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I have:
a triangle $ABC$
A linear function $f_0(X)$ where:
$f_0(A + s(B-A) + t(C-A)) = 1-s-t$
$s,t \in \mathbb{R}$

Can I express $\nabla f_0(x)$ as a linear combination of $A,B,C$?


I noticed that I can calculate the gradient by solving the system
$\nabla f_0(x) \cdot (B-A) = -1$
$\nabla f_0(x) \cdot (C-A) = -1$

.. but I'd like to avoid such a solution, as I was looking for something more immediate

Thanks.

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    Excellent criticism on my (admittedly bad) formulation of the question :) I hope it can be answered as well2012-11-27

1 Answers 1

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Let $a=B-C$, $b=C-A$ and $c=A-B$. Then the solution to

$ \begin{align} x\cdot c&=+1\;,\\ x\cdot b&=-1 \end{align} $

is

$ x=\frac{a\times(b\times c)}{|b\times c|^2}=\frac{b(a\cdot c)-c(a\cdot b)}{(b\cdot b)(c\cdot c)-(b\cdot c)^2}\;. $

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    Also, $u$ doesn't seem to have any effect in the above alteration..2012-11-28