Determine if a polygon is convex

Question

Write a program to determine if the input polygon is convex. The polygon is specified with one line containing N, the number of vertices, then N lines containing the x and y coordinates of each vertex. The vertices will be listed clockwise starting from an arbitrary vertex.

example 1

input

4
0 0
0 1
1 1
1 0

output

convex

example 2

input

4
0 0
2 1
1 0
2 -1

output

concave

example 3

input

8
0 0
0 1
0 2
1 2
2 2
2 1
2 0
1 0

output

convex

x and y are integers, N<1000, and |x|,|y|<1000. You may assume that the input polygon is simple (none of the edges cross, only 2 edges touch each vertex). Shortest program wins.

Answer 1

J, 105

echo>('concave';'convex'){~1=#=(o.1)([:>-.~)(o.2)|3([:-/12 o.-@-/@}.,-/@}:)\(,2&{.)j./"1}.0&".;._2(1!:1)3

Passes all three tests above.

Edit: (111->115) Handle co-linear points by eliminating angles of pi. Gained a few characters elsewhere.

Edit: (115->105) Less dumb.

Explanation for the J-impaired:

• (1!:1)3 read STDIN to EOF. (I think.)
• 0&".;._2 is a nice idiom for parsing this kind of input.
• j./"1}. lop off first line of input (N 0) and convert pairs to complexes.
• (,2&{.) tack first two points onto the end of the list.
• 3(f)\ applies f to sliding window of length 3 (3 points for an angle)
• [:-/12 o.-@-/@}.,-/@}: is a verb that transforms each 3 points into an angle between -pi and pi.
  • -@-/@}.,-/@}: produces (p1 - p2),(p3 - p2). (Recall that these are complexes.)
  • 12 o. gives an angle for each complex.
  • [:-/(...) gives the difference of the two angles.
• (o.1)([:>-.~)(o.2)| mod 2 pi, eliminate angles of pi (straight segments), and compare to pi (greater than, less than, doesn't matter unless the points are supposed to be wound in one direction).
• 1=#= if all those comparison result 1 or 0 (With self-classify. This seems dumb.)
• echo>('concave';'convex'){~ print convex.

Answer 2

Python - 149 chars

p=[map(int,raw_input().split())for i in[0]*input()]*2
print'ccoonncvaevxe'[all((a-c)*(d-f)<=(b-d)*(c-e)for(a,b),(c,d),(e,f)in zip(p,p[1:],p[2:]))::2]

Answer 3

Ruby 1.9, 147 133 130 124 123

gets
puts ($<.map{|s|s.split.map &:to_i}*2).each_cons(3).any?{|(a,b),(c,d),(e,f)|(e-c)*(d-b)<(d-f)*(a-c)}?:concave: :convex

Answer 4

scala: 297 chars

object C{class D(val x:Int,val y:Int)
def k(a:D,b:D,c:D)=(b.y-a.y)*(c.x-b.x)>=(c.y-b.y)*(b.x-a.x) 
def main(a:Array[String]){val s=new java.util.Scanner(System.in)
def n=s.nextInt
val d=for(x<-1 to n)yield{new D(n,n)}print((true/:(d:+d.head).sliding(3,1).toList)((b,t)=>b&&k(t(0),t(1),t(2))))}}