Computing oriented triangle area from coordinates

determinant projective-geometry richter-gerbert

Oriented triangle area: you’re in \(\mathbb{R}^2\), and you have the coordinates of the points of the triangle \(A\), \(B\), \(C\). The oriented area of this triangle can be computed using a determinant! (Reminder, orientation is counterclockwise. If the order of \(A\), \(B\), \(C\) is counterclockwise, the area will be \(\leq 0\)).

\[\text{area} = \frac{1}{2} \det\begin{pmatrix} a_{x} & b_{x} & c_{x}\\ a_{y} & b_{y} & c_{y}\\ 1 & 1 & 1 \end{pmatrix} \label{eq:det}\]

Where I learned this:

Jürgen Richter-Gebert’s Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry 2011, ISBN 9783642172854, page 9 in the softcover version.

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