[Reference] Projection of point onto line using distance from two reference points

January 09, 2015

Consider two known points A, and B, an unknown point C. If we know the distances between C and the other two points, we can recover \(\pi_{AB}(C)\), the projection of C onto the line AB.

The distance between \(\pi_{AB}(C)\) and A is

\[ \begin{align} d(A, \pi_{AB}(C)) = d(A,B) (1+g) / 2 \text{, where} \\ g &= \left ( d(A,C)^2 - d(B,C)^2 \right ) / d(A,B)^2 \end{align} \]

This can be derived using the pythagorean theorem and fact triangles AC\pi(C) and BC\pi(C) share a side.

This can be used to derive an expression for \(\pi_{AB}(C)\) using a weighted sum of A and B:

\[ \begin{align} \pi_{AB}(C) &= (A(1-g) + B(1+g) ) / 2 \end{align} \]
Posted by Kyle Simek
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