Determine if a polygon is convex
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.
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]
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