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Statement and convention. For two oriented planar lines whose unit directions differ, let $S(\lambda_1,\lambda_2)$ consist of rotations taking the first onto the second with its orientation. This is a horizontal spatial line. Conversely every horizontal spatial line arises this way. Directions opposite to one another are permitted; equal directions are excluded.

Source. Mathialagan, published 2021 PDF, pp. 14--15, Proposition 28. We propagate the sign convention in Proposition 27: the horizontal coordinate is $z=-\cot(\theta/2)$ for the counterclockwise angle $\theta\in(0,2\pi)$ from the first direction to the second.

Proof. Write $\lambda_i=a_i+\mathbb R v_i$, with unit oriented directions $v_i$. The rotation matrix is forced to be $R=R_\theta$. A motion with center $o$ sends $\lambda_1$ to $Ra_1+(I-R)o+\mathbb R v_2$. Thus precisely the allowed centers satisfy

$$(I-R)o\in a_2-Ra_1+\mathbb R v_2. \tag{1}$$

Since $I-R$ is invertible, (1) is an affine line of centers. The angle is fixed, so its spatial image is a horizontal line.

For the converse, let a horizontal spatial line have height $z$ and center projection $o_0+\mathbb R w$, where $w\ne0$. Select the unique $\theta$ with $z=-\cot(\theta/2)$, put $R=R_\theta$, and take $v_2$ to be a unit vector in the direction $(I-R)w$. Put $v_1=R^{-1}v_2$. They are unequal because $\theta\ne0$. Choose any $a_1$, and set $a_2=Ra_1+(I-R)o_0$. Then (1) is exactly the prescribed center line.

Application. For fixed $z$ and fixed $p$, the centers of rotations sending $p$ onto a prescribed oriented line $\lambda$ form $S(\lambda',\lambda)$, where $\lambda'$ is the unique line through $p$ with direction $R_\theta^{-1}v_\lambda$. As $z$ runs over $\mathbb R$, these give the horizontal ruling in Proposition 42.

Verification scope. Verified within the independently reviewed Theorem 3 chain, retained in the final review; this complete linear-algebra version of the source argument belongs to the living Theorem 3 record.

Bears on. Problem 661.