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...read Section 2
String testPoint(verts, x, y){ int N = lengthof(verts); int cnt = 0; double x2 = random()*1000+1000; double y2 = random()*1000+1000; for(int i = 0; i<N; i++){ if(distPointToSegment(verts[i],verts[(i+1)%N],x,y) == 0) return "BOUNDARY"; if(segmentsIntersect((verts[i],verts[(i+1)%N],{x,y},{x2,y2})) cnt++; } if(cnt%2 == 0)return "EXTERIOR"; else return "INTERIOR"; } TVTower(SRM 183) Requires: Finding a Circle From 3 Points In problems like this, the first thing to figure out is what sort of solutions might work. In this case, we want to know what sort of circles we should consider. If a circle only has two points on it, then, in most cases, we can make a slightly smaller circle, that still has those two points on it. The only exception to this is when the two points are exactly opposite each other on the circle. Three points, on the other hand, uniquely define a circle, so if there are three points on the edge of a circle, we cannot make it slightly smaller, and still have all three of them on the circle. Therefore, we want to consider two different types of circles: those with two points exactly opposite each other, and those with three points on the circle. Finding the center of the first type of circle is trivial — it is simply halfway between the two points. For the other case, we can use the method for Finding a Circle From 3 Points. Once we find the center of a potential circle, it is then trivial to find the minimum radius. int[] x, y; int N; double best = 1e9; void check(double cx, double cy){ double max = 0; for(int i = 0; i< N; i++){ max = max(max,dist(cx,cy,x[i],y[i])); } best = min(best,max); } double minRadius(int[] x, int[] y){ this.x = x; this.y = y; N = lengthof(x); if(N==1)return 0; for(int i = 0; i<N; i++){ for(int j = i+1; j<N; j++){ double cx = (x[i]+x[j])/2.0; double cy = (y[i]+y[j])/2.0; check(cx,cy); for(int k = j+1; k<N; k++){ //center gives the center of the circle with //(x[i],y[i]), (x[j],y[j]), and (x[k],y[k]) on //the edge of the circle. double[] c = center(i,j,k); check(c[0],c[1]); } } } return best; } Satellites (SRM 180) Requires: LinePoint Distance This problem actually requires an extension of the LinePoint Distance method discussed previously. It is the same basic principle, but the formula for the cross product is a bit different in three dimensions. The first step here is to convert from spherical coordinates into (x,y,z) triples, where the center of the earth is at the origin. double x = sin(lng/180*PI)*cos(lat/180*PI)*alt; double y = cos(lng/180*PI)*cos(lat/180*PI)*alt; double z = sin(lat/180*PI)*alt;Now, we want to take the cross product of two 3D vectors. As I mentioned earlier, the cross product of two vectors is actually a vector, and this comes into play when working in three dimensions. Given vectors (x_{1},y_{1},z_{1}) and (x_{2},y_{2},z_{2}) the cross product is defined as the vector (i,j,k) where i = y_{1}z_{2}  y_{2}z_{1}; j = x_{2}z_{1}  x_{1}z_{2}; k = x_{1}y_{2}  x_{2}y_{1};Notice that if z_{1} = z_{2} = 0, then i and j are 0, and k is equal to the cross product we used earlier. In three dimensions, the cross product is still related to the area of the parallelogram with two sides from the two vectors. In this case, the area of the parallelogram is the norm of the vector: sqrt(i*i+j*j+k*k). Hence, as before, we can determine the distance from a point (the center of the earth) to a line (the line from a satellite to a rocket). However, the closest point on the line segment between a satellite and a rocket may be one of the end points of the segment, not the closest point on the line. As before, we can use the dot product to check this. However, there is another way which is somewhat simpler to code. Say that you have two vectors originating at the origin, S and R, going to the satellite and the rocket, and that X represents the norm of a vector X. Then, the closest point to the origin is R if R^{2} + RS^{2} ≤ S^{2} and it is S if S^{2} + RS^{2} ≤ R^{2} Naturally, this trick works in two dimensions also. Further Problems Once you think you've got a handle on the three problems above, you can give these ones a shot. You should be able to solve all of them with the methods I've outlined, and a little bit of cleverness. I've arranged them in what I believe to ascending order of difficulty. ConvexPolygon (SRM 166) Requires: Polygon Area Surveyor (TCCC '04 Qual 1) Requires: Polygon Area Travel (TCI '02) Requires: Dot Product Parachuter (TCI '01 Round 3) Requires: Point In Polygon, LineLine Intersection PuckShot (SRM 186) Requires: Point In Polygon, LineLine Intersection ElectronicScarecrows (SRM 173) Requires: Convex Hull, Dynamic Programming Mirrors (TCI '02 Finals) Requires: Reflection, LineLine Intersection Symmetry (TCI '02 Round 4) Requires: Reflection, LineLine Intersection Warehouse (SRM 177) Requires: LinePoint Distance, LineLine Intersection The following problems all require geometry, and the topics discussed in this article will be useful. However, they all require some additional skills. If you got stuck on them, the editorials are a good place to look for a bit of help. If you are still stuck, there has yet to be a problem related question on the round tables that went unanswered. DogWoods (SRM 201) ShortCut (SRM 215) SquarePoints (SRM 192) Tether (TCCC '03 W/MW Regional) TurfRoller (SRM 203) Watchtower (SRM 176)

