some unclear stuff about barycentrid coords

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Hi, i'm trying to get a good understanding of baricentric coordinates (I've been writing a raytracer and hate to just write over equations, want to understand it). As far as I know, if you have a tri ABC and a point H inside the tri, then 1)ABH, ACH, BCH (ie the connect H and A, B, C, you get these three tris) - ABH+ACH+BCH = ABC. there's nothing that couldn't be understood (see right picture at http://mathworld.wolfram.com/BarycentricCoordinates.html, where alpha,beta and gamma stand for t1, t2 and t3) 2)but why t1 + t2 + t3 = ABC in the left picture? 3)also, I've seen that if you take vectors from AB and AC and gamma and beta, then (for the X for which gamma and beta, ie the coordinates, were computed) X = A + (B-A) * gamma + (C-A) * beta. I can see IFF beta + gamma <= 1 -> X lies inside ABC. But I solved it just for two numbers whose sum is in the range 0-1. But which properties of baricentric coordinates allow for this? I hope I wasn't too annoying

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mathworld.wolfram.com is not really good for learning. It is good for a)checking if you remembered something right, and b) you can post link to mathworld and then use same naming as mathworld, therefore avoiding long explanations like "Let P is a point, ....".

1: what is ABC in equation? Area of triangle?
2: There's aint no "t1 + t2 + t3 = ABC" on mathworld. Read equation "(1)" on mathworld. As about areas, just compute areas and see that "areas is proportional to cordinates".

3:
From definition, for triangle with vertices A1 A2 A3 , point with baricentric coordinates t1 t2 t3 have cartesian coordinates
P=(A1t1+A2t2+A3t3)/(t1+t2+t3)
If baricentric coordinates is normalized,
t1+t2+t3=1
Therefore,
t1=1-t2-t3
and
P=A1t1+A2t2+A3t3=A1(1-t2-t3)+A2t2+A3t3=A1+A2t2-A1t2+A3t3-A1t3=A1+(A2-A1)t2+(A3-A1)t3

In summary, try to write your questions clearer. Don't try to write in good english, rather try to make questions be clearer.

In this case, instead of truing to write question in words, write equation you don't understand, using naming from mathworld, and one question, "why?". So 3: becomes "P=A1+(A2-A1)t2+(A3-A1)t3 , why?"

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very nice reply, thanks especially for helping me out with the 3rd question.

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Rutin
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