Thank you very much for attending. General equation for a plane in 3D: ! Conf. We use cookies to help provide and enhance our service and tailor content and ads.
In this technique 4-connected pixels are used as shown in the figure. Name each intersection point of the polygon.
Note that the shared endpoint between sequential edges does not count as a non-simple intersection point, and the intersection test routine must check for that. By continuing you agree to the use of cookies. 17-th Ann. Both methods are used in SVG, where the value of attribute 'fill-rule' is either "nonzero" or "evenodd". Next, do surgery at these new vertices to remove the crossover. Clarkson & P.W. Finally, note that this method can be modified to the case when Q is convex and P is non-convex polygon. Algorithm of Weiler-Atherton Polygon Clipping Step 1: . One wants to find intersections between the sets, but not within the same set; that is, red-blue intersections, but not red-red or blue-blue ones. If so, both vertices will be those of the intersection.
There are basically two ways of conducting PIP in Shapely: Notice: even though we are talking here about Point in Polygon
Sometimes an application needs to find the set of intersection points for a collection of many line segments. Does anyone help me with the fast algorithm to determine the intersection of a polygon (rotated rectangle) and a line (definite by 2 points)? Polygon is an ordered list of vertices as shown in the following figure. This algorithm picks a point inside an object and starts to fill until it hits the boundary of the object. In addition to this, we are also putting pixels in diagonals so that entire area of the current pixel is covered. {\displaystyle 2\pi } objects, such as the PIP as weâre interested here. In this technique 8-connected pixels are used as shown in the figure. There are two methods by which we can identify whether particular point is inside an object or outside. If the point is on the outside of the polygon the ray will intersect its edge an even number of times. To do this, one must also output the edges that persist in a linked list, whose connected sublists will be the new simple polygons. Further, when an intersection-event on EQ is processed during the sweep, then it causes a re-ordering of the SL list, and the intersection is also added to the output list .
one of the most typical applications where Point in Polygon (PIP) query When two such planar graphs are overlaid (or superimposed), then their combined graph defines a subdivision refinement of each one. From both ends, the number of interaction points is odd, so the point is considered within the object. Also, the more complicated optimal algorithms are which is the same as the simple brute-force one.
Further, we want to reduce the clutter// accompanying the essential sweep line algorithm.//===================================================================, #define FALSE 0#define TRUE 1#define LEFT 0#define RIGHT 1extern void qsort(void*, unsigned, unsigned, int(*)(const void*,const void*));// xyorder(): determines the xy lexicographical order of two points// returns: (+1) if p1 > p2; (-1) if p1 < p2; and 0 if equal int xyorder( Point* p1, Point* p2 ) { // test the x-coord first if (p1->x > p2->x) return 1; if (p1->x < p2->x) return (-1); // and test the y-coord second if (p1->y > p2->y) return 1; if (p1->y < p2->y) return (-1); // when you exclude all other possibilities, what remains is... return 0; // they are the same point }// isLeft(): tests if point P2 is Left|On|Right of the line P0 to P1.// returns: >0 for left, 0 for on, and <0 for right of the line.// (see Algorithm 1 on Area of Triangles)inline floatisLeft( Point P0, Point P1, Point P2 ){ return (P1.x - P0.x)*(P2.y - P0.y) - (P2.x - P0.x)*(P1.y - P0.y);}//===================================================================, // EventQueue Class// Event element data structtypedef struct _event Event;struct _event { int edge; // polygon edge i is V[i] to V[i+1] int type; // event type: LEFT or RIGHT vertex Point* eV; // event vertex};int E_compare( const void* v1, const void* v2 ) // qsort compare two events{ Event** pe1 = (Event**)v1; Event** pe2 = (Event**)v2; return xyorder( (*pe1)->eV, (*pe2)->eV );}// the EventQueue is a presorted array (no insertions needed)class EventQueue { int ne; // total number of events in array int ix; // index of next event on queue Event* Edata; // array of all events Event** Eq; // sorted list of event pointerspublic: EventQueue(Polygon P); // constructor ~EventQueue(void) // destructor { delete Eq; delete Edata;} Event* next(); // next event on queue};// EventQueue RoutinesEventQueue::EventQueue( Polygon P ){ ix = 0; ne = 2 * P.n; // 2 vertex events for each edge Edata = (Event*)new Event[ne]; Eq = (Event**)new (Event*)[ne]; for (int i=0; i < ne; i++) // init Eq array pointers Eq[i] = &Edata[i]; // Initialize event queue with edge segment endpoints for (int i=0; i < P.n; i++) { // init data for edge i Eq[2*i]->edge = i; Eq[2*i+1]->edge = i; Eq[2*i]->eV = &(P.V[i]); Eq[2*i+1]->eV = &(P.V[i+1]); if (xyorder( &P.V[i], &P.V[i+1]) < 0) { // determine type Eq[2*i]->type = LEFT; Eq[2*i+1]->type = RIGHT; } else { Eq[2*i]->type = RIGHT; Eq[2*i+1]->type = LEFT; } } // Sort Eq[] by increasing x and y qsort( Eq, ne, sizeof(Event*), E_compare );}Event* EventQueue::next(){ if (ix >= ne) return (Event*)0; else return Eq[ix++];}//===================================================================, // SweepLine Class// SweepLine segment data structtypedef struct _SL_segment SLseg;struct _SL_segment { int edge; // polygon edge i is V[i] to V[i+1] Point lP; // leftmost vertex point Point rP; // rightmost vertex point SLseg* above; // segment above this one SLseg* below; // segment below this one};// the Sweep Line itselfclass SweepLine { int nv; // number of vertices in polygon Polygon* Pn; // initial Polygon BBT Tree; // balanced binary treepublic: SweepLine(Polygon P) // constructor { nv = P.n; Pn = &P; } ~SweepLine(void) // destructor { Tree.freetree();} SLseg* add( Event* ); SLseg* find( Event* ); int intersect( SLseg*, SLseg* ); void remove( SLseg* );};SLseg* SweepLine::add( Event* E ){ // fill in SLseg element data SLseg* s = new SLseg; s->edge = E->edge; // if it is being added, then it must be a LEFT edge event // but need to determine which endpoint is the left one Point* v1 = &(Pn->V[s->edge]); Point* v2 = &(Pn->V[s->edge+1]); if (xyorder( v1, v2) < 0) { // determine which is leftmost s->lP = *v1; s->rP = *v2; } else { s->rP = *v1; s->lP = *v2; } s->above = (SLseg*)0; s->below = (SLseg*)0; // add a node to the balanced binary tree Tnode* nd = Tree.insert(s); Tnode* nx = Tree.next(nd); Tnode* np = Tree.prev(nd); if (nx != (Tnode*)0) { s->above = (SLseg*)nx->val; s->above->below = s; } if (np != (Tnode*)0) { s->below = (SLseg*)np->val; s->below->above = s; } return s;}SLseg* SweepLine::find( Event* E ){ // need a segment to find it in the tree SLseg* s = new SLseg; s->edge = E->edge; s->above = (SLseg*)0; s->below = (SLseg*)0; Tnode* nd = Tree.find(s); delete s; if (nd == (Tnode*)0) return (SLseg*)0; return (SLseg*)nd->val;}void SweepLine::remove( SLseg* s ){ // remove the node from the balanced binary tree Tnode* nd = Tree.find(s); if (nd == (Tnode*)0) return; // not there // get the above and below segments pointing to each other Tnode* nx = Tree.next(nd); if (nx != (Tnode*)0) { SLseg* sx = (SLseg*)(nx->val); sx->below = s->below; } Tnode* np = Tree.prev(nd); if (np != (Tnode*)0) { SLseg* sp = (SLseg*)(np->val); sp->above = s->above; } Tree.remove(nd); // now can safely remove it delete s;}// test intersect of 2 segments and return: 0=none, 1=intersectint SweepLine::intersect( SLseg* s1, SLseg* s2){ if (s1 == (SLseg*)0 || s2 == (SLseg*)0) return FALSE; // no intersect if either segment doesn't exist // check for consecutive edges in polygon int e1 = s1->edge; int e2 = s2->edge; if (((e1+1)%nv == e2) || (e1 == (e2+1)%nv)) return FALSE; // no non-simple intersect since consecutive // test for existence of an intersect point float lsign, rsign; lsign = isLeft(s1->lP, s1->rP, s2->lP); // s2 left point sign rsign = isLeft(s1->lP, s1->rP, s2->rP); // s2 right point sign if (lsign * rsign > 0) // s2 endpoints have same sign relative to s1 return FALSE; // => on same side => no intersect is possible lsign = isLeft(s2->lP, s2->rP, s1->lP); // s1 left point sign rsign = isLeft(s2->lP, s2->rP, s1->rP); // s1 right point sign if (lsign * rsign > 0) // s1 endpoints have same sign relative to s2 return FALSE; // => on same side => no intersect is possible // the segments s1 and s2 straddle each other return TRUE; // => an intersect exists}//===================================================================, // simple_Polygon(): test if a Polygon is simple or not// Input: Pn = a polygon with n vertices V[]// Return: FALSE(0) = is NOT simple// TRUE(1) = IS simpleintsimple_Polygon( Polygon Pn ){ EventQueue Eq(Pn); SweepLine SL(Pn); Event* e; // the current event SLseg* s; // the current SL segment // This loop processes all events in the sorted queue // Events are only left or right vertices since // No new events will be added (an intersect => Done) while (e = Eq.next()) { // while there are events if (e->type == LEFT) { // process a left vertex s = SL.add(e); // add it to the sweep line if (SL.intersect( s, s->above)) return FALSE; // Pn is NOT simple if (SL.intersect( s, s->below)) return FALSE; // Pn is NOT simple } else { // processs a right vertex s = SL.find(e); if (SL.intersect( s->above, s->below)) return FALSE; // Pn is NOT simple SL.remove(s); // remove it from the sweep line } } return TRUE; // Pn IS simple}//===================================================================, Brad Appleton, C++ code for an AVL balanced Tree, see: www.bradapp.com, oopweb.com/Algorithms/Documents/AvlTrees/VolumeFrames.html, www.bradapp.com/ftp/src/libs/C++/AvlTrees.html (1997), I.J.
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