Tangenter mellom to konvekse polygoner
Gitt to konvekse polygoner tar vi sikte på å identifisere de nedre og øvre tangentene som forbinder dem.
Som vist i figuren nedenfor T RL og T LR representerer henholdsvis øvre og nedre tangenter.
Eksempler:
Inndata: Første polygon : [[2 2] [3 3] [5 2] [4 0] [3 1]]
Andre polygon: [[-1 0] [0 1] [1 0] [0 -2]].
Produksjon: Øvre Tangent - linjesammenføyning (01) og (33)
Nedre Tangent - linjesammenføyning (0-2) og (40)
Forklaring: Bildet viser tydelig strukturen og tangentene som forbinder de to polygonene
Nærme:
For å finne den øvre tangenten begynner vi med å velge to punkter: punktet lengst til høyre i polygon en og punktet lengst til venstre i polygon b . Linjen som forbinder disse to punktene er merket som Linje 1 . Siden denne linjen går gjennom polygon b (dvs. den er ikke helt over den) vi beveger oss til neste punkt i retning mot klokken på b forming Linje 2 . Denne linjen er nå over polygon b som er bra. Imidlertid krysser den polygon en så vi går videre til neste punkt på en i retning med klokken skaper Linje 3 . Linje 3 krysser fortsatt polygon en ber om et nytt trekk til Linje 4 . Linje 4 krysser imidlertid polygon b så vi går videre til Linje 5 . Endelig Linje 5 krysser ikke noen av polygonene, noe som gjør den til den riktige øvre tangenten for de gitte polygonene.
For å finne den nedre tangenten må vi bevege oss omvendt gjennom polygonene, det vil si at hvis linjen krysser polygon b, flytter vi til neste med klokken og mot klokken neste hvis linjen krysser polygon a.
Algoritme for øvre tangent:
L ← linje som forbinder punktet lengst til høyre i a og punktet lengst til venstre i b. while (L krysser noen av polygonene) { while(L krysser b) L ← L' : punktet på b flyttes opp. mens (L krysser a) L ← L' : punktet på a beveger seg opp. }
Algoritme for lavere tangent:
L ← linje som forbinder punktet lengst til høyre i a og punktet lengst til venstre i b. mens (L krysser noen av polygonene) { mens (L krysser b) L ← L' : punktet på b flyttes ned. mens (L krysser a) L ← L' : punktet på a flyttes ned. }
Merk at koden ovenfor bare beregner den øvre tangenten. En lignende tilnærming kan også brukes til å finne den nedre tangenten.
CPP #include using namespace std ; // Determines the quadrant of a point relative to origin int quad ( vector < int > p ) { if ( p [ 0 ] >= 0 && p [ 1 ] >= 0 ) return 1 ; if ( p [ 0 ] <= 0 && p [ 1 ] >= 0 ) return 2 ; if ( p [ 0 ] <= 0 && p [ 1 ] <= 0 ) return 3 ; return 4 ; } // Returns the orientation of ordered triplet (a b c) // 0 -> Collinear 1 -> Clockwise -1 -> Counterclockwise int orientation ( vector < int > a vector < int > b vector < int > c ) { int res = ( b [ 1 ] - a [ 1 ]) * ( c [ 0 ] - b [ 0 ]) - ( c [ 1 ] - b [ 1 ]) * ( b [ 0 ] - a [ 0 ]); if ( res == 0 ) return 0 ; if ( res > 0 ) return 1 ; return -1 ; } // Compare function to sort points counter-clockwise around center bool compare ( vector < int > p1 vector < int > q1 vector < int > mid ) { vector < int > p = { p1 [ 0 ] - mid [ 0 ] p1 [ 1 ] - mid [ 1 ]}; vector < int > q = { q1 [ 0 ] - mid [ 0 ] q1 [ 1 ] - mid [ 1 ]}; int one = quad ( p ); int two = quad ( q ); if ( one != two ) return ( one < two ); return ( p [ 1 ] * q [ 0 ] < q [ 1 ] * p [ 0 ]); } // Sorts the polygon points counter-clockwise vector < vector < int >> sortPoints ( vector < vector < int >> polygon ) { vector < int > mid = { 0 0 }; int n = polygon . size (); // Calculate center (centroid) of the polygon for ( int i = 0 ; i < n ; i ++ ) { mid [ 0 ] += polygon [ i ][ 0 ]; mid [ 1 ] += polygon [ i ][ 1 ]; polygon [ i ][ 0 ] *= n ; polygon [ i ][ 1 ] *= n ; } // Sort points based on their angle from the center sort ( polygon . begin () polygon . end () [ mid ]( vector < int > p1 vector < int > p2 ) { return compare ( p1 p2 mid ); }); // Divide back to original coordinates for ( int i = 0 ; i < n ; i ++ ) { polygon [ i ][ 0 ] /= n ; polygon [ i ][ 1 ] /= n ; } return polygon ; } // Finds the upper tangent between two convex polygons a and b // Returns two points forming the upper tangent vector < vector < int >> findUpperTangent ( vector < vector < int >> a vector < vector < int >> b ) { int n1 = a . size () n2 = b . size (); // Find the rightmost point of polygon a and leftmost point of polygon b int maxa = INT_MIN ; for ( auto & p : a ) maxa = max ( maxa p [ 0 ]); int minb = INT_MAX ; for ( auto & p : b ) minb = min ( minb p [ 0 ]); // Sort both polygons counter-clockwise a = sortPoints ( a ); b = sortPoints ( b ); // Ensure polygon a is to the left of polygon b if ( minb < maxa ) swap ( a b ); n1 = a . size (); n2 = b . size (); // Find the rightmost point in a int ia = 0 ib = 0 ; for ( int i = 1 ; i < n1 ; i ++ ) if ( a [ i ][ 0 ] > a [ ia ][ 0 ]) ia = i ; // Find the leftmost point in b for ( int i = 1 ; i < n2 ; i ++ ) if ( b [ i ][ 0 ] < b [ ib ][ 0 ]) ib = i ; // Initialize starting points int inda = ia indb = ib ; bool done = false ; // Find upper tangent using orientation checks while ( ! done ) { done = true ; // Move to next point in a if necessary while ( orientation ( b [ indb ] a [ inda ] a [( inda + 1 ) % n1 ]) > 0 ) inda = ( inda + 1 ) % n1 ; // Move to previous point in b if necessary while ( orientation ( a [ inda ] b [ indb ] b [( n2 + indb - 1 ) % n2 ]) < 0 ) { indb = ( n2 + indb - 1 ) % n2 ; done = false ; } } // Return the points forming the upper tangent return { a [ inda ] b [ indb ]}; } // Main driver code int main () { vector < vector < int >> a = {{ 2 2 } { 3 1 } { 3 3 } { 5 2 } { 4 0 }}; vector < vector < int >> b = {{ 0 1 } { 1 0 } { 0 -2 } { -1 0 }}; vector < vector < int >> tangent = findUpperTangent ( a b ); for ( auto it : tangent ){ cout < < it [ 0 ] < < ' ' < < it [ 1 ] < < ' n ' ; } return 0 ; }
Java import java.util.* ; class GfG { // Determines the quadrant of a point static int quad ( int [] p ) { if ( p [ 0 ] >= 0 && p [ 1 ] >= 0 ) return 1 ; if ( p [ 0 ] <= 0 && p [ 1 ] >= 0 ) return 2 ; if ( p [ 0 ] <= 0 && p [ 1 ] <= 0 ) return 3 ; return 4 ; } // Checks whether the line is crossing the polygon static int orientation ( int [] a int [] b int [] c ) { int res = ( b [ 1 ] - a [ 1 ] ) * ( c [ 0 ] - b [ 0 ] ) - ( c [ 1 ] - b [ 1 ] ) * ( b [ 0 ] - a [ 0 ] ); if ( res == 0 ) return 0 ; if ( res > 0 ) return 1 ; return - 1 ; } // Compare function for sorting static class PointComparator implements Comparator < int []> { int [] mid ; public PointComparator ( int [] mid ) { this . mid = mid ; } public int compare ( int [] p1 int [] p2 ) { int [] p = { p1 [ 0 ] - mid [ 0 ] p1 [ 1 ] - mid [ 1 ] }; int [] q = { p2 [ 0 ] - mid [ 0 ] p2 [ 1 ] - mid [ 1 ] }; int one = quad ( p ); int two = quad ( q ); if ( one != two ) return one - two ; return p [ 1 ] * q [ 0 ] - q [ 1 ] * p [ 0 ] ; } } // Finds upper tangent of two polygons 'a' and 'b' // represented as 2D arrays and stores the result static int [][] findUpperTangent ( int [][] a int [][] b ) { // n1 -> number of points in polygon a // n2 -> number of points in polygon b int n1 = a . length n2 = b . length ; int [] mid = { 0 0 }; int maxa = Integer . MIN_VALUE ; // Calculate centroid for polygon a and adjust points for scaling for ( int i = 0 ; i < n1 ; i ++ ) { maxa = Math . max ( maxa a [ i ][ 0 ] ); mid [ 0 ] += a [ i ][ 0 ] ; mid [ 1 ] += a [ i ][ 1 ] ; a [ i ][ 0 ] *= n1 ; a [ i ][ 1 ] *= n1 ; } // Sorting the points in counter-clockwise order for polygon a Arrays . sort ( a new PointComparator ( mid )); for ( int i = 0 ; i < n1 ; i ++ ) { a [ i ][ 0 ] /= n1 ; a [ i ][ 1 ] /= n1 ; } mid [ 0 ] = 0 ; mid [ 1 ] = 0 ; int minb = Integer . MAX_VALUE ; // Calculate centroid for polygon b and adjust points for scaling for ( int i = 0 ; i < n2 ; i ++ ) { mid [ 0 ] += b [ i ][ 0 ] ; mid [ 1 ] += b [ i ][ 1 ] ; minb = Math . min ( minb b [ i ][ 0 ] ); b [ i ][ 0 ] *= n2 ; b [ i ][ 1 ] *= n2 ; } // Sorting the points in counter-clockwise order for polygon b Arrays . sort ( b new PointComparator ( mid )); for ( int i = 0 ; i < n2 ; i ++ ) { b [ i ][ 0 ] /= n2 ; b [ i ][ 1 ] /= n2 ; } // If a is to the right of b swap a and b if ( minb < maxa ) { int [][] temp = a ; a = b ; b = temp ; n1 = a . length ; n2 = b . length ; } // ia -> rightmost point of a int ia = 0 ib = 0 ; for ( int i = 1 ; i < n1 ; i ++ ) { if ( a [ i ][ 0 ] > a [ ia ][ 0 ] ) ia = i ; } // ib -> leftmost point of b for ( int i = 1 ; i < n2 ; i ++ ) { if ( b [ i ][ 0 ] < b [ ib ][ 0 ] ) ib = i ; } // Finding the upper tangent int inda = ia indb = ib ; boolean done = false ; while ( ! done ) { done = true ; while ( orientation ( b [ indb ] a [ inda ] a [ ( inda + 1 ) % n1 ] ) > 0 ) inda = ( inda + 1 ) % n1 ; while ( orientation ( a [ inda ] b [ indb ] b [ ( n2 + indb - 1 ) % n2 ] ) < 0 ) { indb = ( n2 + indb - 1 ) % n2 ; done = false ; } } // Returning the upper tangent as a 2D array return new int [][] { { a [ inda ][ 0 ] a [ inda ][ 1 ] } { b [ indb ][ 0 ] b [ indb ][ 1 ] } }; } // Driver code public static void main ( String [] args ) { int [][] a = {{ 2 2 }{ 3 1 }{ 3 3 }{ 5 2 }{ 4 0 }}; int [][] b = {{ 0 1 }{ 1 0 }{ 0 - 2 }{ - 1 0 }}; // Get the upper tangent as a 2D array int [][] upperTangent = findUpperTangent ( a b ); // Store or use the result System . out . println ( upperTangent [ 0 ][ 0 ] + ' ' + upperTangent [ 0 ][ 1 ] ); System . out . println ( upperTangent [ 1 ][ 0 ] + ' ' + upperTangent [ 1 ][ 1 ] ); } }
Python from functools import cmp_to_key def quad ( p ): if p [ 0 ] >= 0 and p [ 1 ] >= 0 : return 1 if p [ 0 ] <= 0 and p [ 1 ] >= 0 : return 2 if p [ 0 ] <= 0 and p [ 1 ] <= 0 : return 3 return 4 def orientation ( a b c ): res = ( b [ 1 ] - a [ 1 ]) * ( c [ 0 ] - b [ 0 ]) - ( c [ 1 ] - b [ 1 ]) * ( b [ 0 ] - a [ 0 ]) if res == 0 : return 0 if res > 0 : return 1 return - 1 def compare ( mid ): def cmp ( p1 q1 ): p = [ p1 [ 0 ] - mid [ 0 ] p1 [ 1 ] - mid [ 1 ]] q = [ q1 [ 0 ] - mid [ 0 ] q1 [ 1 ] - mid [ 1 ]] one = quad ( p ) two = quad ( q ) if one != two : return one - two if p [ 1 ] * q [ 0 ] < q [ 1 ] * p [ 0 ]: return - 1 return 1 return cmp def findUpperTangent ( a b ): n1 n2 = len ( a ) len ( b ) mid_a = [ 0 0 ] maxa = float ( '-inf' ) for i in range ( n1 ): maxa = max ( maxa a [ i ][ 0 ]) mid_a [ 0 ] += a [ i ][ 0 ] mid_a [ 1 ] += a [ i ][ 1 ] a = sorted ( a key = cmp_to_key ( compare ( mid_a ))) mid_b = [ 0 0 ] minb = float ( 'inf' ) for i in range ( n2 ): minb = min ( minb b [ i ][ 0 ]) mid_b [ 0 ] += b [ i ][ 0 ] mid_b [ 1 ] += b [ i ][ 1 ] b = sorted ( b key = cmp_to_key ( compare ( mid_b ))) if minb < maxa : a b = b a n1 n2 = n2 n1 ia = 0 for i in range ( 1 n1 ): if a [ i ][ 0 ] > a [ ia ][ 0 ]: ia = i ib = 0 for i in range ( 1 n2 ): if b [ i ][ 0 ] < b [ ib ][ 0 ]: ib = i inda indb = ia ib done = False while not done : done = True while orientation ( b [ indb ] a [ inda ] a [( inda + 1 ) % n1 ]) > 0 : inda = ( inda + 1 ) % n1 while orientation ( a [ inda ] b [ indb ] b [( n2 + indb - 1 ) % n2 ]) < 0 : indb = ( n2 + indb - 1 ) % n2 done = False # return integer coordinates return [[ int ( a [ inda ][ 0 ]) int ( a [ inda ][ 1 ])] [ int ( b [ indb ][ 0 ]) int ( b [ indb ][ 1 ])]] # Driver Code if __name__ == '__main__' : a = [ [ 2 2 ] [ 3 1 ] [ 3 3 ] [ 5 2 ] [ 4 0 ] ] b = [ [ 0 1 ] [ 1 0 ] [ 0 - 2 ] [ - 1 0 ] ] upperTangent = findUpperTangent ( a b ) for point in upperTangent : print ( f ' { point [ 0 ] } { point [ 1 ] } ' )
C# using System ; public class UpperTangentFinder { static int Quad ( int x int y ) { if ( x >= 0 && y >= 0 ) return 1 ; if ( x <= 0 && y >= 0 ) return 2 ; if ( x <= 0 && y <= 0 ) return 3 ; return 4 ; } static int Orientation ( int [] a int [] b int [] c ) { int res = ( b [ 1 ] - a [ 1 ]) * ( c [ 0 ] - b [ 0 ]) - ( c [ 1 ] - b [ 1 ]) * ( b [ 0 ] - a [ 0 ]); if ( res == 0 ) return 0 ; return res > 0 ? 1 : - 1 ; } static bool Compare ( int [] p1 int [] p2 int [] mid ) { int [] p = { p1 [ 0 ] - mid [ 0 ] p1 [ 1 ] - mid [ 1 ] }; int [] q = { p2 [ 0 ] - mid [ 0 ] p2 [ 1 ] - mid [ 1 ] }; int quadP = Quad ( p [ 0 ] p [ 1 ]); int quadQ = Quad ( q [ 0 ] q [ 1 ]); if ( quadP != quadQ ) return quadP < quadQ ; return ( p [ 1 ] * q [ 0 ]) < ( q [ 1 ] * p [ 0 ]); } static int [] SortPoints ( int [] polygon ) { int n = polygon . GetLength ( 0 ); int [] mid = { 0 0 }; for ( int i = 0 ; i < n ; i ++ ) { mid [ 0 ] += polygon [ i 0 ]; mid [ 1 ] += polygon [ i 1 ]; polygon [ i 0 ] *= n ; polygon [ i 1 ] *= n ; } for ( int i = 0 ; i < n - 1 ; i ++ ) { for ( int j = i + 1 ; j < n ; j ++ ) { int [] p1 = { polygon [ i 0 ] polygon [ i 1 ] }; int [] p2 = { polygon [ j 0 ] polygon [ j 1 ] }; if ( ! Compare ( p1 p2 mid )) { int tempX = polygon [ i 0 ] tempY = polygon [ i 1 ]; polygon [ i 0 ] = polygon [ j 0 ]; polygon [ i 1 ] = polygon [ j 1 ]; polygon [ j 0 ] = tempX ; polygon [ j 1 ] = tempY ; } } } for ( int i = 0 ; i < n ; i ++ ) { polygon [ i 0 ] /= n ; polygon [ i 1 ] /= n ; } return polygon ; } static int [] FindUpperTangent ( int [] a int [] b ) { int n1 = a . GetLength ( 0 ); int n2 = b . GetLength ( 0 ); int maxa = int . MinValue ; for ( int i = 0 ; i < n1 ; i ++ ) maxa = Math . Max ( maxa a [ i 0 ]); int minb = int . MaxValue ; for ( int i = 0 ; i < n2 ; i ++ ) minb = Math . Min ( minb b [ i 0 ]); a = SortPoints ( a ); b = SortPoints ( b ); if ( minb < maxa ) { int [] temp = a ; a = b ; b = temp ; n1 = a . GetLength ( 0 ); n2 = b . GetLength ( 0 ); } int ia = 0 ib = 0 ; for ( int i = 1 ; i < n1 ; i ++ ) if ( a [ i 0 ] > a [ ia 0 ]) ia = i ; for ( int i = 1 ; i < n2 ; i ++ ) if ( b [ i 0 ] < b [ ib 0 ]) ib = i ; int inda = ia indb = ib ; bool done = false ; while ( ! done ) { done = true ; while ( Orientation ( new int [] { b [ indb 0 ] b [ indb 1 ] } new int [] { a [ inda 0 ] a [ inda 1 ] } new int [] { a [( inda + 1 ) % n1 0 ] a [( inda + 1 ) % n1 1 ] }) > 0 ){ inda = ( inda + 1 ) % n1 ; } while ( Orientation ( new int [] { a [ inda 0 ] a [ inda 1 ] } new int [] { b [ indb 0 ] b [ indb 1 ] } new int [] { b [( n2 + indb - 1 ) % n2 0 ] b [( n2 + indb - 1 ) % n2 1 ] }) < 0 ){ indb = ( n2 + indb - 1 ) % n2 ; done = false ; } } int [] result = new int [ 2 2 ]; result [ 0 0 ] = a [ inda 0 ]; result [ 0 1 ] = a [ inda 1 ]; result [ 1 0 ] = b [ indb 0 ]; result [ 1 1 ] = b [ indb 1 ]; return result ; } public static void Main ( string [] args ) { int [] a = new int [] { { 2 2 } { 3 1 } { 3 3 } { 5 2 } { 4 0 } }; int [] b = new int [] { { 0 1 } { 1 0 } { 0 - 2 } { - 1 0 } }; int [] tangent = FindUpperTangent ( a b ); for ( int i = 0 ; i < 2 ; i ++ ) { Console . WriteLine ( tangent [ i 0 ] + ' ' + tangent [ i 1 ]); } } }
JavaScript // Determine the quadrant of a point function quad ( p ) { if ( p [ 0 ] >= 0 && p [ 1 ] >= 0 ) return 1 ; if ( p [ 0 ] <= 0 && p [ 1 ] >= 0 ) return 2 ; if ( p [ 0 ] <= 0 && p [ 1 ] <= 0 ) return 3 ; return 4 ; } // Find orientation of triplet (a b c) function orientation ( a b c ) { let res = ( b [ 1 ] - a [ 1 ]) * ( c [ 0 ] - b [ 0 ]) - ( c [ 1 ] - b [ 1 ]) * ( b [ 0 ] - a [ 0 ]); if ( res === 0 ) return 0 ; return res > 0 ? 1 : - 1 ; } // Compare two points based on mid function compare ( p1 p2 mid ) { let p = [ p1 [ 0 ] - mid [ 0 ] p1 [ 1 ] - mid [ 1 ]]; let q = [ p2 [ 0 ] - mid [ 0 ] p2 [ 1 ] - mid [ 1 ]]; let quadP = quad ( p ); let quadQ = quad ( q ); if ( quadP !== quadQ ) return quadP - quadQ ; return ( p [ 1 ] * q [ 0 ]) - ( q [ 1 ] * p [ 0 ]); } // Sort polygon points counter-clockwise function sortPoints ( polygon ) { let n = polygon . length ; let mid = [ 0 0 ]; for ( let i = 0 ; i < n ; i ++ ) { mid [ 0 ] += polygon [ i ][ 0 ]; mid [ 1 ] += polygon [ i ][ 1 ]; polygon [ i ][ 0 ] *= n ; polygon [ i ][ 1 ] *= n ; } polygon . sort (( p1 p2 ) => compare ( p1 p2 mid )); for ( let i = 0 ; i < n ; i ++ ) { polygon [ i ][ 0 ] = Math . floor ( polygon [ i ][ 0 ] / n ); polygon [ i ][ 1 ] = Math . floor ( polygon [ i ][ 1 ] / n ); } return polygon ; } // Find upper tangent between two convex polygons function findUpperTangent ( a b ) { let n1 = a . length ; let n2 = b . length ; let maxa = - Infinity ; for ( let i = 0 ; i < n1 ; i ++ ) maxa = Math . max ( maxa a [ i ][ 0 ]); let minb = Infinity ; for ( let i = 0 ; i < n2 ; i ++ ) minb = Math . min ( minb b [ i ][ 0 ]); a = sortPoints ( a ); b = sortPoints ( b ); if ( minb < maxa ) { let temp = a ; a = b ; b = temp ; n1 = a . length ; n2 = b . length ; } let ia = 0 ib = 0 ; for ( let i = 1 ; i < n1 ; i ++ ) if ( a [ i ][ 0 ] > a [ ia ][ 0 ]) ia = i ; for ( let i = 1 ; i < n2 ; i ++ ) if ( b [ i ][ 0 ] < b [ ib ][ 0 ]) ib = i ; let inda = ia indb = ib ; let done = false ; while ( ! done ) { done = true ; while ( orientation ( b [ indb ] a [ inda ] a [( inda + 1 ) % n1 ]) > 0 ) inda = ( inda + 1 ) % n1 ; while ( orientation ( a [ inda ] b [ indb ] b [( n2 + indb - 1 ) % n2 ]) < 0 ) { indb = ( n2 + indb - 1 ) % n2 ; done = false ; } } return [ a [ inda ] b [ indb ]]; } // Driver code let a = [[ 2 2 ][ 3 1 ][ 3 3 ][ 5 2 ][ 4 0 ]]; let b = [[ 0 1 ][ 1 0 ][ 0 - 2 ][ - 1 0 ]]; let tangent = findUpperTangent ( a b ); for ( let point of tangent ) { console . log ( point [ 0 ] + ' ' + point [ 1 ]); }
Produksjon
upper tangent (01) (33)
Tidskompleksitet: O(n1 log (n1) + n2 log(n2))
Hjelpeplass: O(1)
