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CIRCLES - Editorial






Quadratic Equations
Basic Geometry


Given two circles in 3 dimensions (denoted using 3 points on each circle) find whether they are entangled or not.


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 xy-plane. 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->
1) imaginary roots -> line doesn't intersect the first circle[Not possible]. Two rings can't entangle because plane of second ring doesn’t intersect first(ring in xy plane) ring.

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.


Amrita '07


setter.cpp By Sukhjashan Singh

This question is marked "community wiki".

asked 07 Feb '14, 06:12

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edited 12 Feb '14, 03:29

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question asked: 07 Feb '14, 06:12

question was seen: 3,021 times

last updated: 12 Feb '14, 03:29