;;; BILLIARD.SCM: This file contains code for a very simple billiard ball
;;; simulator. The simulation takes place in two dimensions.
;;; The balls are really disks in that their height is not taken
;;; into account. All interactions are assumed to be
;;; frictionless so spin in irrelevant and not accounted for.
;;; (See section on limitations.)
;;;
;;; NOTES: A simulation is initiated by creating a number of balls and bumpers
;;; and and specifying a duration for the simulation. For each ball,
;;; its mass, radius, initial position, and initial velocity must be
;;; specified. For each bumper, the location of its two ends must be
;;; specified. (Bumpers are assumed to have zero width.)
;;;
;;; A sample run might be started as follows:
;;; (simulate
;;; (list (make-ball 2 1 9 5 -1 -1)
;;; (make-ball 4 2 2 5 1 -1))
;;; (list (make-bumper 0 0 0 10)
;;; (make-bumper 0 0 10 0)
;;; (make-bumper 0 10 10 10)
;;; (make-bumper 10 0 10 10))
;;; 30)
;;;
;;; It would create one billiard ball of mass 2 and radius 1 at position
;;; (9, 5) with initial velocity (-1, -1) and a second ball of mass 4
;;; and radius 2 at position (2, 5) with initial velocity (1, -1). The
;;; table would be a 10X10 square. (See diagram below)
;;;
;;; +---------------------------+
;;; | |
;;; | |
;;; | XXXX |
;;; | XXXXXXXX XX |
;;; |XXXXXX4XXXXX XXX2XX|
;;; | XXXXXXXX /XX |
;;; | XXXX \ |
;;; | |
;;; | |
;;; +---------------------------+
;;;
;;; LIMITATIONS: This simulator does not handle 3 body problems correctly. If
;;; 3 objects interact at one time, only the interactions of 2 of
;;; the bodies will be accounted for. This can lead to strange
;;; effects like balls tunneling through walls and other balls.
;;; It is also possible to get balls bouncing inside of each
;;; other in this way.
;;;
;;MAKE-QUEUE-RECORD returns a queue record with the given next, previous, and
;;value values
;;NEXT = The next record pointer
;;PREV = The previous record pointer
;;REST = A list of values for any optional fields (this can be used for
;; creating structure inheritance)
(define-macro (make-queue-record next prev . rest)
`(vector ,next ,prev ,@rest))
;;QUEUE-RECORD-NEXT returns the next field of the given queue record
;;QUEUE-RECORD = The queue record whose next field is to be returned
(define-macro (queue-record-next queue-record)
`(vector-ref ,queue-record 0))
;;SET-QUEUE-RECORD-NEXT! sets the next field of the given queue record
;;QUEUE-RECORD = The queue record whose next field is to be set
;;VALUE = The value to which the next field is to be set
(define-macro (set-queue-record-next! queue-record value)
`(vector-set! ,queue-record 0 ,value))
;;QUEUE-RECORD-PREV returns the prev field of the given queue record
;;QUEUE-RECORD = The queue record whose prev field is to be returned
(define-macro (queue-record-prev queue-record)
`(vector-ref ,queue-record 1))
;;SET-QUEUE-RECORD-PREV! sets the prev field of the given queue record
;;QUEUE-RECORD = The queue record whose prev field is to be set
;;VALUE = The value to which the prev field is to be set
(define-macro (set-queue-record-prev! queue-record value)
`(vector-set! ,queue-record 1 ,value))
;;QUEUE-RECORD-LEN returns the length of a queue record which has no optional
;;fields
(define-macro (queue-record-len) 2)
;;QUEUE-HEAD returns a dummy record at the end of the queue with the record
;;with the smallest key.
;;QUEUE = the queue whose head record is to be returned
(define-macro (queue-head queue)
`(vector-ref ,queue 0))
;;QUEUE-TAIL returns a dummy record at the end of the queue with the record
;;with the largest key.
;;QUEUE = the queue whose tail record is to be returned
(define-macro (queue-tail queue)
`(vector-ref ,queue 1))
;;QUEUE-<? returns the less-than comparitor to be used in sorting
;;records into the queue
;;QUEUE = The queue whose comparitor is to be returned
(define-macro (queue-<? queue)
`(vector-ref ,queue 2))
;;MAKE-SORTED-QUEUE returns a queue object. A queue header is a vector which
;;contains a head pointer, a tail pointer, and a less-than comparitor.
;;QUEUE-<? = A predicate for sorting queue items
(define (make-sorted-queue queue-<?)
(let ((queue
(vector
(make-queue-record ;The queue head record has no initial
'() ;next, previous, or value values
'())
(make-queue-record ;The queue tail record has no intial
'() ;next, previous, or value values
'())
queue-<?)))
(set-queue-record-next!
(queue-head queue)
(queue-tail queue))
(set-queue-record-prev!
(queue-tail queue)
(queue-head queue))
queue))
;;MAKE-EVENT-QUEUE-RECORD returns an event queue record with the given next,
;;previous, object, and collision-time values
;;NEXT = The next record pointer
;;PREV = The previous record pointer
;;OBJECT = The simulation object associated with this record
;;COLLISION-TIME = The collision time for this object
(define-macro (make-event-queue-record next prev object collision-time)
`(make-queue-record ,next ,prev ,object ,collision-time))
;;EVENT-QUEUE-RECORD-OBJECT returns the object associated with the given record
;;QUEUE-RECORD = The queue record whose object field is to be returned
(define-macro (event-queue-record-object queue-record)
`(vector-ref ,queue-record ,(queue-record-len)))
;;EVENT-QUEUE-COLLISION-TIME returns the collision time associated with the
;;given queue record
;;QUEUE-RECORD = The queue record whose collision time field is to be returned
(define-macro (event-queue-record-collision-time queue-record)
`(vector-ref ,queue-record ,(1+ (queue-record-len))))
;;SET-EVENT-QUEUE-COLLISION-TIME! sets the collision time associated with the
;;given queue record
;;QUEUE-RECORD = The queue record whose collision time field is to be returned
;;VALUE = The value to which it is to be set
(define-macro (set-event-queue-record-collision-time! queue-record value)
`(vector-set! ,queue-record ,(1+ (queue-record-len)) ,value))
;;QUEUE-INSERT inserts the given record in the given queue based on its value
;;QUEUE = The queue into which the record is to be inserted
;;QUEUE-RECORD = The record to be inserted in the queue
(define (queue-insert queue queue-record)
(define (actual-insert insert-record next-record)
(if (or ;If the insert position has been found
(eq? next-record ;or the end on the queue has been
(queue-tail queue)) ;reached
((queue-<? queue)
insert-record
next-record))
(sequence ;Link the insert record into the queue
(set-queue-record-next! ;just prior to next-record
(queue-record-prev
next-record)
insert-record)
(set-queue-record-prev!
insert-record
(queue-record-prev
next-record))
(set-queue-record-next!
insert-record
next-record)
(set-queue-record-prev!
next-record
insert-record))
(actual-insert ;Else, continue searching for the
insert-record ;insert position
(queue-record-next
next-record))))
(actual-insert ;Search for the correct position to
queue-record ;perform the insert starting at the
(queue-record-next ;queue head and perform the insert
(queue-head queue)))) ;once this position has been found
;;QUEUE-REMOVE removes the given queue record from its queue
;;QUEUE-RECORD = The record to be removed from the queue
(define (queue-remove queue-record)
(set-queue-record-next!
(queue-record-prev
queue-record)
(queue-record-next
queue-record))
(set-queue-record-prev!
(queue-record-next
queue-record)
(queue-record-prev
queue-record)))
;;QUEUE-SMALLEST returns the queue record with the smallest key on the given
;;queue
;;QUEUE = The queue from which the smallest record is to be extracted
(define (queue-smallest queue)
(queue-record-next
(queue-head queue)))
;;CLEAR-QUEUE! clears the given queue by destructively removing all the records
;;QUEUE = The queue to be cleared
(define (clear-queue queue)
(set-queue-record-next!
(queue-head queue)
(queue-tail queue))
(set-queue-record-prev!
(queue-tail queue)
(queue-head queue)))
;;EMPTY-QUEUE? returns true if the given queue is empty
;;QUEUE = The queue to be tested for emptiness
(define (empty-queue? queue)
(eq? (queue-record-next
(queue-head queue))
(queue-tail queue)))
;;MAKE-SIMULATION-OBJECT returns a simulation object containing the given
;;fields
;;COLLISION-PROCEDURE = A function for processing information about a potential
;; collision between this object and some ball
;;REST = A list of values for any optional fields (this can be used for
;; creating structure inheritance)
(define-macro (make-simulation-object collision-procedure . rest)
`(vector ,collision-procedure ,@rest))
;;SIMULATION-OBJECT-COLLLISION-PROCEDURE returns the collision procedure for
;;the given simulation object
;;OBJECT = The object whose collision procedure is to be returned
(define-macro (simulation-object-collision-procedure object)
`(vector-ref ,object 0))
;;SIMULATION-OBJECT-LEN returns the length of a simulation object which has no
;;optional fields
(define-macro (simulation-object-len) 1)
;;ACTUAL-MAKE-BALL returns a ball object
;;BALL-NUMBER = An index into the ball vector for this ball
;;MASS = The ball's mass
;;RADIUS = The ball's radius
;;PX = The x-coordinate of the ball's initial position
;;PY = The y-coordinate of the ball's initial position
;;VX = The x-coordinate of the ball's initial velocity
;;VY = The y-coordinate of the ball's initial velocity
(define-macro (actual-make-ball ball-number mass radius px py vx vy)
`(make-simulation-object
ball-collision-procedure ;The collision procedure for a ball
,ball-number
,mass
,radius
(make-sorted-queue ;The event queue
collision-time-<?)
0 ;Time of last collision
,px ;Position of last collision
,py ; "
,vx ;Velocity following last colliosion
,vy ; "
'() ;No vector of queue records for ball's
;with smaller numbers
'() ;No vector of queue records for bumpers
'() ;No list of balls with larger numbers
'())) ;No global event queue record, yet
(define (make-ball mass radius px py vx vy)
(actual-make-ball '() mass radius px py vx vy))
;;BALL-NUMBER returns the index of the given ball
;;BALL = The ball whose index is to be returned
(define-macro (ball-number ball)
`(vector-ref ,ball ,(simulation-object-len)))
;;SET-BALL-NUMBER! set the index of the given ball to the given value
;;BALL = The ball whose index is to be set
;;VALUE = The value to which it is to be set
(define-macro (set-ball-number! ball value)
`(vector-set! ,ball ,(simulation-object-len) ,value))
;;BALL-MASS returns the mass of the given ball
;;BALL = The ball whose mass is to be returned
(define-macro (ball-mass ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 1)))
;;BALL-RADIUS returns the radius of the given ball
;;BALL = The ball whose radius is to be returned
(define-macro (ball-radius ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 2)))
;;BALL-EVENT-QUEUE returns the sort queue of collision events for the given
;;ball
;;BALL = The ball whose event is to be returned
(define-macro (ball-event-queue ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 3)))
;;BALL-COLLISION-TIME returns the time of the last collision for the given ball
;;BALL = The ball whose collision time is to be returned
(define-macro (ball-collision-time ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 4)))
;;SET-BALL-COLLISION-TIME! sets the time of the last collision for the given
;;ball
;;BALL = The ball whose collision time is to be set
;;VALUE = The value to which the ball's collision time is to be set
(define-macro (set-ball-collision-time! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 4) ,value))
;;BALL-COLLISION-X-POSITION returns the x-coordinate of the position of the
;;last collision for the given ball
;;BALL = The ball whose collision position is to be returned
(define-macro (ball-collision-x-position ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 5)))
;;SET-BALL-COLLISION-X-POSITION! sets the x-coordinate of the position of the
;;last collision for the given ball
;;BALL = The ball whose collision position is to be set
;;VALUE = The value to which the ball's collision position is to be set
(define-macro (set-ball-collision-x-position! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 5) ,value))
;;BALL-COLLISION-Y-POSITION returns the y-coordinate of the position of the
;;last collision for the given ball
;;BALL = The ball whose collision position is to be returned
(define-macro (ball-collision-y-position ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 6)))
;;SET-BALL-COLLISION-Y-POSITION! sets the y-coordinate of the position of the
;;last collision for the given ball
;;BALL = The ball whose collision position is to be set
;;VALUE = The value to which the ball's collision position is to be set
(define-macro (set-ball-collision-y-position! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 6) ,value))
;;BALL-X-VELOCITY returns the x-coordinate of the velocity of the given ball
;;following its last collision
;;BALL = The ball whose velocity is to be returned
(define-macro (ball-x-velocity ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 7)))
;;SET-BALL-X-VELOCITY! sets the x-coordinate of the velocity of the given ball
;;BALL = The ball whose velocity is to be set
;;VALUE = The value to which the ball's velocity is to be set
(define-macro (set-ball-x-velocity! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 7) ,value))
;;BALL-Y-VELOCITY returns the y-coordinate of the velocity of the given ball
;;following its last collision
;;BALL = The ball whose velocity is to be returned
(define-macro (ball-y-velocity ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 8)))
;;SET-BALL-Y-VELOCITY! sets the y-coordinate of the velocity of the given ball
;;BALL = The ball whose velocity is to be set
;;VALUE = The value to which the ball's velocity is to be set
(define-macro (set-ball-y-velocity! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 8) ,value))
;;BALL-BALL-VECTOR returns the vector of queue records for balls with smaller
;;ball numbers
;;BALL = The ball whose ball vector is to be returned
(define-macro (ball-ball-vector ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 9)))
;;SET-BALL-BALL-VECTOR! sets the vector of queue records for balls with smaller
;;ball numbers
;;BALL = The ball whose ball vector is to be set
;;VALUE = The vector to which the field is to be set
(define-macro (set-ball-ball-vector! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 9) ,value))
;;BALL-BUMPER-VECTOR returns the vector of queue records for bumpers
;;BALL = The ball whose bumper vector is to be returned
(define-macro (ball-bumper-vector ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 10)))
;;SET-BALL-BUMPER-VECTOR! sets the vector of queue records for bumpers
;;BALL = The ball whose bumper vector is to be set
;;VALUE = The vector to which the field is to be set
(define-macro (set-ball-bumper-vector! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 10) ,value))
;;BALL-BALL-LIST returns a list of balls with larger ball numbers than the
;;given ball
;;BALL = The ball whose ball list is to be returned
(define-macro (ball-ball-list ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 11)))
;;SET-BALL-BALL-LIST! sets the list of balls with larger ball numbers than the
;;given ball
;;BALL = The ball whose ball list is to be set
;;VALUE = The value to which the ball list is to be set
(define-macro (set-ball-ball-list! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 11) ,value))
;;BALL-GLOBAL-EVENT-QUEUE-RECORD returns the global event queue record for the
;;given ball
;;BALL = The ball whose global event queue record is to be returned
(define-macro (ball-global-event-queue-record ball)
`(vector-ref ,ball ,(+ (simulation-object-len) 12)))
;;SET-BALL-GLOBAL-EVENT-QUEUE-RECORD! set the global event queue record for the
;;given ball to the given value
;;BALL = The ball whose global event queue record is to be set
;;VALUE = The value to which the global event queue record field is to be set
(define-macro (set-ball-global-event-queue-record! ball value)
`(vector-set! ,ball ,(+ (simulation-object-len) 12) ,value))
;;ACTUAL-MAKE-BUMPER returns a bumper object
;;BUMPER-NUMBER = An index into the bumper vector for this bumper
;;X1 = The x-coordiante of one end of the bumper
;;Y1 = The y-coordiante of one end of the bumper
;;X2 = The x-coordiante of the other end of the bumper
;;Y2 = The y-coordiante of the other end of the bumper
(define-macro (actual-make-bumper bumper-number x1 y1 x2 y2)
`(make-simulation-object
bumper-collision-procedure ;The collision procedure for a bumper
,bumper-number
,x1 ;The bumper endpoints
,y1
,x2
,y2))
(define (make-bumper x1 y1 x2 y2)
(actual-make-bumper '() x1 y1 x2 y2))
;;BUMPER-NUMBER returns the index of the given bumper
;;BUMPER = The bumper whose index is to be returned
(define-macro (bumper-number bumper)
`(vector-ref ,bumper ,(simulation-object-len)))
;;SET-BUMPER-NUMBER! set the index of the given bumper to the given value
;;BUMPER = The bumper whose index is to be set
;;VALUE = The value to which it is to be set
(define-macro (set-bumper-number! bumper value)
`(vector-set! ,bumper ,(simulation-object-len) ,value))
;;BUMPER-X1 returns the x-coordinate of one end of the given bumber
;;BUMPER = the bumper whose x-coordinate is to be returned
(define-macro (bumper-x1 bumper)
`(vector-ref ,bumper ,(1+ (simulation-object-len))))
;;SET-BUMPER-X1! sets the x-coordinate of one end of the given bumber
;;BUMPER = the bumper whose x-coordinate is to be set
;;VALUE = The value to which the bumpers x-coordinate is to be set
(define-macro (set-bumper-x1! bumper value)
`(vector-set! ,bumper ,(1+ (simulation-object-len)) ,value))
;;BUMPER-Y1 returns the y-coordinate of one end of the given bumber
;;BUMPER = the bumper whose y-coordinate is to be returned
(define-macro (bumper-y1 bumper)
`(vector-ref ,bumper ,(+ (simulation-object-len) 2)))
;;SET-BUMPER-Y1! sets the y-coordinate of one end of the given bumber
;;BUMPER = the bumper whose y-coordinate is to be set
;;VALUE = The value to which the bumpers y-coordinate is to be set
(define-macro (set-bumper-y1! bumper value)
`(vector-set! ,bumper ,(+ (simulation-object-len) 2) ,value))
;;BUMPER-X2 returns the x-coordinate of the other end of the given bumber
;;BUMPER = the bumper whose x-coordinate is to be returned
(define-macro (bumper-x2 bumper)
`(vector-ref ,bumper ,(+ (simulation-object-len) 3)))
;;SET-BUMPER-X2! sets the x-coordinate of the other end of the given bumber
;;BUMPER = the bumper whose x-coordinate is to be set
;;VALUE = The value to which the bumpers x-coordinate is to be set
(define-macro (set-bumper-x2! bumper value)
`(vector-set! ,bumper ,(+ (simulation-object-len) 3) ,value))
;;BUMPER-Y2 returns the y-coordinate of the other end of the given bumber
;;BUMPER = the bumper whose y-coordinate is to be returned
(define-macro (bumper-y2 bumper)
`(vector-ref ,bumper ,(+ (simulation-object-len) 4)))
;;SET-BUMPER-Y2! sets the y-coordinate of the other end of the given bumber
;;BUMPER = the bumper whose y-coordinate is to be set
;;VALUE = The value to which the bumpers y-coordinate is to be set
(define-macro (set-bumper-y2! bumper value)
`(vector-set! ,bumper ,(+ (simulation-object-len) 4) ,value))
;;COLLISION-TIME-<? is a predicate which returns true if the first event queueu
;;record represents a collision that will take place at an earlier time than
;;the one for the second event queue record
;;EVENT-QUEUE-RECORD1 = The first event queue record
;;EVENT-QUEUE-RECORD2 = The second event queue record
(define (collision-time-<? event-queue-record1 event-queue-record2)
(time-<?
(event-queue-record-collision-time
event-queue-record1)
(event-queue-record-collision-time
event-queue-record2)))
;;TIME-<? is a predicate which returns true if the first time is smaller than
;;the second. '() represents a time infinitly large.
(define (time-<? time1 time2)
(if (null? time1)
#f
(if (null? time2)
#t
(< time1 time2))))
;;SQUARE returns the square of its argument
(define (square x)
(* x x))
;;BALL-BALL-COLLISION-TIME returns the time at which the two given balls would
;;collide if neither interacted with any other objects, '() if never. This
;;calculation is performed by setting the distance between the balls to the sum
;;of their radi and solving for the contact time.
;;BALL1 = The first ball
;;BALL2 = The second ball
(define (ball-ball-collision-time ball1 ball2)
(let ((delta-x-velocity ;Cache the difference in the ball's
( - (ball-x-velocity ball2) ;velocities,
(ball-x-velocity ball1)))
(delta-y-velocity
( - (ball-y-velocity ball2)
(ball-y-velocity ball1)))
(radius-sum ;the sum of their radi,
(+ (ball-radius ball1)
(ball-radius ball2)))
(alpha-x ;and common subexpressions in the time
(- ;equation
(- (ball-collision-x-position
ball2)
(ball-collision-x-position
ball1))
(-
(* (ball-x-velocity ball2)
(ball-collision-time
ball2))
(* (ball-x-velocity ball1)
(ball-collision-time
ball1)))))
(alpha-y
(-
(- (ball-collision-y-position
ball2)
(ball-collision-y-position
ball1))
(-
(* (ball-y-velocity ball2)
(ball-collision-time
ball2))
(* (ball-y-velocity ball1)
(ball-collision-time
ball1))))))
(let* ((delta-velocity-magnitude-squared
(+ (square
delta-x-velocity)
(square
delta-y-velocity)))
(discriminant
(- (* (square radius-sum)
delta-velocity-magnitude-squared)
(square
(- (* delta-y-velocity
alpha-x)
(* delta-x-velocity
alpha-y))))))
(if (or (negative? discriminant) ;If the balls don't colloide:
(zero?
delta-velocity-magnitude-squared))
'() ;Return infinity
(let ((time ;Else, calculate the collision time
(/
(- 0
(+ (sqrt discriminant)
(+
(* delta-x-velocity
alpha-x)
(* delta-y-velocity
alpha-y))))
(+ (square
delta-x-velocity)
(square
delta-y-velocity)))))
(if (and ;If the balls collide in the future:
(time-<?
(ball-collision-time
ball1)
time)
(time-<?
(ball-collision-time
ball2)
time))
time ;Return the collision time
'())))))) ;Else, return that they never collide
;;BALL-BUMPER-COLLISION-TIME returns the time at which the given ball would
;;collide with the given bumper if the ball didn't interacted with any other
;;objects, '() if never. This is done by first calculating the time at which
;;the ball would collide with a bumper of infinite length and then checking if
;;the collision position represents a portion of the actual bumper.
;;BALL = The ball
;;BUMPER = The bumper
(define (ball-bumper-collision-time ball bumper)
(let ((delta-x-bumper ;Collision time with the bumper of
(- (bumper-x2 bumper) ;infinite extent is calculated by
(bumper-x1 bumper))) ;setting the distance between the ball
(delta-y-bumper ;and the bumper to be the radius of the
(- (bumper-y2 bumper) ;ball and solving for the time. The
(bumper-y1 bumper)))) ;distance is calculated by |aXb|/|a|,
(let ((bumper-length-squared ;where 'a' is the vector from one end
(+ (square delta-x-bumper) ;of the bumper to the other and 'b' is
(square delta-y-bumper))) ;the vector from the first end of the
(denominator ;bumper to the center of the ball
(- (* (ball-y-velocity ball)
delta-x-bumper)
(* (ball-x-velocity ball)
delta-y-bumper))))
(if (zero? denominator) ;If the ball's motion is parallel to
;the bumper:
'() ;Return infinity
(let ((delta-t ;Calculate the collision time
(-
(/
(+
(*
(- (ball-collision-x-position
ball)
(bumper-x1 bumper))
delta-y-bumper)
(*
(- (ball-collision-y-position
ball)
(bumper-y1 bumper))
delta-x-bumper))
denominator)
(/
(* (ball-radius
ball)
(sqrt
bumper-length-squared))
(abs denominator)))))
(if (not (positive? ;If the ball is moving away from the
delta-t)) ;bumper:
'() ;Return infinity
(let ((ball-x-contact ;Whether the ball contacts the actual
(+ (ball-collision-x-position ;bumper of limited extent
ball) ;will be determined by comparing |b.a|
(* (ball-x-velocity ;with |a|^2
ball)
delta-t)))
(ball-y-contact
(+ (ball-collision-y-position
ball)
(* (ball-y-velocity
ball)
delta-t))))
(let ((delta-x-ball
(- ball-x-contact
(bumper-x1
bumper)))
(delta-y-ball
(- ball-y-contact
(bumper-y1
bumper))))
(let ((dot-product
(+
(* delta-x-ball
delta-x-bumper)
(* delta-y-ball
delta-y-bumper))))
(if (or ;If the ball misses the bumper on
(negative? ;either end:
dot-product)
(> dot-product
bumper-length-squared))
'() ;Return infinity
(+ delta-t ;Else, return the contact time
(ball-collision-time
ball))))))))))))
;;BALL-COLLISION-PROCEDURE calculates the new velocities of the given balls
;;based on their collision at the given time. Also, tells all other balls
;;about the new trajectories of these balls so they can update their event
;;queues
;;BALL1 = The first ball
;;BALL2 = The second ball
;;COLLISION-TIME = The collision time
;;GLOBAL-EVENT-QUEUE = The global queue of earliest events for each ball
(define (ball-collision-procedure ball1 ball2 collision-time
global-event-queue)
(queue-remove ;Remove the earliest event associated
(ball-global-event-queue-record ;with each ball from the global event
ball1)) ;queue
(queue-remove
(ball-global-event-queue-record
ball2))
(let ((ball1-collision-x-position ;Calculate the positions of both balls
(+ (ball-collision-x-position ;when they collide
ball1)
(* (ball-x-velocity
ball1)
(- collision-time
(ball-collision-time
ball1)))))
(ball1-collision-y-position
(+ (ball-collision-y-position
ball1)
(* (ball-y-velocity
ball1)
(- collision-time
(ball-collision-time
ball1)))))
(ball2-collision-x-position
(+ (ball-collision-x-position
ball2)
(* (ball-x-velocity
ball2)
(- collision-time
(ball-collision-time
ball2)))))
(ball2-collision-y-position
(+ (ball-collision-y-position
ball2)
(* (ball-y-velocity
ball2)
(- collision-time
(ball-collision-time
ball2))))))
(let ((delta-x ;Calculate the displacements of the
(- ball2-collision-x-position ;centers of the two balls
ball1-collision-x-position))
(delta-y
(- ball2-collision-y-position
ball1-collision-y-position)))
(let* ((denominator ;Calculate the angle of the line
(sqrt (+ (square ;joining the centers at the collision
delta-x) ;time with the x-axis (this line is
(square ;the normal to the balls at the
delta-y)))) ;collision point)
(cos-theta
(/ delta-x denominator))
(sin-theta
(/ delta-y denominator)))
(let ((ball1-old-normal-velocity ;Convert the velocities of the balls
(+ (* (ball-x-velocity ;into the coordinate system defined by
ball1) ;the normal and tangential lines at
cos-theta) ;the collision point
(* (ball-y-velocity
ball1)
sin-theta)))
(ball1-tang-velocity
(- (* (ball-y-velocity
ball1)
cos-theta)
(* (ball-x-velocity
ball1)
sin-theta)))
(ball2-old-normal-velocity
(+ (* (ball-x-velocity
ball2)
cos-theta)
(* (ball-y-velocity
ball2)
sin-theta)))
(ball2-tang-velocity
(- (* (ball-y-velocity
ball2)
cos-theta)
(* (ball-x-velocity
ball2)
sin-theta)))
(mass1 (ball-mass
ball1))
(mass2 (ball-mass
ball2)))
(let ((ball1-new-normal-velocity ;Calculate the new velocities
(/ ;following the collision (the
(+ ;tangential velocities are unchanged
(* ;because the balls are assumed to be
(* 2 ;frictionless)
mass2)
ball2-old-normal-velocity)
(*
(- mass1 mass2)
ball1-old-normal-velocity))
(+ mass1 mass2)))
(ball2-new-normal-velocity
(/
(+
(*
(* 2
mass1)
ball1-old-normal-velocity)
(*
(- mass2 mass1)
ball2-old-normal-velocity))
(+ mass1 mass2))))
(set-ball-x-velocity! ;Store data about the collision in the
ball1 ;structure for each ball after
(- (* ball1-new-normal-velocity ;converting the information back
cos-theta) ;to the x,y frame
(* ball1-tang-velocity
sin-theta)))
(set-ball-y-velocity!
ball1
(+ (* ball1-new-normal-velocity
sin-theta)
(* ball1-tang-velocity
cos-theta)))
(set-ball-x-velocity!
ball2
(- (* ball2-new-normal-velocity
cos-theta)
(* ball2-tang-velocity
sin-theta)))
(set-ball-y-velocity!
ball2
(+ (* ball2-new-normal-velocity
sin-theta)
(* ball2-tang-velocity
cos-theta)))
(set-ball-collision-time!
ball1
collision-time)
(set-ball-collision-time!
ball2
collision-time)
(set-ball-collision-x-position!
ball1
ball1-collision-x-position)
(set-ball-collision-y-position!
ball1
ball1-collision-y-position)
(set-ball-collision-x-position!
ball2
ball2-collision-x-position)
(set-ball-collision-y-position!
ball2
ball2-collision-y-position))))))
(newline)
(display "Ball ")
(display (ball-number ball1))
(display " collides with ball ")
(display (ball-number ball2))
(display " at time ")
(display (ball-collision-time ball1))
(newline)
(display " Ball ")
(display (ball-number ball1))
(display " has a new velocity of ")
(display (ball-x-velocity ball1))
(display ",")
(display (ball-y-velocity ball1))
(display " starting at ")
(display (ball-collision-x-position ball1))
(display ",")
(display (ball-collision-y-position ball1))
(newline)
(display " Ball ")
(display (ball-number ball2))
(display " has a new velocity of ")
(display (ball-x-velocity ball2))
(display ",")
(display (ball-y-velocity ball2))
(display " starting at ")
(display (ball-collision-x-position ball2))
(display ",")
(display (ball-collision-y-position ball2))
(recalculate-collisions ball1 global-event-queue)
(recalculate-collisions ball2 global-event-queue))
;;BUMPER-COLLISION-PROCEDURE calculates the new velocity of the given ball
;;following its collision with the given bumper at the given time. Also, tells
;;other balls about the new trajectory of the given ball so they can update
;;their event queues.
;;BALL = The ball
;;BUMPER = The bumper
;;COLLISION-TIME = The collision time
;;GLOBAL-EVENT-QUEUE = The global queue of earliest events for each ball
(define (bumper-collision-procedure ball bumper collision-time
global-event-queue)
(queue-remove ;Remove the earliest event associated
(ball-global-event-queue-record ;with the ball from the global event
ball)) ;queue
(let ((delta-x-bumper ;Compute the bumper's delta-x
(- (bumper-x2 bumper)
(bumper-x1 bumper)))
(delta-y-bumper ;delta-y
(- (bumper-y2 bumper)
(bumper-y1 bumper))))
(let ((bumper-length ;length
(sqrt
(+ (square
delta-x-bumper)
(square
delta-y-bumper)))))
(let ((cos-theta ;and cosine and sine of its angle with
(/ delta-x-bumper ;respect to the positive x-axis
bumper-length))
(sin-theta
(/ delta-y-bumper
bumper-length))
(x-velocity ;Cache the ball's velocity in the x,y
(ball-x-velocity ball)) ;frame
(y-velocity
(ball-y-velocity ball)))
(let ((tang-velocity ;Calculate the ball's velocity in the
(+ (* x-velocity ;bumper frame
cos-theta)
(* y-velocity
sin-theta)))
(normal-velocity
(- (* y-velocity
cos-theta)
(* x-velocity
sin-theta))))
(set-ball-collision-x-position! ;Store the collision position
ball
(+ (ball-collision-x-position
ball)
(* (- collision-time
(ball-collision-time
ball))
(ball-x-velocity
ball))))
(set-ball-collision-y-position!
ball
(+ (ball-collision-y-position
ball)
(* (- collision-time
(ball-collision-time
ball))
(ball-y-velocity
ball))))
(set-ball-x-velocity! ;Calculate the new velocity in the
ball ;x,y frame based on the fact that
(+ (* tang-velocity ;tangential velocity is unchanged and
cos-theta) ;the normal velocity is inverted when
(* normal-velocity ;the ball collides with the bumper
sin-theta)))
(set-ball-y-velocity!
ball
(- (* tang-velocity
sin-theta)
(* normal-velocity
cos-theta)))
(set-ball-collision-time!
ball
collision-time)))))
(newline)
(display "Ball ")
(display (ball-number ball))
(display " collides with bumper ")
(display (bumper-number bumper))
(display " at time ")
(display (ball-collision-time ball))
(newline)
(display " Ball ")
(display (ball-number ball))
(display " has a new velocity of ")
(display (ball-x-velocity ball))
(display ",")
(display (ball-y-velocity ball))
(display " starting at ")
(display (ball-collision-x-position ball))
(display ",")
(display (ball-collision-y-position ball))
(recalculate-collisions ball global-event-queue))
;;RECALCULATE-COLLISIONS removes all old collisions for the given ball from
;;all other balls' event queues and calcultes new collisions for these balls
;;and places them on the event queues. Also, updates the global event queue if
;;the recalculation of the collision effects the earliest collision for any
;;other balls.
;;BALL = The ball whose collisions are being recalculated
;;GLOBAL-EVENT-QUEUE = The global queue of earliest events for each ball
(define (recalculate-collisions ball global-event-queue)
(clear-queue (ball-event-queue ;Clear the queue of events for this
ball)) ;ball as they have all changed
(let ((event-queue ;Calculate all ball collision events
(ball-event-queue ball))) ;with balls of lower number
(let ((ball-vector
(ball-ball-vector ball)))
(do ((i (-1+ (ball-number ball))
(-1+ i)))
((negative? i))
(let ((ball2-queue-record
(vector-ref
ball-vector
i)))
(set-event-queue-record-collision-time!
ball2-queue-record
(ball-ball-collision-time
ball
(event-queue-record-object
ball2-queue-record)))
(queue-insert
event-queue
ball2-queue-record))))
(let ((bumper-vector ;Calculate all bumper collision events
(ball-bumper-vector ball)))
(do ((i (-1+ (vector-length
bumper-vector))
(-1+ i)))
((negative? i))
(let ((bumper-queue-record
(vector-ref
bumper-vector
i)))
(set-event-queue-record-collision-time!
bumper-queue-record
(ball-bumper-collision-time
ball
(event-queue-record-object
bumper-queue-record)))
(queue-insert
event-queue
bumper-queue-record))))
(let ((global-queue-record ;Get the global event queue record
(ball-global-event-queue-record ;for this ball
ball)))
(set-event-queue-record-collision-time! ;Set the new earliest event time
global-queue-record ;for this ball
(if (empty-queue? event-queue)
'()
(event-queue-record-collision-time
(queue-smallest event-queue))))
(queue-insert ;Enqueue on the global event queue
global-event-queue ;the earliest event between this ball
global-queue-record))) ;and any ball of lower number or any
;bumper
(for-each ;For each ball on the ball list:
(lambda (ball2)
(let ((ball2-event-queue
(ball-event-queue ball2)))
(let ((alter-global-event-queue? ;Set flag to update global event queue
(and ;if the earliest event for ball2 was
(not (empty-queue? ;with the deflected ball
ball2-event-queue))
(eq? ball
(event-queue-record-object
(queue-smallest
ball2-event-queue)))))
(ball-event-queue-record ;Get the queue record for the deflected
(vector-ref ;ball for this ball
(ball-ball-vector
ball2)
(ball-number ball))))
(queue-remove ;Remove the queue record for the
ball-event-queue-record) ;deflected ball
(set-event-queue-record-collision-time! ;Recalculate the collision
ball-event-queue-record ;time for this ball and the deflected
(ball-ball-collision-time ;ball
ball
ball2))
(queue-insert ;Enqueue the new collision event
ball2-event-queue
ball-event-queue-record)
(if (or alter-global-event-queue? ;If the earliest collision event for
(eq? ball ;this ball has changed:
(event-queue-record-object
(queue-smallest
ball2-event-queue))))
(let ((queue-record ;Remove the old event from the global
(ball-global-event-queue-record ;event queue and replace it
ball2))) ;with the new event
(set-event-queue-record-collision-time!
queue-record
(event-queue-record-collision-time
(queue-smallest
ball2-event-queue)))
(queue-remove
queue-record)
(queue-insert
global-event-queue
queue-record))))))
(ball-ball-list ball)))
;;SIMULATE performs the billiard ball simulation for the given ball list and
;;bumper list until the specified time.
;;BALL-LIST = A list of balls
;;BUMPER-LIST = A list of bumpers
;;END-TIME = The time at which the simulation is to terminate
(define (simulate ball-list bumper-list end-time)
(let ((num-of-balls ;Cache the number of balls and bumpers
(length ball-list))
(num-of-bumpers
(length bumper-list))
(global-event-queue ;Build the global event queue
(make-sorted-queue
collision-time-<?)))
(let ((complete-ball-vector ;Build a vector for the balls
(make-vector
num-of-balls)))
(let loop ((ball-num 0) ;For each ball:
(ball-list ball-list))
(if (not (null? ball-list))
(let ((ball (car ball-list)))
(set-ball-number! ;Store the ball's number
ball
ball-num)
(vector-set! ;Place it in the ball vector
complete-ball-vector
ball-num
ball)
(set-ball-ball-list! ;Save the list of balls with ball
ball ;numbers greater than the current ball
(cdr ball-list))
(display-ball-state
ball)
(loop
(1+ ball-num)
(cdr ball-list)))))
(let loop ((bumper-num 0) ;For each bumper:
(bumper-list
bumper-list))
(if (not (null? bumper-list))
(sequence
(set-bumper-number! ;Store the bumper's number
(car bumper-list)
bumper-num)
(display-bumper-state
(car bumper-list))
(loop
(1+ bumper-num)
(cdr bumper-list)))))
(do ((ball-num 0 (1+ ball-num))) ;For each ball:
((= ball-num num-of-balls))
(let* ((ball (vector-ref ;Cache a reference to the ball
complete-ball-vector
ball-num))
(ball-vector ;Build a vector for the queue records
(make-vector ;of balls with smaller numbers than
ball-num)) ;this ball
(bumper-vector ;Build a vector for the queue records
(make-vector ;of bumpers
num-of-bumpers))
(event-queue ;Build an event queue for this ball
(ball-event-queue
ball)))
(set-ball-ball-vector! ;Install the vector of ball queue
ball ;records
ball-vector)
(do ((i 0 (1+ i))) ;For each ball of smaller number than
((= i ball-num)) ;the current ball:
(let* ((ball2 ;Cache the ball
(vector-ref
complete-ball-vector
i))
(queue-record ;Create a queue record for this ball
(make-event-queue-record ;based on the collision time
'() ;of the two balls
'()
ball2
(ball-ball-collision-time
ball
ball2))))
(vector-set! ;Install the queue record in the ball
ball-vector ;vector for this ball
i
queue-record)
(queue-insert ;Insert the queue record into the event
event-queue ;queue for this ball
queue-record)))
(set-ball-bumper-vector! ;Install the vector of bumper queue
ball ;records
bumper-vector)
(let loop ((bumper-num 0)
(bumper-list
bumper-list))
(if (not (null? bumper-list))
(let* ((bumper ;Cache the bumper
(car
bumper-list))
(queue-record ;Create a queue record for this bumper
(make-event-queue-record ;based on the collision time
'() ;of the current ball and this bumper
'()
bumper
(ball-bumper-collision-time
ball
bumper))))
(vector-set! ;Install the queue record in the bumper
bumper-vector ;vector for this ball
bumper-num
queue-record)
(queue-insert ;Insert the queue record into the event
event-queue ;queue for this ball
queue-record)
(loop
(1+ bumper-num)
(cdr bumper-list)))))
(let ((queue-record ;Build a global event queue record for
(make-event-queue-record ;the earliest event on this ball's
'() ;event queue
'()
ball
(if (empty-queue?
event-queue)
'()
(event-queue-record-collision-time
(queue-smallest
event-queue))))))
(set-ball-global-event-queue-record! ;Store this queue record in
ball ;the frame for this ball
queue-record)
(queue-insert ;Insert this queue record in the global
global-event-queue ;event queue
queue-record)))))
(actually-simulate ;Now that all of the data structures
global-event-queue ;have been built, actually start the
end-time))) ;simulation
;;DISPLAY-BALL-STATE displays the ball number, mass, radius, position, and
;;velocity of the given ball
;;BALL = The ball whose state is to be displayed
(define (display-ball-state ball)
(newline)
(display "Ball ")
(display (ball-number ball))
(display " has mass ")
(display (ball-mass ball))
(display " and radius ")
(display (ball-radius ball))
(newline)
(display " Its position at time ")
(display (ball-collision-time ball))
(display " was ")
(display (ball-collision-x-position ball))
(display ",")
(display (ball-collision-y-position ball))
(display " and its velocity is ")
(display (ball-x-velocity ball))
(display ",")
(display (ball-y-velocity ball)))
;;DISPLAY-BUMPER-STATE displays the bumper number and position of the given
;;bumper
;;BUMPER = The bumper whose state is to be displayed
(define (display-bumper-state bumper)
(newline)
(display "Bumper ")
(display (bumper-number bumper))
(display " extends from ")
(display (bumper-x1 bumper))
(display ",")
(display (bumper-y1 bumper))
(display " to ")
(display (bumper-x2 bumper))
(display ",")
(display (bumper-y2 bumper)))
;;ACTUALLY-SIMULATE performs the actual billiard ball simulation
;;GLOBAL-EVENT-QUEUE = The global queue of earliest events for each ball.
;; Contains a single event for each ball which is the
;; earliest collision it would have with a ball of a
;; smaller number or a bumper, if no other collisions took
;; place first.
;;END-TIME = The time at which the simulation should be terminated
(define (actually-simulate global-event-queue end-time)
(letrec ((loop
(lambda ()
(let* ((record ;Get the globally earliest event and
(queue-smallest ;its time
global-event-queue))
(collision-time
(event-queue-record-collision-time
record)))
(if (not ;If this event happens before the
(time-<? ;simulation termination time:
end-time
collision-time))
(let* ((ball ;Get the ball involved in the event,
(event-queue-record-object
record))
(ball-queue ;the queue of events for that ball,
(ball-event-queue
ball))
(other-object ;and the first object with which the
(event-queue-record-object ;ball interacts
(queue-smallest
ball-queue))))
((simulation-object-collision-procedure ;Process this
other-object) ;globally earliest collision
ball
other-object
collision-time
global-event-queue)
(loop))))))) ;Process the next interaction
(loop)))
(require 'cscheme)
(set! autoload-notify? #f)
(simulate
(list (make-ball 2 1 9 5 -1 -1)
(make-ball 4 2 2 5 1 -1))
(list (make-bumper 0 0 0 10)
(make-bumper 0 0 10 0)
(make-bumper 0 10 10 10)
(make-bumper 10 0 10 10))
100)
(newline)