CIRCLES - Editorial

cdcrft14
editorial
geometry
hard

#1

PROBLEM LINK:

Contest

DIFFICULTY:

HARD

PREREQUISITES:

Circumcenter
Quadratic Equations
Basic Geometry

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 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.

RELATED PROBLEMS:

Amrita '07

CODE

setter.cpp
By Sukhjashan Singh