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It's been a long time since I did computer graphics and algebra, and I don't remember how to correctly manipulate linear transformations.

My scenario: I have 3 points, $a,b,c \in \mathbb{R}^3$, forming a triangle embedded in $\mathbb{R}^3$. I need a transformation (i.e., matrix), that correctly takes the points to $xy$-plane. So, formally with all specifications:

Find $a'$, $b'$, and $c'$ such that: the vector $a'b'$ lies in the $y$-axis and $a'$ must be the origin. All three points must lie in $xy$-plane, and $\|a'b'\| = \|ab\|$, $\|a'c'\| = \|ac\|$, and $\|b'c'\| = \|bc\|$, so that $\operatorname{Area}(abc) = \operatorname{Area}(a'b'c')$. Remember that $a$, $b$, and $c$ form a valid triangle. I need the projection of $\operatorname{Triangle}(a,b,c)$ given this constraints.

I know I need to do a couple rotations and translations and find a combination of then, but can't figure out if it is possible to put all the conditions in a single matrix.

Thanks in advance.

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    Welcome to math.SE! I've edited your post a bit, hopefully making it a bit clearer, but if I accidentally changed what you wanted to say, I apologize, and please do feel free to fix it.2012-05-15
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    You must, in general, have a translation. The 3 points define a hyperplane. Either it passes through the origin or not. If not, no invertible linear transformation will map it to a hyperplane through the origin.2012-05-16

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