Check whether two 2D lines are coincident.

To check whether two 2D lines are coincident, the distance of $C$ and $D$ to $AB$ need to be calculated. For a coincident line both need to be (almost) zero.

The z component of the cross vector can be used for this quite elegantly.

Note: The cross vector is noted as an x e.g. $A$ x $B$

To check whether the two lines $AB$ and $CD$ are coincident, the z-component of two cross vectors need to be calculated. Both calculate the distance of the points on line $CD$ to $AB$ .

$AB$ x $AC$ is the area of the parallelogram $AB AC$ , dividing it with the length of $AB$ ( $|AB|$ ) gives the shortest distance of point $C$ to line $AB$ .

$AB$ x $AD$ gives the area of the parallelogram $AB AD$

After dividing with $|AB|$ this is also the shortest distance of $D$ to line $AB$ . If both distances are small enough the lines are coincident.

$dist_{abc} = \overline{AB}_x·\overline{AC}_y - \overline{AB}_y·\overline{AC}_x$
$dist_{abd} = \overline{AB}_x·\overline{AD}_y - \overline{AB}_y·\overline{AD}_x$

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Stuff I need to remember

Check whether two 2D lines are coincident.

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