Decide whether two lines are parallel

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I have two lines which I´d like to know whether they are parallel or not in 3D space. Each line is defined using two points $(x_1,y_1,z_1)$,$(x_2,y_2,z_2)$. Important condition is that there should be a slight rotation threshold allowed, i.e. if the angle between the two lines is < 5 degrees then they are still parallel.

My idea is to compare the slopes of the two line segments somehow? Another way is to find the direction/normal of the line segment, and compare the two directions using the dot product

Any hints?

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3 Answers

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Create vectors pointing along each line by computing $(x_2,y_2,z_2)-(x_1,y_1,z_1)$ for both pairs. Make them into unit vectors by dividing them by their lengths. Call these two unit vectors $u$ and $v$.

Then you can use the inner product identity: $\langle u,v\rangle=\cos(\theta)$, where $\theta$ is the angle between the two vectors.

You want to create two small thresholds around 1 and -1. When the dot product is close to 1, this means that the vectors are very nearly pointing in the same direction, and when the dot product is nearly -1, they are very close to pointing in opposite directions. In both cases, they are "nearly parallel".

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Natural approaches would be to find two vectors $\vec{a}, \vec{b}$ on the lines ant then either check $\Big|\frac{\vec{a} \cdot \vec{b}}{|\vec{a}| |\vec{b}|}\Big| > \cos \epsilon$ or $\Big|\frac{\vec{a} \times \vec{b}}{|\vec{a}| |\vec{b}|}\Big| < \sin \epsilon$, where $\cdot$ is the dot product, $\times$ is the cross product and $\epsilon$ is the angle of error.

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You can also do this using direction cosines. A vector is defined as r= x i + y j + z k. The direction cosines of r are l=cosα= x/|r|, m=cosβ=y/|r| n=cosγ=z/|r|. Find direction cosines for both vectors and if the direction cosines are equal, then you've just proved the lines are parallel.

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