Waste less time on Facebook — follow Brilliant.
×

Computational Geometry

Computers are being used more and more to solve geometric problems, like modeling physical objects such as brains and bridges.

Shapes, Polygons, Trigonometry

       

Given an array of \(n\) points, write an algorithm to compute the closest pair points.

In the figure above the closest pair of points have been marked in red. Given a set of points shown below, let \(d\) be the distance between the closest pair of point. Then what is the value of \(\left\lfloor 100d \right\rfloor\)?

\[S=\{(4,0), (6,2), (9,4), (8,9), (1,2), (3,5), (3, 1)\}\]

Let the two closest points in the following set of coordinate points be \(P_1=(x_1,y_1)\) and \(P_2 = (x_2,y_2)\). What is the value of \(x_1+y_1+x_2+y_2\)?

Given a polygon \(P\) and a point \(p\) implement an algorithm to check whether the point lies inside the polygon or not.

The algorithm should output \(1\) if the point is inside the polygon, and \(0\) if it isn't. Consider the following pairs of polygon and point shown below, if \(l_{n}\) in the value output by the algorithm for the \(n\)th pair of polygon and point, what is the value of the string \([l_{1}l_{2}l_{3}l_{4}]\)?

\(P_1=[(-2, 1), (1, 3), (3, -3)], \\ p_1=(1,1)\)

\(P_2=[(2, 4), (4, 2), (6, 8), (8,6)], \\ p_2=(3,3)\)

\(P_3=[(-5,2), (8,2), (8,-4), (-5,-4)], \\ p_3=(-6,1)\)

\(P_4= [(-2,2), (2,2), (-6,0), (6,0)], \\ p_4=(3,1)\)

Details and assumptions

The points lying on the border of the polygon are considering to be inside the polygon

How many of the \(24\) triangles below enclose the origin within their perimeter?

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
((86, 74, 44), (38, 47, 68), (48, 92, 81))
((37, 59, 35), (-84, -71, -23), (57, -71, -34))
((-16, -22, -71), (-85, -74, 8), (-54, 46, 92))
((58, -60, 83), (27, -2, 77), (72, -47, 28))
((62, 17, 89), (-32, 84, 59), (61, -85, -9))
((8, 80, -80), (8, -73, -30), (40, -6, 36))
((1, -13, 34), (34, -12, -63), (62, 70, 22))
((-40, -20, -39), (63, -2, -28), (-86, 13, -60))
((54, -30, 0), (63, -77, -27), (71, 8, -5))
((-90, -35, 25), (-64, 22, 36), (-34, -21, 11))
((57, 54, -96), (-33, 53, 93), (91, -23, 32))
((-17, -73, -85), (-71, 93, 83), (-36, -59, -48))
((-49, -29, 18), (-60, 55, 9), (37, -78, -92))
((-69, 21, 60), (98, -81, -49), (89, 31, -87))
((-5, -97, -45), (29, 36, -98), (70, -39, -10))
((-74, 89, -64), (-8, 36, 59), (-5, 4, -29))
((74, -50, -58), (-13, 60, 61), (98, 4, 99))
((-25, 75, 55), (84, 29, 34), (-50, -41, 84))
((93, -97, -88), (-50, 27, 9), (-100, 16, 26))
((86, -82, 36), (83, -52, -74), (43, 16, 4))
((-1, -17, 53), (-6, 56, 44), (15, -7, 29))
((-63, -7, -34), (60, 100, 91), (-13, 20, -65))
((-33, 64, -18), (28, 5, -81), (-78, 52, 41))
((21, -20, -76), (51, -85, -28), (-91, 90, 70))
×

Problem Loading...

Note Loading...

Set Loading...