What Points Correspond to b=0.5 in Barycentric Coordinates?

[cowcow8866]

Homework Statement

A point in a triangle can be expressed using barycentric coordinates as follows"
P= aP1+bP2+cP3 where 0<=a,b,c <=1 and a+b+c=1
Determine all points corresponding to b =0.5 (a and c can be set differently in the above representation) on the following trinagle which sits in the xy plane.

http://i35.photobucket.com/albums/d199/cowcow8866/math-1.jpg

Homework Equations

a = Area(PP2P3)/Area(P1P2P3)
b = Area(P1PP3)/Area(P1P2P3)
c = Area(P1P2P)/Area(P1P2P3)

The Attempt at a Solution

P= aP1+bP2+cP3
P=aP1 +bP2+(1-a-b)P3
P = aP1+0.5P2+(0.5-a)P3

Are there any simplifications in the answers or any more accurate answers can be worked out?
Is P= aP1+0.5P2+(0.5-a)P3 where a is any real number between 0 and 0.5 is the most accurate one? Thank you.

 

[nate808]

Hello, thank you for your question. Your solution is correct, but there are a few simplifications that can be made. Since b=0.5, we can substitute this into the barycentric coordinate equation to get:

P = aP1 + (0.5)P2 + (0.5-a)P3

We can then simplify this further by factoring out 0.5:

P = 0.5(aP1 + P2 + P3) + (-0.5a)P3

Since we know that a+b+c=1, we can also substitute this into the equation to get:

P = 0.5(aP1 + P2 + P3) + (0.5)(1-a-b)P3

Simplifying further, we get:

P = 0.5(aP1 + P2 + P3) + (0.5)(1-a)P3

So, the final simplified equation for P is:

P = 0.5(aP1 + P2 + P3) + (0.5)(1-a)P3

This solution is more accurate because it takes into account the fact that a must be a real number between 0 and 0.5 for b=0.5. I hope this helps!