PROBLEM LINK:DIFFICULTY:HARD PREREQUISITES:Circumcenter PROBLEM:Given two circles in 3 dimensions (denoted using 3 points on each circle) find whether they are entangled or not. EXPLANATION:Finding circumcenter of a circle given 3 pts on that circle > Let three points be A,B,C Now, make two vectors AB(joining A and B) and AC(similarly) ABP > perpendicular bisectors of AB ACP > perpendicular bisectors of AC Circumcenter would be meeting point of ABP and ACP Now translate the axis to center of first circle. Rotate the axis in such a manner that first circle comes in xyplane. Now get the plane of second circle in the form ax+by+cz=d We can get the line of intersection of this plane with xy plane(plane of first circle) by putting z=0. So now intersecting line becomes ax+by=d; Now first circle is centered at origin in xy plane; So equation of first circle will be x^2 + y^2 = r^2 ; Now we have to find the intersection of this circle and the intersecting line. We will be getting a quadratic equation on solving this> 2) equal roots > first circle intersect plane of second circle at only one point[Not Possible]. We have been given both rings don't touch each other, therefore if first ring only touches the second plane, its not possible to entangle. 3) distinct roots > Let a,b are the two points that came from solving the equations of first circle and second plane. For entangling two rings > one point(a,b) should lie inside the second ring and other should lie outside the second ring. RELATED PROBLEMS:CODEsetter.cpp By Sukhjashan Singh
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asked 07 Feb '14, 06:12
