can we insert elements[inserted_id] and remove elements[removed_id] without breaking independence?

can we insert elements[id] without breaking independence?

can we insert elements[id] without breaking independence?

centroid of a (possibly non-convex) polygon, assuming that the coordinates are listed in a clockwise or counterclockwise fashion. Note that the centr

O(nlogn)

O(nlogn)

returns center of circle passing through three non-colinear and co-planer points a, b and c

returns area of intersection between two circles

5 - outside and do not intersect 4 - intersect outside in one point 3 - intersect in 2 points 2 - intersect inside in one point 1 - inside and do not

0 if outside, 1 if on circumference, 2 if inside circle

0 if outside, 1 if on circumference, 2 if inside circle

returns the sphere passing through four points

compare two points by their orthogonal projection on this line a projection point comes before another if it comes first according to vector v

compare points by their projection on the line so you can sort points on the line using this

combine node l and r to form t by updating corresponding queries

p(g(x)) O(nlogn * sqrt(nlogn))

p(g(x)) O(n^2 logn)

radius and height

given a convex polygon p, and a line ab and the top vertex of the polygon returns the intersection of the line with the polygon it returns the indices

From three points P, Q, R on the plane

count of numbers in [l, r] equal to k

number of length n sequences with elements in [0, 2^k) such that its basis has size r same as the above function

ways to select k vectors of length n having basis size r

number of length i sequences with elements in [0, 2^k) such that no subset has xor 1 for all 1 <= i <= n O(n logn)

number of subsets having xor < x

building a 3D convex hull

radius and height

positive: p3 in the same direction

depth in the rep. tree

recursively search all faces that should be removed from the convex hull