Pursuit Curves Molly Severdia May 15, 2008 Molly Severdia Pursuit - PowerPoint PPT Presentation

Pursuit Curves Molly Severdia May 15, 2008 Molly Severdia Pursuit Curves

Assumptions y ( x 0 , V m t ) ◮ At t = 0, merchant at ( x 0 , 0), pirate at (0 , 0). V m t − y ◮ Merchant’s speed is V m . ◮ Pirate’s speed is V p . ◮ Merchant travels along ( x, y ) x 0 − x vertical line x = x 0 . ◮ At time t ≥ 0, pirate at y = y ( x ) ( x , y ). x x 0 Figure: Geometry of pirate pursuit Molly Severdia Pursuit Curves

y ( x 0 , V m t ) dy dx = V m t − y x 0 − x V m t − y ( x, y ) � x 0 − x � x � 2 � dy V p t = 1 + dz dz y = y ( x ) 0 x x 0 Figure: Geometry of pirate pursuit Molly Severdia Pursuit Curves

Differential Equation for Pirate Pursuit ( x − x 0 ) dp � dx = − n 1 + p 2 ( x ) n = V m , p ( x ) = dy V p dx Molly Severdia Pursuit Curves

Separable Equation 1 + p 2 = − n dx dp � x − x 0 � ln( p + 1 + p 2 ) + C = − n ln( x 0 − x ) �� � n � � − n dx = 1 dy 1 − x � 1 − x − 2 x 0 x 0 Molly Severdia Pursuit Curves

Separable Equation 1 + p 2 = − n dx dp � x − x 0 � ln( p + 1 + p 2 ) + C = − n ln( x 0 − x ) �� � n � � − n dy dx = 1 1 − x � 1 − x − 2 x 0 x 0 � (1 − x / x 0 ) n − (1 − x / x 0 ) − n y ( x ) = 1 � n 2( x − x 0 ) + 1 − n 2 x 0 1 + n 1 − n Molly Severdia Pursuit Curves

Results n=0.3 4 3.5 3 2.5 y(x) 2 1.5 1 0.5 0 0 5 10 15 x-axis Molly Severdia Pursuit Curves Figure: Results using ode45

Circular Pursuit ”A dog at the center of a circular pond makes straight for a duck which is swimming [counterclockwise] along the edge of the pond. If the rate of swimming of the dog is to the rate of swimming of the duck as n : 1, determine the equation of the curve of pursuit...” Molly Severdia Pursuit Curves

Generic Case y duck ρ ( t ) ρ ρ d ( t ) hound h ( t ) O x d ( t ) = h ( t ) + ρ ρ ρ ( t ) d ( t ) = x d ( t ) + iy d ( t ) h ( t ) = x h ( t ) + iy h ( t ) Molly Severdia Pursuit Curves

Duck ◮ Duck’s position vector given by d ( t ) = x d ( t ) + iy d ( t ) Molly Severdia Pursuit Curves

Duck ◮ Duck’s position vector given by d ( t ) = x d ( t ) + iy d ( t ) ◮ Duck’s velocity vector given by d d ( t ) = dx d dt + i dy d dt dt Molly Severdia Pursuit Curves

Duck ◮ Duck’s position vector given by d ( t ) = x d ( t ) + iy d ( t ) ◮ Duck’s velocity vector given by d d ( t ) = dx d dt + i dy d dt dt ◮ Duck’s speed is �� dx d � 2 � 2 � � � dy d d d ( t ) � � � = + � � dt dt dt � Molly Severdia Pursuit Curves

Hound ◮ Hound’s position vector given by h ( t ) = x h ( t ) + iy h ( t ) Molly Severdia Pursuit Curves

Hound ◮ Hound’s position vector given by h ( t ) = x h ( t ) + iy h ( t ) ◮ Hound’s velocity vector is given by � � d h ( t ) d h ( t ) � · ρ ρ ρ ( t ) � � = (1) � � dt dt | ρ ρ ρ ( t ) | � Molly Severdia Pursuit Curves

Hound ◮ Hound’s position vector given by h ( t ) = x h ( t ) + iy h ( t ) ◮ Hound’s velocity vector is given by � � d h ( t ) d h ( t ) � · ρ ρ ρ ( t ) � � = (1) � � dt dt | ρ ρ ρ ( t ) | � ◮ Hound’s speed is n times that of the duck, �� dx d � 2 � 2 � � � dy d d h ( t ) � � � = n + � � dt dt dt � Molly Severdia Pursuit Curves

◮ Equation (1) becomes �� dx d � 2 � 2 d h ( t ) � dy d · d ( t ) − h ( t ) = n + | d ( t ) − h ( t ) | dt dt dt Molly Severdia Pursuit Curves

◮ Equation (1) becomes �� dx d � 2 � 2 d h ( t ) � dy d · d ( t ) − h ( t ) = n + | d ( t ) − h ( t ) | dt dt dt ◮ In Cartesian Coordinates, �� dx d � 2 � 2 � dy d ( x d − x h ) + i ( y d − y h ) dx h dt + i dy h dt = n + · ( x d − x h ) 2 + ( y d − y h ) 2 � dt dt Molly Severdia Pursuit Curves

◮ Equation (1) becomes �� dx d � 2 � 2 d h ( t ) � dy d · d ( t ) − h ( t ) = n + | d ( t ) − h ( t ) | dt dt dt ◮ In Cartesian Coordinates, �� dx d � 2 � 2 � dy d ( x d − x h ) + i ( y d − y h ) dx h dt + i dy h dt = n + · ( x d − x h ) 2 + ( y d − y h ) 2 � dt dt ◮ Equating real and imaginary parts leads to... Molly Severdia Pursuit Curves

Equations for General Pursuit �� dx d � 2 � 2 dx h � dy d x d − x h dt = n + · ( x d − x h ) 2 + ( y d − y h ) 2 dt dt � �� dx d � 2 � 2 � dy d y d − y h dy h dt = n + · ( x d − x h ) 2 + ( y d − y h ) 2 dt dt � Molly Severdia Pursuit Curves

◮ If the duck swims counterclockwise around a unit circle, x d ( t ) = cos( t ) , y d ( t ) = sin( t ) . Molly Severdia Pursuit Curves

◮ If the duck swims counterclockwise around a unit circle, x d ( t ) = cos( t ) , y d ( t ) = sin( t ) . ◮ Also, �� dx d � 2 � 2 � dy d � sin 2 ( t ) + cos 2 ( t ) = n + = n n dt dt Molly Severdia Pursuit Curves

Circle Pursuit dx h cos( t ) − x h dt = n (cos( t ) − x h ) 2 + (sin( t ) − y h ) 2 � dy h sin( t ) − y h dt = n (cos( t ) − x h ) 2 + (sin( t ) − y h ) 2 � Molly Severdia Pursuit Curves

n = 0 . 3 n = 0 . 3 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 Molly Severdia Pursuit Curves

n = 0 . 5 n = 0 . 5 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 Molly Severdia Pursuit Curves

n = 0 . 2 n = 0 . 7 1 1 0.5 0.5 0 0 −0.5 −0.5 −1 −1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 Molly Severdia Pursuit Curves

y duck ρ a ( x, y ) ω θ x x 0 ( a, 0) Molly Severdia Pursuit Curves

◮ Equation of tangent line: y cos( ω ) − x sin( ω ) = − a sin( ω − θ ) ◮ Equation of normal line: x cos( ω ) + y sin( ω ) = a cos( ω − θ ) − ρ Molly Severdia Pursuit Curves

Differentiate tangent line d θ cos( ω )+ d ω � d ω � dx d θ sin( ω ) − dy d θ [ x cos( ω )+ y sin( ω )] = a cos( ω − θ ) d θ − 1 Molly Severdia Pursuit Curves

Differentiate tangent line d θ cos( ω )+ d ω � d ω � dx d θ sin( ω ) − dy d θ [ x cos( ω )+ y sin( ω )] = a cos( ω − θ ) d θ − 1 ρ d ω d θ = a cos( ω − θ ) Molly Severdia Pursuit Curves

Differentiate normal line dx d θ cos( ω ) − x sin( ω ) d ω d θ + dy d θ sin( ω ) + y cos( ω ) d ω d θ � d ω � − d ρ = − a sin( ω − θ ) d θ − 1 d θ Molly Severdia Pursuit Curves

Differentiate normal line dx d θ cos( ω ) − x sin( ω ) d ω d θ + dy d θ sin( ω ) + y cos( ω ) d ω d θ � d ω � − d ρ = − a sin( ω − θ ) d θ − 1 d θ d ρ d θ = a [sin( ω − θ ) − n ] Molly Severdia Pursuit Curves

ρ d ω d ρ d θ = a cos( ω − θ ) d θ = a [sin( ω − θ ) − n ] φ = ω − θ d ω d θ = d φ d θ + 1 ρ d 2 ρ d θ 2 + a ρ cos( φ ) = a 2 cos 2 ( φ ) d ρ d θ = a sin( φ ) − an Molly Severdia Pursuit Curves

ρ d 2 ρ d θ 2 + a ρ cos( φ ) = a 2 cos 2 ( φ ) d ρ d θ = a sin( φ ) − an y ◮ lim θ →∞ ρ = c d θ = d 2 ρ ◮ d ρ d θ 2 = 0 R x ρ ◮ As θ → ∞ , ρ = a cos( φ ) a ◮ As θ → ∞ , sin( φ ) = n duck’s position hound’s limit cycle Molly Severdia Pursuit Curves

As θ → ∞ ... � ρ � = a 2 [1 − sin 2 ( φ )] = a 2 (1 − n 2 ) a ρ a � θ →∞ ρ = a lim 1 − n 2 Molly Severdia Pursuit Curves

The Limit Cycle Letting R be the radius of the limit cycle, R 2 + ρ 2 = a 2 R = na y R x ρ a duck’s position hound’s limit cycle Molly Severdia Pursuit Curves