For example, the intrinsic equation of a circle of radius ρ can be written s = ρψ.
ρ
C
Ψ=0
P
s=0
If we always measure ψ from the line x = 0, then the intrinsic and cartesian equations are
dy
connected by: tanψ =
dx
2
Also
dy
ds
dx
⎛ dy ⎞
= sinψ
= 1 + ⎜ ⎟ = 1 + tan 2 ψ = secψ so that
= cosψ and
ds
dx
ds
⎝ dx ⎠
These results can be remembered by the ‘differential triangle’
ds
dy
ψ
dx
3
We can convert from the intrinsic equation to the parametric or cartesian equation as
follows:
For example, if s =
3
π
a sin 2 ψ , where 0 ≤ ψ ≤
2
2
ds
= 3a sinψ cosψ
dψ
So
dx dx ds
=
x
= cosψ x 3a sinψ cosψ = 3a cos 2 ψ sinψ
dψ ds dψ
and
dy dy ds
=
x
= sinψ x 3a sinψ cosψ = 3a sin 2 ψ cosψ
dψ ds dψ
Integrating wrt ψ gives: x = − a cos3 ψ and y = a sin 3 ψ if x = − a and y = 0 when ψ = 0
2
2
2
The Cartesian equation is: x 3 + y 3 = a 3
4
Curvature
The curvature at a point P is denoted by κ and defined by: κ =
dψ
ds
If the intrinsic equation is known then finding curvature is straightforward.
For example, as given above, a circle of radius ρ has intrinsic equation s = ρψ
ds
1
= ρ , and κ =
dψ
ρ
As expected, this circle has constant curvature, inversely proportional to the radius ρ.
If the intrinsic equation is not known and the curve is defined in terms of a parameter p
things are a little more complicated but not difficult!
For brevity, let x / stand for
dx
d 2x
dψ ψ /
and x // for
.
In
this
notation
=
= /
κ
dp
dp 2
ds
s
We know s / = x / 2 + y / 2 and tanψ =
y/
from the differential triangle
x/
x / y // − y / x //
x/ 2
x / y // − y / x //
ψ / = /2 2
x sec ψ
Differentiating wrt p gives sec 2 ψψ / =
So
=
=
=
Hence
κ=
x / y // − y / x //
x / 2 (1 + tan 2ψ )
x / y // − y / x //
x / 2 (1 +
y/ 2
x/ 2
/ //
)
x / y // − y x
x/ 2 + y/ 2
x / y // − y / x //
3
( x/ 2 + y/ 2 ) 2
d2y
dx 2
If y is given explicitly in terms of x, this reduces to κ =
⎡ ⎡ dy ⎤ 2 ⎤
⎢1 + ⎢ ⎥ ⎥
⎣⎢ ⎣ dx ⎦ ⎦⎥
5
Circle of curvature
The circle of curvature that ‘best’ fits a given curve at a point P is defined as the one with
the same curvature at P and its radius ρ is the radius of curvature at P. Since the
1
1 ds
curvature of a circle of radius ρ is κ = it follows that ρ = =
ρ
κ dψ
This means that if κ is negative then so is ρ.
^
^
⎛ cosψ ⎞
⎛ -sinψ ⎞
If we define t and n to represent the unit vectors ⎜
⎟ and ⎜
⎟ respectively,
⎝ sinψ ⎠
⎝ cosψ ⎠
^
Then, if c is the position vector of the centre of curvature, c = r + ρ n see diagrams
below.
As the point P moves along a given curve, the centre of curvature C also moves.
The locus of the centre of curvature is called the evolute of the curve.
C
κ > 0, ρ > 0
n
t
P
r
O
t
κ < 0, ρ < 0
n
r
O
P
C
6
Consider again the part of the astroid in the second quadrant with intrinsic equation
s=
3
π
a sin 2 ψ , where 0 ≤ ψ ≤
2
2
and parametric equations
x = − a cos3 ψ and y = a sin 3 ψ if x = − a and y = 0 when ψ = 0
The curvature is given by:
κ=
=
ρ=
Also
1
κ
=
dψ
ds
1
3a sinψ cosψ
3a sin 2ψ
2
and the position vector of the centre of curvature is given by:
^
c=r+ρn
⎛ −a cos3 ψ ⎞ 3a sin 2ψ ⎛ − sinψ ⎞
=⎜
⎟+
⎜
⎟
3
2
⎝ cosψ ⎠
⎝ a sin ψ ⎠
⎛ − cos3 ψ − 3sin 2 ψ cosψ ⎞
= a⎜
⎟
3
2
⎝ sin ψ + 3sinψ cos ψ ⎠
This locus appears complicated but if we consider it from a different axis system (axes
parallel to y = -x and y = x), the new position vector c/ is given by:
1
c =
2
/
=
⎡1 −1⎤ ⎛ − cos3 ψ − 3sin 2 ψ cosψ ⎞
⎟
⎢1 1 ⎥ a ⎜
3
2
⎣
⎦ ⎝ sin ψ + 3sinψ cos ψ ⎠
a ⎛ − cos3 ψ − 3sin 2 ψ cosψ − sin 3 ψ − 3sinψ cos 2 ψ ⎞
⎜
⎟
2 ⎝ − cos3 ψ − 3sin 2 ψ cosψ + sin 3 ψ + 3sinψ cos 2 ψ ⎠
⎛ − cos3 (ψ − π4 ) ⎞
= 2a ⎜
⎟
3
π
⎝ sin (ψ − 4 ) ⎠
So the evolute of this astroid is another astroid, twice as large!
7
Diagram of original astroid and ‘astroid evolute’
NB as in the above diagram, when the curvature is greatest or least the evolute has a
cusp.
8
Envelopes
If f ( x, y, p) = 0 represents a family of curves (one for each value of a parameter p), then,
when these curves are drawn, the outline, which all the curves touch, forms another curve
called an envelope.
If the family consists of curves which are ‘reasonably smooth’, the condition for these
∂
f ( x, y , p ) = 0
curves to form an envelope is that
∂p
Consider a line of fixed length a moving with its end points on the coordinate axes.
y
a
θ
x
O
The equation of the family is f ( x, y, θ ) = x sec θ + y cos ecθ − a = 0
and
∂f
= x sec θ tan θ − y cos ecθ cot θ = 0
∂θ
x sec θ tan θ = y cos ecθ cot θ
x
y
= 3
3
cos θ sin θ
So, if x = λ cos3 θ then y = λ sin 3 θ
Substituting in f ( x, y,θ ) = 0 gives λ cos 2 θ +λ sin 2 θ − a = 0
So λ = a and the parametric equations are x = a cos3 θ , y = a sin 3 θ (the astroid again!)’
9
The evolute as the envelope of normals
^
If we differentiate c = r + ρ n with respect to s
^
Then
dc dr
d n dρ ^
n
=
+ρ
+
ds ds
ds ds
^
ds d n d ρ ^
n
= t+
+
dψ ds ds
^
^
^
dρ ^
dn
=
n since
= −t
ds
dψ
Since the LHS is a vector tangential to the evolute at C and the RHS is a vector in the
direction of the normal PC, this normal touches the evolute at C and so the evolute is the
envelope of normals.
Hence the evolute of the astroid is an envelope of normals to an envelope!
10
Solids of revolution
δs
y2
l1
y1
l2
The curved surface area of the solid formed by rotating about the x axis the line segment
joining points ( x1 , y1 ) and ( x2 , y2 ) where the distance between the points is δs (a frustum
of a cone) is given by:
δ s = π y2l2 − π y1l1 ie the difference between the curved surface area of two cones.
But
y2l1 = y1l2
Hence
δ s = π y2l2 − π y1l1 + π y2l1 − π y1l2
(similar triangles)
= π ( y2 + y1 )(l2 − l1 )
= 2π yδ s where y =
1
( y1 + y2 ) is the average radius of the frustum.
2
Hence, in general, the curved surface area of a solid of revolution rotated about the x-axis
is given by:
B
S = lim ∑ 2π yδ s = ∫ 2π yds
δ s →0
A
B
A
If the curve is defined in terms of a parameter p then:
β
S = ∫ 2π y
α
ds
dp with p = α at A and p = β at B
dp
11
Appendix: meaning of differentials
dy
is regarded as a single entity, not as a ratio of two separate
dx
quantities dy and dx. In fact, dy and dx can be given separate meanings in such a way that
their ratio is equal to the derivative.
In the Leibnitz notation,
Let y = f (x) where f is a differentiable function of x.
The differential dx is defined as an independent variable, ie dx can take any real value.
dy
The differential dy is then defined by the equation: dy = f’(x)dx or dy = dx
dx
(So dy is also a variable, but is dependent on x and dx.)
dy
and the ‘derivative’
can be interpreted as a ratio of differentials.
dx
In 2 dimensions it is easy to interpret the differentials geometrically.
Consider points P ( x, y ) and Q ( x + δ x, y + δ y )
y
R
dy
Q
δy
P
S
δx
=dx
x
dy RS
=
dx PS
dy
dy
dy
RS = PS = δ x = dx = dy
dx
dx
dx
Gradient at P = slope of tangent PR ie.
The differential dy is the change in y to stay on the tangent line when x changes by dx.
δ y is the change in y to stay on the curve when x changes by δ x (= dx). JGC: 2.7.08
12