problem analysis
Title meaning find the minimum distance between two projections packet.
A very violent start to write a Minkowski addition, later found stuck rotation variant of a turn like QAQ
Minkowski and the idea is very simple, look at the following rotation stuck practice:
Difficult to find the shortest distance between two convex hull as a certain image on FIG. So we just need to enumerate the edge of a convex hull, looking like the point of the heel on the convex hull of another. This process needs to be performed twice.
Note that the line segment is determined in parallel to the details and request points of the line segment.
Reference program
Minkowski addition edition:
#include <cstdio>
#include <algorithm>
#include <cmath>
using namespace std;
const int Maxn = 10010;
const double Eps = 1e-12;
struct point {
double x, y;
point() {}
point( double _x, double _y ) : x( _x ), y( _y ) {}
inline point operator + ( const point Other ) const {
return point( x + Other.x, y + Other.y );
}
inline point operator - ( const point Other ) const {
return point( x - Other.x, y - Other.y );
}
inline point operator * ( const double Other ) const {
return point( x * Other, y * Other );
}
inline double operator * ( const point Other ) const {
return x * Other.y - Other.x * y;
}
inline double operator / ( const point Other ) const {
return x * Other.x + y * Other.y;
}
inline double Dis() const { return sqrt( x * x + y * y ); }
};
int N, M, L;
point A[ Maxn ], B[ Maxn ], C[ Maxn << 1 ], Base;
inline int Cmp( double x, double y ) {
if( fabs( x - y ) <= Eps ) return 0;
if( x - y > Eps ) return 1;
return -1;
}
inline bool Cmp1( point x, point y ) {
return Cmp( ( x - Base ) * ( y - Base ), 0.0 ) == 1 ||
( Cmp( ( x - Base ) * ( y - Base ), 0.0 ) == 0 && Cmp( ( x - Base ).Dis(), ( y - Base ).Dis() ) == -1 );
}
void Get( point *A, int &N ) {
for( int i = 2; i <= N; ++i )
if( Cmp( A[ i ].y, A[ 1 ].y ) == -1 || ( Cmp( A[ i ].y, A[ 1 ].y ) == 0 && Cmp( A[ i ].x, A[ 1 ].x ) == -1 ) )
swap( A[ i ], A[ 1 ] );
Base = A[ 1 ]; sort( A + 2, A + N + 1, Cmp1 );
L = 1; C[ 1 ] = A[ 1 ];
for( int i = 2; i <= N; ++i ) {
for( ; L > 1 && Cmp( ( A[ i ] - C[ L - 1 ] ) * ( C[ L ] - C[ L - 1 ] ), 0.0 ) >= 0; --L );
C[ ++L ] = A[ i ];
}
N = L; for( int i = 1; i <= L; ++i ) A[ i ] = C[ i ];
return;
}
void Merge( point *A, int N, point *B, int M, point *C ) {
L = 1; C[ 1 ] = A[ 1 ] + B[ 1 ];
A[ ++N ] = A[ 1 ]; B[ ++M ] = B[ 1 ];
int i1 = 1, i2 = 1;
for( ; i1 < N && i2 < M; ) {
if( Cmp( ( A[ i1 + 1 ] - A[ i1 ] ) * ( B[ i2 + 1 ] - B[ i2 ] ), 0.0 ) >= 0 ) {
C[ L + 1 ] = C[ L ] + ( A[ i1 + 1 ] - A[ i1 ] ); ++L; ++i1;
} else {
C[ L + 1 ] = C[ L ] + ( B[ i2 + 1 ] - B[ i2 ] ); ++L; ++i2;
}
}
for( ; i1 < N; ++i1 ) C[ L + 1 ] = C[ L ] + ( A[ i1 + 1 ] - A[ i1 ] ), ++L;
for( ; i2 < M; ++i2 ) C[ L + 1 ] = C[ L ] + ( B[ i2 + 1 ] - B[ i2 ] ), ++L;
return;
}
bool Check() {
for( int i = 1; i < L; ++i )
if( Cmp( C[ i + 1 ] * C[ i ], 0.0 ) > 0 ) return false;
return true;
}
double GetDis( point A, point B, point C ) {
point D = C + point( -( A - B ).y, ( A - B ).x );
double T = ( D - A ) * ( C - A ) / ( ( C - B ) * ( D - B ) );
if( Cmp( T, 0.0 ) <= 0 ) return min( A.Dis(), B.Dis() );
point O = A * ( 1 / ( T + 1.0 ) ) + B * ( T / ( T + 1.0 ) );
return ( O - C ).Dis();
}
int main() {
scanf( "%d%d", &N, &M );
while( N != 0 || M != 0 ) {
for( int i = 1; i <= N; ++i ) scanf( "%lf%lf", &A[ i ].x, &A[ i ].y );
for( int i = 1; i <= M; ++i ) scanf( "%lf%lf", &B[ i ].x, &B[ i ].y );
for( int i = 1; i <= M; ++i ) B[ i ].x = -B[ i ].x, B[ i ].y = -B[ i ].y;
Get( A, N ); Get( B, M );
Merge( A, N, B, M, C );
if( Check() ) printf( "0.000000\n" );
else {
double Ans = 1000000000.0;
for( int i = 1; i < L; ++i ) Ans = min( Ans, GetDis( C[ i ], C[ i + 1 ], point( 0.0, 0.0 ) ) );
printf( "%.6lf\n", Ans );
}
scanf( "%d%d", &N, &M );
}
return 0;
}Rotating stuck version:
#include <cstdio>
#include <cmath>
#include <algorithm>
using namespace std;
const int Maxn = 10010;
const double Eps = 1e-12;
struct point {
double x, y;
point() {}
point( double _x, double _y ) : x( _x ), y( _y ) {}
inline point operator + ( const point Other ) const {
return point( x + Other.x, y + Other.y );
}
inline point operator - ( const point Other ) const {
return point( x - Other.x, y - Other.y );
}
inline double operator * ( const point Other ) const {
return x * Other.y - Other.x * y;
}
inline point operator * ( const double Other ) const {
return point( x * Other, y * Other );
}
inline double Mod() const { return sqrt( x * x + y * y ); }
};
int N, M, Size;
point A[ Maxn ], B[ Maxn ], Base, Stack[ Maxn ];
int Cmp( double x, double y ) {
if( fabs( x - y ) <= Eps ) return 0;
if( x - y > Eps ) return 1;
return -1;
}
bool Cmp1( point X, point Y ) {
return Cmp( ( X - Base ) * ( Y - Base ), 0.0 ) == 1 ||
( Cmp( ( X - Base ) * ( Y - Base ), 0.0 ) == 0 && Cmp( ( X - Base ).Mod(), ( Y - Base ).Mod() ) == -1 );
}
void Graham( point *A, int &N ) {
for( int i = 2; i <= N; ++i )
if( Cmp( A[ i ].y, A[ 1 ].y ) == -1 ||
( Cmp( A[ i ].y, A[ 1 ].y ) == 0 && Cmp( A[ i ].x, A[ 1 ].x ) == -1 ) )
swap( A[ i ], A[ 1 ] );
Base = A[ 1 ]; sort( A + 2, A + N + 1, Cmp1 );
Size = 1; Stack[ 1 ] = A[ 1 ];
for( int i = 2; i <= N; ++i ) {
for( ; Size > 1 && Cmp( ( A[ i ] - Stack[ Size - 1 ] ) * ( Stack[ Size ] - Stack[ Size - 1 ] ), 0.0 ) >= 0; --Size );
Stack[ ++Size ] = A[ i ];
}
N = Size; for( int i = 1; i <= N; ++i ) A[ i ] = Stack[ i ];
return;
}
inline int Pre( int N, int x ) { return ( x - 1 < 1 ) ? N : x - 1; }
inline int Suc( int N, int x ) { return ( x + 1 > N ) ? 1 : x + 1; }
inline double GetDis( point A, point B, point C ) {
point D = C + point( -( B - A ).y, ( B - A ).x );
double K = ( D - A ) * ( C - A ) / ( ( C - B ) * ( D - B ) );
if( Cmp( K, 0.0 ) <= 0 ) return min( ( C - A ).Mod(), ( C - B ).Mod() );
point O = A * ( 1.0 / ( K + 1.0 ) ) + B * ( K / ( K + 1.0 ) );
return ( C - O ).Mod();
}
double Dis( point *A, int N, point *B, int M ) {
int i1 = 1, i2;
for( i2 = 1; i2 <= M; ++i2 )
if( Cmp( ( A[ Suc( N, i1 ) ] - A[ i1 ] ) * ( B[ i2 ] - B[ Pre( M, i2 ) ] ), 0.0 ) == 1 &&
Cmp( ( A[ Suc( N, i1 ) ] - A[ i1 ] ) * ( B[ Suc( M, i2 ) ] - B[ i2 ] ), 0.0 ) <= 0 )
break;
double Ans = GetDis( A[ Suc( N, i1 ) ], A[ i1 ], B[ i2 ] );
if( Cmp( ( A[ Suc( N, i1 ) ] - A[ i1 ] ) * ( B[ Suc( M, i2 ) ] - B[ i2 ] ), 0.0 ) == 0 ) {
i2 = Suc( M, i2 );
Ans = min( Ans, GetDis( A[ Suc( N, i1 ) ], A[ i1 ], B[ i2 ] ) );
}
for( ++i1; i1 <= N; ++i1 ) {
for( ; Cmp( ( A[ Suc( N, i1 ) ] - A[ i1 ] ) * ( B[ i2 ] - B[ Pre( M, i2 ) ] ), 0.0 ) <= 0 ||
Cmp( ( A[ Suc( N, i1 ) ] - A[ i1 ] ) * ( B[ Suc( M, i2 ) ] - B[ i2 ] ), 0.0 ) == 1; i2 = Suc( M, i2 ) );
Ans = min( Ans, GetDis( A[ Suc( N, i1 ) ], A[ i1 ], B[ i2 ] ) );
if( Cmp( ( A[ Suc( N, i1 ) ] - A[ i1 ] ) * ( B[ Suc( M, i2 ) ] - B[ i2 ] ), 0.0 ) == 0 ) {
i2 = Suc( M, i2 );
Ans = min( Ans, GetDis( A[ Suc( N, i1 ) ], A[ i1 ], B[ i2 ] ) );
}
}
return Ans;
}
int main() {
scanf( "%d%d", &N, &M );
for( ; N != 0 || M != 0; scanf( "%d%d", &N, &M ) ) {
for( int i = 1; i <= N; ++i ) scanf( "%lf%lf", &A[ i ].x, &A[ i ].y );
for( int i = 1; i <= M; ++i ) scanf( "%lf%lf", &B[ i ].x, &B[ i ].y );
Graham( A, N ); Graham( B, M );
printf( "%.6lf\n", min( Dis( A, N, B, M ), Dis( B, M, A, N ) ) );
}
return 0;
}