Theoretical Aspects of Intruder Search

Theoretical Aspects of Intruder Search

Course Wintersemester 2015/16

Geometric Firefighting – Lower Bound and FF Curve

Elmar Langetepe

University of Bonn

December 22nd, 2015

Elmar Langetepe Theoretical Aspects of Intruder Search

Lower bound construction, spiralling strategies!

Start at the fire!

Spiralling strategies!

Visit four axes in cyclic order Visit axes in increasing distance

A

Theorem 58:Each “spiralling” strategy must have speed v>1.618. . .(golden ratio) to be successful.

Proof of lower speed bound: suppose v ≤ 1 . 618

x pi−1

pi

A A

By induction:

On reaching pi,

interval of lengthA below pi−1 is on fire.

(Induction base!)

Elmar Langetepe Theoretical Aspects of Intruder Search

Proof of lower speed bound: suppose v ≤ 1 . 618

x pi−1

pi

A A p_i+1

y

Inductive Step:

After arrivingp_i+1 fire moves at least x+A

Proof of lower speed bound: suppose v ≤ 1 . 618

x pi−1

pi

A A p_i+1

y

A

Inductive Step:

After arrivingp_i+1 fire moves at least x+A

Elmar Langetepe Theoretical Aspects of Intruder Search

Proof of lower speed bound: suppose v ≤ 1 . 618

x pi−1

pi

A A A

x/v

p_i+1

y/v

y

A pi−3

x/v

On reaching p_i+1: 1.A+^x_v ≤pi ≤x and 2.A+^x_v +^y_v ≤p_i+1 ≤y

=⇒ v(v−1)¹ x+_v−1¹ A≤ ^y_v

=⇒x+A≤ ^y_v fromv²−v ≤1

FollowFire Strategy for v = 5 . 27!

Logarithmic spiral of excentricityα aroundZ (_v¹ = cos(α))!

(First Part)

Elmar Langetepe Theoretical Aspects of Intruder Search

FollowFire Strategy for v = 5 . 27!

Logarithmic spiral of excentricityα aroundp₀ (¹_v = cos(α))!

(Second Part)

FollowFire Strategy for v = 5 . 27!

Excentricityα around wrapping center Z1 (_v¹ = cos(α))!

(Third part!)

Elmar Langetepe Theoretical Aspects of Intruder Search

FollowFire: Free String Wrapping!

v = 5.27 (α= 1.38) Log(p₀,p₁),Log(p₁,p₂) Free string: F₁(l):

Wrapping around Log(p0,p1)

v = 3.07 (α= 1.24) Wrappingaround Log(p₁,p₂)

Wrapping aroundwrappings!

FollowFire: Free String Wrapping!

v = 5.27 (α= 1.38) Log(p₀,p₁),Log(p₁,p₂) Free string: F₁(l):

Wrapping around Log(p0,p1)

v = 3.07 (α= 1.24) Wrappingaround Log(p₁,p₂)

Wrapping aroundwrappings!

Elmar Langetepe Theoretical Aspects of Intruder Search

FollowFire: Free String Wrapping!

v = 5.27 (α= 1.38) Log(p₀,p₁),Log(p₁,p₂) Free string: F₁(l):

Wrapping around Log(p0,p1)

v = 3.07 (α= 1.24) Wrappingaround Log(p₁,p₂)

Wrapping aroundwrappings!

Experimental approach!

(Spiral Generator Appet!)

Elmar Langetepe Theoretical Aspects of Intruder Search

FollowFire: Successful?

v= 2.69 (α= 1.19):

8 rounds!

v= 2.593 (α= 1.175):

Simulation did not succeed!

Successful for which v∈(1,∞)?

Lower and upper bounds onv! Proofs!

Upper bound by FollowFire

Theorem 59:FollowFire strategy is successful ifv >v_c≈2.6144

Sketch! When gets the free string to zero?

1 Parameterize free strings for coil j (Linkage)

2 Structural properties

3 Successive interacting differential equations

4 Inserting end of parameter interval

5 Coefficients of power series

6 Ph. Flajolet: Singularities

7 Pringsheim’s Theorem and Cauchy’s Residue Theorem

Elmar Langetepe Theoretical Aspects of Intruder Search

Upper bound by FollowFire

Theorem 59:FollowFire strategy is successful ifv >v_c≈2.6144 Sketch! When gets the free string to zero?

1 Parameterize free strings for coil j (Linkage)

2 Structural properties

3 Successive interacting differential equations

4 Inserting end of parameter interval

5 Coefficients of power series

6 Ph. Flajolet: Singularities

7 Pringsheim’s Theorem and Cauchy’s Residue Theorem

Upper bound: 1. Parameterize the free string

FollowFireWrapping process!

Free stringsFj/φj parameterized by lenght of starting spirals!

|Log(p₀,p₁)|=l₁

|Log(p0,p1)|+|Log(p1,p2)|=l2

Fj:l ∈[0,l1] φj:l ∈[l1,l2]

Elmar Langetepe Theoretical Aspects of Intruder Search

Upper bound: 1. Parameterize the free string (Linkage)

FollowFireDrawing backwards tagents!

Free stringsF_j/φ_j parameterized by lenght of starting spirals!

Fj

L_j−1 Lj

Fj−1

F0

α α

α p

l

F_j:l ∈[0,l₁] φ_j:l ∈[l₁,l₂]

l₁= _cos(α)^A ·(e^2πcot(α)−1) l2= _cos^A_α(e^2πcotα−1)e^αcotα

F_j+1(l₁) = φ_j+1(l₁) Fj+1(0) = φj(l2)

Upper bound: 1. Parameterize the free string (Linkage)

FollowFireDrawing backwards tagents!

Free stringsF_j/φ_j parameterized by lenght of starting spirals!

p

φ0

φj−1

φj

α α

l p0

F_j:l ∈[0,l₁] φ_j:l ∈[l₁,l₂]

l₁= _cos(α)^A ·(e^2πcot(α)−1) l2= _cos^A_α(e^2πcotα−1)e^αcotα

F_j+1(l₁) = φ_j+1(l₁) Fj+1(0) = φj(l2)

Elmar Langetepe Theoretical Aspects of Intruder Search

Upper bound: 1. Parameterize the free string

FollowFireWrapping process!

Free stringsFj/φj parameterized by lenght of starting spirals!

|Log(p₀,p₁)|=l₁

|Log(p0,p1)|+|Log(p1,p2)|=l2

Fj:l ∈[0,l1] φ_j:l ∈[l1,l2]

F_j+1(l₁) = φ_j+1(l₁) Fj+1(0) = φj(l2)

F0(l) = A+ cos(α)l

2. Linkage: Structural Properties

Parameterized by lenghtl of starting spirals!

L_j(l) length of the curve!F_j(l) (and φ_j(l))length of the free string!

Fj

L_j−1 Lj

F_j−1

F0

α α

α p

l

Lemma 60:

L_j−1+F_j = cosαL_j Lemma 61:

L⁰_j

L⁰_j−1

=

j−1

Elmar Langetepe Theoretical Aspects of Intruder Search

Helping Lemmata

Lemma 60: L_j−1+F_j = cosαL_j

Fire and fire fighter, reach endpoint atF_j(l) at the same time Unit-speed fire, geodesic distance of Lj−1(l) +F_j(l)

Fighter distance of Lj(l) at speed 1/cosα

Helping Lemmata

Lemma 61: ^L

0 j

L⁰_j−1

=

j−1

Lemma 62: String of lengthF is tangent to point t on smooth curve C. End of string moves distancein direction α. For the curve lengthC_t^t between t and the new tangent point, t, we have

→0lim C_t^t

= rsinα F

where r denotes radius of osculating circle at t.

α

F F

r

t t

C sin(α)

r

a a φ/2

φ

s

Elmar Langetepe Theoretical Aspects of Intruder Search

Helping Lemmata

r sin(φ/2) =s =acos(φ/2) gives a=r tan(φ/2)

2aapproximates c :=C_t^t: c

2a = rφ

2r tan(φ/2) ≈ cos²(φ/2) → 1

sin(α)

sin(φ) = F+a

sin(π/2) gives sin(φ)

= sin(α)

F+a → sin(α) F sin(φ/2)/ → sin(α)/(2F)

α

F F

r

t t sin(α)

r

a a φ/2

φ

s