# Projecting 3D point to 2D image coordinates

I am trying to replicate the projections of a 3D point to 2D using the pinhole camera model and the formula $$x = K * [R|-RC] * X$$, where $$x = s\begin{bmatrix} u\\ v\\ 1 \end{bmatrix}$$, $$K$$ is the intrinsic matrix: $$\begin{bmatrix} 35 & 0 & 810\\ 0 & 35 & 540 \\ 0 & 0 & 1 \end{bmatrix}$$ as the focal length of my ideal pinhole camera is $$35mm$$ and the image size is $$1620px$$ by $$1080px$$.

My camera is looking straight in the middle of the object and therefore has a translation vector $$C=\begin{pmatrix} 0 & 0 & -distance \end{pmatrix}$$ and $$R = I_{3}$$.

My first question is, if I know the width of the world the camera is capturing, which would be e.g. $$200mm$$ can I use the formula mentioned here, to receive the missing measurements (distance and width of world)?

After obtaining the distance of the camera to the object I can complete $$C=\begin{pmatrix} 0 & 0 & -448.72mm \end{pmatrix}$$ and should be able to calculate $$x = P*X$$, so $$\begin{pmatrix} 363463.2 & 242308.8 & 448.72 \end{pmatrix} = \begin{bmatrix} 35 & 0 & 810 & 363463\\ 0 &35 & 540 &242308 \\ 0 &0 &1 & 448.72 \end{bmatrix} * \begin{pmatrix} 0 & 0 & 0 & 1 \end{pmatrix}$$ which correctly gives me the center of the image plane if I divide the first two rows of $$x$$ by the last.

BUT, now I wanted to project the top left corner of the world which would be $$\begin{pmatrix} -294.87 / 2 & 200 / 2 & 0 \end{pmatrix}$$. This should give me $$\begin{pmatrix} 0 & 0 \end{pmatrix}$$ as the image coordinates as the coordinate system starts in the top left corner. But unfortunately it does not. What am I doing wrong.