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CRESC4.SIF
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***************************
* SET UP THE INITIAL DATA *
***************************
NAME CRESC4
* Problem :
* *********
* This problem consists in finding the crescent of smallest area containing
* a set of points given in the plane. This problem arises as a subproblem
* in pattern recognition and has been suggested by J.P. Rasson. It
* originates in the detection of "salt domes" (with the potential presence of
* oil pockets!) from geological data.
* The present problem is a simplified version where the crescent is entirely
* determined by the only four data points.
* The problem is not convex.
* A crescent is defined as follows. Assume one has two circles of center
* C1 and C2 and of radii r1 and r2 respectively. Assume furthermore that
* r1 >= r2 and that C2 is within C1. Assume finally that the distance from
* C1 to C2 is >= r1 - r2. Then the crescent is the part of the plane
* contained in circle 2 but not in circle 1.
* In order to preserve feasibility at all stages (ensuring that the
* crescent exists and that its area can be computed), the following
* parametrization is used:
* ( C2x, C2y ) = ( C1x, C1y ) + a * d * ( cos(t), sin(t) )
* r1 = a * d + r
* r2 = ( a + 1 ) * d + r
* with the bounds
* a >= 1, 0 <= t <= 2 * pi, r2 >= 0 , 0 <= d <= 1.
* SIF input: Ph. Toint, June 1993.
* classification OOR2-MY-6-8
* number of points to be included in the crescent.
* the number of constraints is 2*NP
IE NP 4
* data points
RE X1 1.0
RE Y1 0.0
RE X2 0.0
RE Y2 1.0
RE X3 0.0
RE Y3 -1.0
RE X4 0.5
RE Y4 0.0
* useful constants
IE 1 1
VARIABLES
X V1
X W1
X D
X A
X T
X R
GROUPS
XN OBJ
DO I 1 NP
XL IS2(I)
XG OS1(I)
ND
BOUNDS
XR CRESC4 V1
XR CRESC4 W1
XL CRESC4 R 0.39
XL CRESC4 A 1.0
XL CRESC4 T 0.0
XU CRESC4 T 6.2831852
XL CRESC4 D 1.0D-8
START POINT
XV CRESC4 V1 -40.0
XV CRESC4 W1 5.0
XV CRESC4 R 0.75
XV CRESC4 A 2.0
XV CRESC4 T 1.5
XV CRESC4 D 1.0
* Global solution
*XV CRESC4 V1 -0.75
*XV CRESC4 W1 0.0
*XV CRESC4 R 0.5
*XV CRESC4 A 1.5
*XV CRESC4 T 0.0
*XV CRESC4 D 0.5
ELEMENT TYPE
EV SQR1 A R
EV SQR1 D
EV SQR2 D R
IV SQR2 RAD
EV SC AZ BZ
EV SC DZ
EV DIST X
EP DIST P
EV DISTX X T
EV DISTX A D
EP DISTX P
EV DISTY X T
EV DISTY A D
EP DISTY P
ELEMENT USES
XT OB SC
ZV OB AZ A
ZV OB BZ R
ZV OB DZ D
XT R2SQ SQR2
ZV R2SQ D D
ZV R2SQ R R
XT R1SQ SQR1
ZV R1SQ D D
ZV R1SQ A A
ZV R1SQ R R
DO I 1 NP
XT XV1(I) DIST
ZV XV1(I) X V1
ZP XV1(I) P X(I)
XT XV2(I) DISTX
ZV XV2(I) X V1
ZV XV2(I) A A
ZV XV2(I) D D
ZV XV2(I) T T
ZP XV2(I) P X(I)
XT YW1(I) DIST
ZV YW1(I) X W1
ZP YW1(I) P Y(I)
XT YW2(I) DISTY
ZV YW2(I) X W1
ZV YW2(I) A A
ZV YW2(I) D D
ZV YW2(I) T T
ZP YW2(I) P Y(I)
ND
GROUP USES
XE OBJ OB
DO I 1 NP
XE IS2(I) XV2(I) YW2(I)
XE IS2(I) R2SQ -1.0
XE OS1(I) XV1(I) YW1(I)
XE OS1(I) R1SQ -1.0
ND
OBJECT BOUND
* Solution 0.87189692
ENDATA
***********************
* SET UP THE FUNCTION *
* AND RANGE ROUTINES *
***********************
ELEMENTS CRESC4
TEMPORARIES
R A
R AB
R AD
R B
R BT
R BA
R BB
R BD
R BAD
R C
R CQ
R CT
R CA
R CB
R CD
R CH
R CHH
R CAA
R CAB
R CAD
R CBB
R CBD
R CDD
R D
R D2
R DA
R DD
R DAD
R E
R EA
R EB
R ED
R EAB
R EAD
R EBD
R EDD
R F
R FA
R FB
R FD
R FAA
R FAB
R FAD
R FBB
R FBD
R FDD
R G
R SG
R CG
R GF
R GA
R GB
R GD
R GFF
R GAA
R GAB
R GAD
R GBB
R GBD
R GDD
R H
R HA
R HB
R HD
R HAA
R HAB
R HAD
R HBB
R HBD
R HDD
R I
R IA
R IB
R ID
R IAA
R IAB
R IAD
R IBB
R IBD
R IDD
R J
R JA
R JB
R JD
R JAA
R JAB
R JAD
R JBB
R JBD
R JDD
R K
R KX
R KY
R KA
R KB
R KD
R L
R LA
R LB
R LD
R LAA
R LAB
R LAD
R LBB
R LBD
R LDD
R M
R MA
R MB
R MD
R N
R NB
R ND
R O
R OD
R P
R PA
R PB
R PD
R PAA
R PAB
R PAD
R PBD
R PDD
R Q
R QL
R QLL
R QA
R QB
R QD
R QAA
R QAB
R QAD
R QBB
R QBD
R QDD
R R
R RA
R RB
R RD
R RAA
R RAB
R RAD
R RBB
R RBD
R RDD
R S
R SQ
R ST
R SA
R SB
R SD
R SAA
R SAB
R SAD
R SBB
R SBD
R SDD
R U
R UA
R UB
R UD
R UAA
R UAB
R UAD
R UBB
R UBD
R UDD
R V
R VA
R VB
R VD
R VAA
R VAB
R VAD
R VBB
R VBD
R VDD
R W
R WA
R WB
R WD
R WAA
R WAB
R WAD
R WBB
R WBD
R WDD
M SQRT
M SIN
M COS
M ACOS
M ACOS
INDIVIDUALS
T SQR1
A Q A * D + R
F Q * Q
G A 2.0 * Q * D
G D 2.0 * Q * A
G R 2.0 * Q
H A A 2.0 * D * D
H A D 2.0 * ( A * D + Q )
H A R 2.0 * D
H D D 2.0 * A * A
H D R 2.0 * A
H R R 2.0
T SQR2
R RAD D 1.0 R 1.0
F RAD * RAD
G RAD RAD + RAD
H RAD RAD 2.0
T SC
A A DZ + BZ
A AB 1.0
A AD 1.0
A B AZ * DZ + BZ
A BA DZ
A BB 1.0
A BD AZ
A BAD 1.0
A D AZ * DZ
A DA DZ
A DD AZ
A DAD 1.0
A E 2.0 * A * D
A EA 2.0 * A * DA
A EB 2.0 * AB * D
A ED 2.0 * ( A * DD + AD * D )
A EAB 2.0 * AB * DA
A EAD 2.0 * ( AD * DA + A * DAD )
A EBD 2.0 * AB * DD
A EDD 2.0 * ( AD * DD + AD * DD )
A P 2.0 * B * D
A PA 2.0 * ( B * DA + BA * D )
A PB 2.0 * BB * D
A PD 2.0 * ( B * DD + BD * D )
A PAA 2.0 * ( BA * DA + BA * DA )
A PAB 2.0 * BB * DA
A PAD 2.0 * ( BD * DA + B * DAD + BAD * D
A+ + BA * DD )
A PBD 2.0 * BB * DD
A PDD 2.0 * ( BD * DD + BD * DD )
A F D * D + B * B - A * A
A FA 2.0 * ( D * DA + B * BA )
A FB 2.0 * ( B * BB - A * AB )
A FD 2.0 * ( D * DD + B * BD - A * AD )
A FAA 2.0 * ( DA * DA + BA * BA )
A FAB 2.0 * BB * BA
A FAD 2.0 * ( DD * DA + D * DAD + BD * BA
A+ + B * BAD )
A FBB 2.0 * ( BB * BB - AB * AB )
A FBD 2.0 * ( BD * BB - AD * AB )
A FDD 2.0 * ( DD * DD + BD * BD - AD * AD )
A G D * D - B * B + A * A
A GA 2.0 * ( D * DA - B * BA )
A GB 2.0 * ( - B * BB + A * AB )
A GD 2.0 * ( D * DD - B * BD + A * AD )
A GAA 2.0 * ( DA * DA - BA * BA )
A GAB - 2.0 * BB * BA
A GAD 2.0 * ( DD * DA + D * DAD - BD * BA
A+ - B * BAD )
A GBB 2.0 * ( - BB * BB + AB * AB )
A GBD 2.0 * ( - BD * BB + AD * AB )
A GDD 2.0 * ( DD * DD - BD * BD + AD * AD )
A H F / P
A I FA - H * PA
A J FB - H * PB
A K FD - H * PD
A HA I / P
A HB J / P
A HD K / P
A IA FAA - HA * PA - H * PAA
A IB FAB - HB * PA - H * PAB
A ID FAD - HD * PA - H * PAD
A JB FBB - HB * PB
A JD FBD - HD * PB - H * PBD
A KD FDD - HD * PD - H * PDD
A HAA ( IA - HA * PA ) / P
A HAB ( IB - HA * PB ) / P
A HAD ( ID - HA * PD ) / P
A HBB ( JB - HB * PB ) / P
A HBD ( JD - HB * PD ) / P
A HDD ( KD - HD * PD ) / P
A L - G / E
A M - GA - L * EA
A N - GB - L * EB
A O - GD - L * ED
A LA M / E
A LB N / E
A LD O / E
A MA - GAA - LA * EA
A MB - GAB - LB * EA - L * EAB
A MD - GAD - LD * EA - L * EAD
A NB - GBB - LB * EB
A ND - GBD - LD * EB - L * EBD
A OD - GDD - LD * ED - L * EDD
A LAA ( MA - LA * EA ) / E
A LAB ( MB - LA * EB ) / E
A LAD ( MD - LA * ED ) / E
A LBB ( NB - LB * EB ) / E
A LBD ( ND - LB * ED ) / E
A LDD ( OD - LD * ED ) / E
A C ACOS( H )
A CH - 1.0 / SQRT( 1.0 - H * H )
A CHH CH * H / ( 1.0 - H * H )
A CA CH * HA
A CB CH * HB
A CD CH * HD
A CAA CHH * HA * HA + CH * HAA
A CAB CHH * HA * HB + CH * HAB
A CAD CHH * HA * HD + CH * HAD
A CBB CHH * HB * HB + CH * HBB
A CBD CHH * HB * HD + CH * HBD
A CDD CHH * HD * HD + CH * HDD
A Q ACOS( L )
A QL - 1.0 / SQRT( 1.0 - L * L )
A QLL QL * L / ( 1.0 - L * L )
A QA QL * LA
A QB QL * LB
A QD QL * LD
A QAA QLL * LA * LA + QL * LAA
A QAB QLL * LA * LB + QL * LAB
A QAD QLL * LA * LD + QL * LAD
A QBB QLL * LB * LB + QL * LBB
A QBD QLL * LB * LD + QL * LBD
A QDD QLL * LD * LD + QL * LDD
A R B * B * C
A RA 2.0 * B * BA * C + B * B * CA
A RB 2.0 * B * BB * C + B * B * CB
A RD 2.0 * B * BD * C + B * B * CD
A RAA 2.0 * ( BA * BA * C + B * BA * CA
A+ + B * BA * CA ) + B * B * CAA
A RAB 2.0 * ( BB * BA * C + B * BA * CB
A+ + B * BB * CA ) + B * B * CAB
A RAD 2.0 * ( BD * BA * C + B * BAD * C
A+ + B * BA * CD + B * BD * CA )
A+ + B * B * CAD
A RBB 2.0 * ( BB * BB * C + B * BB * CB
A+ + B * BB * CB ) + B * B * CBB
A RBD 2.0 * ( BD * BB * C + B * BB * CD
A+ + B * BD * CB ) + B * B * CBD
A RDD 2.0 * ( BD * BD * C + B * BD * CD
A+ + B * BD * CD ) + B * B * CDD
A S A * A * Q
A SA A * A * QA
A SB 2.0 * A * AB * Q + A * A * QB
A SD 2.0 * A * AD * Q + A * A * QD
A SAA A * A * QAA
A SAB 2.0 * A * AB * QA + A * A * QAB
A SAD 2.0 * A * AD * QA + A * A * QAD
A SBB 2.0 * ( AB * AB * Q + A * AB * QB
A+ + A * AB * QB ) + A * A * QBB
A SBD 2.0 * ( AD * AB * Q + A * AB * QD
A+ + A * AD * QB ) + A * A * QBD
A SDD 2.0 * ( AD * AD * Q + A * AD * QD
A+ + A * AD * QD ) + A * A * QDD
A SQ SIN( Q )
A CQ L
A W 0.5 * E * SQ
A WA 0.5 * ( EA * SQ + E * CQ * QA )
A WB 0.5 * ( EB * SQ + E * CQ * QB )
A WD 0.5 * ( ED * SQ + E * CQ * QD )
A WAA 0.5 * ( EA * CQ * QA + EA * CQ * QA
A+ - E * SQ * QA * QA + E * CQ * QAA )
A WAB 0.5 * ( EAB * SQ + EA * CQ * QB
A+ + EB * CQ * QA - E * SQ * QB * QA
A+ + E * CQ * QAB )
A WAD 0.5 * ( EAD * SQ + EA * CQ * QD
A+ + ED * CQ * QA - E * SQ * QD * QA
A+ + E * CQ * QAD )
A WBB 0.5 * ( EB * CQ * QB
A+ + EB * CQ * QB - E * SQ * QB * QB
A+ + E * CQ * QBB )
A WBD 0.5 * ( EBD * SQ + EB * CQ * QD
A+ + ED * CQ * QB - E * SQ * QD * QB
A+ + E * CQ * QBD )
A WDD 0.5 * ( EDD * SQ + ED * CQ * QD
A+ + ED * CQ * QD - E * SQ * QD * QD
A+ + E * CQ * QDD )
A V S - R + W
A VA SA - RA + WA
A VB SB - RB + WB
A VD SD - RD + WD
A VAA SAA - RAA + WAA
A VAB SAB - RAB + WAB
A VAD SAD - RAD + WAD
A VBB SBB - RBB + WBB
A VBD SBD - RBD + WBD
A VDD SDD - RDD + WDD
F V
G AZ VA
G BZ VB
G DZ VD
H AZ AZ VAA
H AZ BZ VAB
H AZ DZ VAD
H BZ BZ VBB
H BZ DZ VBD
H DZ DZ VDD
T DIST
F ( X - P )**2
G X 2.0 * ( X - P )
H X X 2.0
T DISTY
A ST SIN( T )
A CT COS( T )
A B X + A * D * ST - P
A BA D * ST
A BD A * ST
A BT A * D * CT
F B * B
G X 2.0 * B
G A 2.0 * B * BA
G D 2.0 * B * BD
G T 2.0 * B * BT
H X X 2.0
H X A 2.0 * BA
H X D 2.0 * BD
H X T 2.0 * BT
H A A 2.0 * BA * BA
H A D 2.0 * ( BD * BA + B * ST )
H A T 2.0 * ( BT * BA + B * D * CT )
H D D 2.0 * BD * BD
H D T 2.0 * ( BT * BD + B * A * CT )
H T T 2.0 * ( BT * BT - B * A * D * ST )
T DISTX
A ST SIN( T )
A CT COS( T )
A B X + A * D * CT - P
A BA D * CT
A BD A * CT
A BT - A * D * ST
F B * B
G X 2.0 * B
G A 2.0 * B * BA
G D 2.0 * B * BD
G T 2.0 * B * BT
H X X 2.0
H X A 2.0 * BA
H X D 2.0 * BD
H X T 2.0 * BT
H A A 2.0 * BA * BA
H A D 2.0 * ( BD * BA + B * CT )
H A T 2.0 * ( BT * BA - B * D * ST )
H D D 2.0 * BD * BD
H D T 2.0 * ( BT * BD - B * A * ST )
H T T 2.0 * ( BT * BT - B * A * D * CT )
ENDATA