A Space of Siamese Contours in Time-dependent Complex Vector Fields

A Space of Siamese Contours in Time-dependent Complex Vector Fields

John Gill March 2014

Abstract: Siamese Contours – having the same initial points - arise in two or more time-dependent complex vector

fields and can be combined into simple sums and products that incorporate the features of the vector fields. These

elementary classroom notes illustrate the processes.

Imagine the following scenario: We know the various paths a particle in the complex plane

would follow over one time-unit due to individual underlying vector fields. How can we

combine those paths into a single path the particle will actually follow? Answering this simple

question, one is led to a certain vector space having sums, scalar products, products, projection

products, and compositions easily defined, correlated with a simple norm and metric.

Zeno (or equivalent parametric) contours are defined algorithmically as a distribution of points

{ }, 0

n

k n kz

= by the iterative procedure: ( ), 1, , 1,

, kk n k n k n k n nz z zη ϕ− −= + ⋅ which arises from the

following composition structure:

( ), , , 1, , , 1, 1,( ) lim ( ) , ( ) ( ) , ( ) ( , ) , ( ) ( )k

n n k n k n k n k n k n n nnn

G z G z G z g G z g z z z G z g zη ϕ−→∞

= = = + ⋅ = .

Usually ,

1k n

nη = , providing a partition of the unit time interval. ( )zγ is the continuous arc

from z to ( )G z that results as n→ ∞ (Euler’s Method is a special case of a Zeno contour).

When ( , )z tϕ is well-behaved an equivalent closed form of the contours , ( )z t , has the

property ( , )dz

z tdt

ϕ= , with vector field ( , ) ( , )f z t z t zϕ= + . Siamese contours are streamlines

or pathlines joined at their origin and arising from different vector fields.

Contours will be abbreviated, using the iterative algorithm, as

(i) ( )1 , 1, , 1 1, 1,: ( , )k

k n k n k n k n k nnz z f z zγ η− − −= + − and ( )2 , 1, , 2 1, 1,

: ( , )kk n k n k n k n k nnz z f z zγ η− − −= + −

Or 11 1 1

: ( , )d

tdt

γγ ϕ γ= and 2

2 2 2: ( , )d

tdt

γγ ϕ γ=

A parametric form of ( )mzγ : ( )z z t= , exists when the equation ( , )

m

dzz t

dtϕ= admits a

closed solution. For example 22

1 1 0 1: ( ) ( , ) (1 4 )t itz t z e z t z itγ ϕ+= ⇒ = + .

If a point in the plane is moved simultaneously by infinitesimal actions of two vector fields -

such as two force fields acting on a particle - the combined action is given by the following, in

which contours represent vector fields:

A commutative Contour Sum: 1 2

γ γ γ= ⊕�

where

(ii) ( ), 1, , 1, 1,: ( , )k

k n k n k n k n k nnz z F z zγ η− − −= + − ,

1 2( , ) ( , ) ( , )F z t f z t f z t z= + − .

Or 1 2

: ( , ) ( , ) ( , )d

t t tdt

γγ ϕ γ ϕ γ ϕ γ= = + . It is assumed that contours begin at the same point

of origin: 0z , defining a Siamese contour space. In the following images vector clusters

represent vector fields over the unit time interval, from black (t=0) to light color (t=1).

Suppose that we are given two parametrically-defined contours:

Example 1 (One VF is time-dependent) 2

1 1 0: ( ) ( 1)z t z t itγ = + + and 0

2 2

0

: ( )1

zz t

tzγ =

− .

How do we find 1 2

γ γ γ= ⊕�

? Not by simple addition: 2 0

3 0

0

( ) ( 1)1

zz t z t it

tz= + + +

−.

Rather, we first find the vector fields 1f ,

2f in which these contours are embedded:

11 1

( 2)( , ) ( , )

1

dz z it tz t f z t z

dt tϕ

+ += = = −

+ and

222 2( , ) ( , )

dzz z t f z t z

dtϕ= = = − ,

then solve 2

1 2

( 1) ( 2)( , ) ( , )

1

dz t z z it tz t z t

dt tϕ ϕ

+ + + += + =

+, or apply the algorithm:

( ), 1, , 1 1, 2 1,

2

1,

1, ,

2

1,

1, ,

: ( , ) ( , )

( 1) ( 2)

1

( ) ( 2 )

k kk n k n k n k n k nn n

k k kn n nk n

k n k n kn

knk n

k n k n

z z z z

z z iz

z k n z n i k nz

k n

γ η ϕ ϕ

η

η

− − −

−

−

−

−

= + +

+ + + += +

+

+ + ⋅ + += +

+

The two vector fields are shown in red (time-dependent VF) and green (normal VF), and the

contour sum is in purple: 0

1z i= + .

As seen above, it may be that 1( , )

dzz t

dtϕ= and

2( , )

dzz t

dtϕ= each have simple closed

solutions 1( )z t and

2( )z t , whereas

1 2( , ) ( , )

dzz t z t

dtϕ ϕ= + is intractable.

Example 2 (Both VFs are time-dependent) 22

1 1 0 1: ( ) ( , ) (1 4 )t itz t z e z t z itγ ϕ+= ⇒ = + and

20

2 2 22

0

2: ( ) ( , )

2

zz t z t z it

z itγ ϕ= ⇒ =

−. Here

2( , ) (1 4 )dz

z t z it z itdt

ϕ= = + + admits no easy

solution. However, as seen in Example 1, the corresponding Zeno contour provides a

constructive approach: ( )2

, 1, 1, 1,

1: (1 4 )k k

k n k n k n k nn nz z z i z i

nγ − − −= + + + ,

The TDVF of the sum (shown) is 2( , ) 2 4F z t z zit z it= + + . It is tempting to assume the

resultant contour (in green) is somehow an average or a sum of the two parametric curves (in

red), but that is not, in general, the case. The sum 1 2

γ γ γ= ⊕�

represents a path wherein each

point is moved forward by a simultaneous infinitesimal application of the two (or more) vector

fields and depends less on the original contours and more on the VFs underlying them. But the

sum of contours notation is useful.

If two (or more) TDVFs admit parametric formulations of their contours, and

1 2( , ) ( , ) ( , )

dzz t z t z t

dtϕ ϕ ϕ= = + is solvable in closed form ( )z t , then it may be possible to

demonstrate how ( )z t and 1( )z t ,

2( )z t are related, as seen in this next example:

Example 3 1( , )f z t z izt= + ,

2

2( , ) 2f z t z z t= + ⇒

2( , ) 2dz

z t z t iztdt

ϕ= = + . Thus

2/2

1 0( ) itz t z e= and 0

2 2

0

( )1

zz t

t z=

− , with

2

2

/2

0 00/2

00

( ) , 21 2

it

it

i e zz t

z ie

ωω

ω= =

+−.

Tedious algebra reveals:

( )

( )

2

1 2

2 2

2 1 1

( ) 1 ( )( )

( ) 2 2 ( ) 2 ( )

iz t t z tz t

z t it t z t i z t

+=

+ − + −.

1( )z t is red,

2( )z t is blue, and the contour ( )z t is in green.

Continuing with algebraic analogies, we define a Scalar Product:

1

γ α γ= � : 1( , )

dt

dt

γα ϕ γ= ⋅ ,

That provides a distributive feature:

1 2 1 2

( ) ( ) ( )γ α γ γ α γ α γ= ⊕ = ⊕� �

� � � : 1 2 1 2

( ( , ) ( , )) ( , ) ( , )d

t t t tdt

γα ϕ γ ϕ γ αϕ γ αϕ γ= + = +

Example 4 0 1

: ( ) (1+it) ( , )1

izz t z z t

itγ ϕ= ⇒ =

+. Thus

0: ( ) (1+it)z t z ααγ =

For =2α , 0

1z i= + . The γ - TDVF is 1 (1 )

( , )1

i tf z t z

it

+ += ⋅

+. γ is green and αγ is red.

A Contour Product is defined: 1 2

γ γ γ= ⊗�

: 1 2( , ) ( , )

dt t

dt

γϕ γ ϕ γ= ⋅ , from which one derives

1 2 2 1

( ) ( )γ α γ γ α γ γ= ⊗ = ⊗� �

� �

Example 5 1 0 1 1

1 (1 ): ( ) (1+it) ( , ) ( , )

1 1

iz i tz t z z t f z t z

it itγ ϕ

+ + = ⇒ = ⇒ =

+ + ,

02 2 2

: ( ) ( , ) ( , )1 1 1

z z ztz t z t f z t

t it tγ ϕ

−= ⇒ = ⇒ =

+ + +. Therefore

2

: ( , )(1 )(1 )

dz izz t

dt it tγ ϕ

−= =

+ + and 0

0

1 1: ( ) , ( )

1 ( ) 2 1

z i itz t t z Ln

t tγ ω

ω

− + = = ⋅

+ + .

02z i= + . Contour (1) is green, contour (2) is red, the product contour is purple.

Another concept is the Projection Product of two contours: Suppose a point is moving along a

particular pathline arising from a TDVF 1( , )f z t and a secondary vector or force field

2( , )f z t is

applied to the point in such a way that only the projection of infinitesimal vectors from the

secondary field apply to the point. Let 1

γ and 2

γ be two contours (primary and secondary):

1 1, , , 1 , 2 1, , , 2 ,: ( , ) , : ( , )k kk n k n k n k n k n k n k n k nn nz z z z z zγ η ϕ γ η ϕ+ += + = +

Define a non-commutative Projection Product:

( )1 2 1, , , 1 , ,: ( , ) 1 ( , )k k

k n k n k n k n k nn nz z z zγ γ γ η ϕ λ+= ⊗ = + ⋅ ⋅ +

�,where 1 2

,

2 2

( , )kk n nz

ϕ ϕλ

ϕ ϕ=i

i. Thus

( )1( , ) 1 ( , )

dzz t z t

dtϕ λ= + . . . rarely solvable in closed form.

Example 6 Suppose 1 1

: ( ): ( , ) ( ( ) 1) ( ( ) 1)z t f z t x Cos x yt iy Sin y xtγ = + + + + + and

2 2: ( ): ( , ) (2 ) ( )z t f z t xt x y i x yt yγ = + − + − + ,

04 3z i= + , defined by Zeno contours.

(Purple rings around initial point, green contour is primary, red contour is the projection

product contour ending in yellow rings. The green vector clusters are primary VF, red clusters

are secondary VF.)

The first image is of the paths in the TDVFs The second image is of the stable VFs, t=1

In a stable VF the projection product contour lies on a streamline, but the pathlines diverge in a

TDVF. If one has control over the secondary VF - an on-off switch, say – then a simple

adjustment assures only a positive boost: Set , 1 , 2 ,( , ) 0 if ( , ) ( , ) 0k k kk n k n k nn n nz z zλ ϕ ϕ= <� :

Define Contour Composition: 1 2 1, , , 1 2: ( ( , ), )k n k n k nz z z t tγ γ γ η ϕ ϕ+= = + ⋅� or 1 2

dz

dtϕ ϕ= �

Example 7 1 0 1

: ( ) (1 ) 1

izz t z it

itγ ϕ= + ⇒ =

+,

2 0 2: ( ) itz t z e izγ ϕ= ⇒ = ,

1 21

z

itϕ ϕ ϕ

−= =

+� . Therefore

( )tan( ) 2 tan( ) 2

1 2: ( ) ( 1 ) ( 1 )Arc t Arc t

oz t z e Cos Ln t ie Sin Ln tγ γ γ − −= = + + +�

1

γ is green, 2

γ is red, and 1 2

γ γ γ= � is purple. 4 2oz i= +

The relation between a contour ( )z z t= and its vector field : ( , )f z tF has been discussed.

Here are a few simple correspondences. (0) 0α = and (0) 0β = .

(1) 0

( ) ( ) ( , ) ( )z t z t f z t z tα α ′= + ↔ = +

(2) ( )( )

0( ) ( , ) 1 ( )tz t z e f z t z tα α ′= ↔ = +

(3) ( ) ( )0( ) 1 ( ) ( , ) 1 ( ) ( )1 ( )

zz t z t f z t t z t

tα α α

α′= + ↔ = + +

+

(4) ( )2

0

0

( ) ( ) ( ) ( ) ( ) ( )( )( ) ( , )

1 ( ) 1 ( ) ( )

z t z t t t t tz tz t f z t z

z t t t

β α β α β αα

β α β

′ ′ ′ ′+ − ++= ↔ = +

− +

(5) ( )2 2

0

2

0

( ) ( ) ( ) ( ) ( )( )( ) ( , )

1 ( ) ( )

z t t z t t tz tz t f z t

z t t

α β α α αα

β α

′ ′+ += ↔ =

−, (0) 1α =

Example 8 ( )2

2 0

2

0

(1 )( , ) ( 1) ( )

1 1

z z tf z t z t z t

t z t

+= + + ↔ =

+ − .

The floating vessel problem: given a target point (1)z , find an initial point (0)z for

(1) 3 2z i= − + . Thus 7 4

(0)5 5

z i= + from (5)

A norm γ of a contour in a 0z - based space can be formulated as 0

[0,1]

( , )t

Sup z tγ ϕ∈

= ,

giving rise to a metric: ( )1 2 1 2 1 2, ( 1)d γ γ γ γ γ γ= − = ⊕ − ⋅

�.

Example 9 1 1 0

: ( , ) (2 ) (2 ), 1.5f z t x yt i y xt z iγ = − + + = − + ,

2 2

2 2 0: ( , ) (1 ( )) ( (1 ( )) ), 1.5f z t x Cos t i y Cos t t z iγ = + + + + = − +