Is it correct to use the pinhole camera model to get the mapping of a 3D line to a line in an image? I've been working on a computer vision project in which I'm using the pinhole camera model to map one line of the world to one line in an image.

From previous research on the topic, I know that it is possible to map a point in the 3D world to a point in one image using the projection matrix obtained by calibrating the camera you're using to get the images of the 3D world. This mapping can be done as follows:

$ \lambda \begin{bmatrix}u \\ v \\ 1\end{bmatrix} = P \begin{bmatrix}X \\ Y \\ Z \\ 1\end{bmatrix}$,

in which $(u,v,1)$ are the coordinates of the point in the image, $(X,Y,Z,1)$ are the coordinates of the same point in the world and $P$ is the projection matrix.

A line in the image can be defined as:

$\begin{bmatrix}A &amp; B &amp; C\end{bmatrix}\begin{bmatrix}u \\ v \\ 1\end{bmatrix}$,

in which $(A,B,C)$ are the parameters used to describe the line. My research also taught me that perspective projections preserve lines, i.e., a straight line in the world will map to a straight line in the image.

Using the description of a line and the method used to get the mapping of points, I got this model:

$ \lambda \begin{bmatrix}A &amp; B &amp; C\end{bmatrix} \begin{bmatrix}u \\ v \\ 1\end{bmatrix} = \begin{bmatrix}A &amp; B &amp; C\end{bmatrix} P \begin{bmatrix}X \\ Y \\ Z \\ 1\end{bmatrix}$.

I've done some additional research, but so far I haven't found anything that can support this affirmation.

Is it correct to say that the mapping of a 3D line in the world to a line in an image of the world is done using the previous model?

Could anyone please point me in the right direction?

Thank you.
The assumption is correct, if(f) you project the 3d line into the rectified image. The idea of projecting a 3d-line to a pixel-line is most often used in a stereo setup with two cameras: if the left camera sees a features at a certain pixel, then you know that this object is on a 3d-ray that corresponds to this pixel. If you want to search this object in the right camera's image, you only have to search on the pixel line that corresponds to this 3d-line. 
Keywords are here epipolar geometry or epipolar line: 
https://www.robots.ox.ac.uk/~vgg/hzbook/hzbook1/HZepipolar.pdf