I'm using Mathematica 10 (but also have Mathematica 9) and I've tried entering triangle coordinates both in clockwise and counterclockwise directions but obtain only positive areas with the following:
tCounterClockwise = Triangle[{{0, 0, 0}, {1, 0, 0}, {0, 1, 0}}]
tClockwise = Triangle[{{0, 0, 0}, {0, 1, 0}, {1, 0, 0}}]
RegionMeasure[tCounterClockwise]
RegionMeasure[tClockwise]
Is there a way to obtain the signed area of a triangle?
My ultimate goal is to test for whether a point is in a 3D convex hull without having to use SignedRegionDistance (from Mathematica 10) and the TetGenLink package (from Mathematica 9) because neither is available for a CDF (which is the output format I need). My rationale is that if I can obtain the signed area and then arrange the triangle vertices on the surface of a convex hull all in a counterclockwise manner, then points outside the convex hull will all have the same sign of volume (which I can calculate) when the 3-simplex is formed with the triangle and that point. (The points to be tested are selected by the user in a CDF document and are not known ahead of time.)
If I'm all wet with that rationale for determining whether a point is in the convex hull or not, please comment on that, too. (This is also a plea for updating CDF to version 10 so I wouldn't have to figure this out.)
Thanks!