So I got the four points of a 2 dimensional square exactly like it is displayed on this image:
Note: black&white squares displays the corners.
(!) Note: (!) Keep in mind that the corner coordinates are already sorted (Point 1 -> TOP LEFT, Point 2 - TOP RIGHT, Point 3 - BOTTOM RIGHT, Point 4 - BOTTOM LEFT) in every case (!).
The first step was drawing a green line starting at Point 4 into direction of Point 1 and drawing a red line from Point 4 into direction of Point 3 as you can see in this image:
To my question: How can I determine the missing
blue lineto kind ofpose estimatethe squares position as being displayed in the image below?
Summary: What is given are only the 4 corner points of the square shape (x, y - coordinates)! Any help would be really appreciated and what I'd like to calculate is the end-point of the blue line (marked with a red cross).
You didn't mention how you were mapping 3D points and vectors to 2D. I will show you how to solve the problem assuming an orthogonal projection, as the calculation is significantly more straightforward than perspective projection.
If the two (unknown) vectors making up the side of your square in 3D are $\bar u$ and $\bar v$, and the two vectors in 2D are $u=P\bar u$ and $P \bar v$, we can try to recover $\bar u$ and $\bar v$ by using the fact that they must be orthogonal and equal length in $\mathbb{R}^3$, and that $$\bar u = R(u + \alpha \hat{z})$$ for some unknown rotation matrix $R$ and scalar $\alpha$, and likewise for $v$, $$\bar v = R(v + \beta \hat{z}).$$
Since \begin{align*} \bar u \cdot \bar v &= 0\\ \|\bar u\| &= \|\bar v\| \end{align*} we have that $$u\cdot v + \alpha \beta = 0$$ $$\|u\|^2 + \alpha^2 = \|v\|^2 + \beta^2.$$ This is a system of two nonlinear equation in two variables, which can be directly solved to yield $$\beta = \frac{-(u\cdot v)}{\alpha}$$ $$\alpha^4 + (\|u\|^2-\|v\|^2)\alpha^2 - (u\cdot v)^2 = 0$$ $$\alpha = \sqrt{ \frac{\|v\|^2-\|u\|^2 + \sqrt{(\|v\|^2-\|u\|^2)^2 + 4(u\cdot v)^2}}{2}}$$ (Note that the negative option in the quadratic formula is not possible since $\alpha$ must be real.)
So now you know $\alpha$ and $\beta$. We still don't know $R$, but the vector $\bar w = \bar u \times \bar v$ perpendicular to $\bar u$ and $\bar v$ in 3D satisfies \begin{align*}\bar u \times \bar v &= R\left([u+\alpha \hat{z}] \times [v + \beta \hat{z}]\right)\\ &= R\left(\alpha v^{\perp} - \beta u^{\perp} + \|u\|\|\|v\|\hat{z}\right), \end{align*} where $u^{\perp} = \hat{z}\times u$ is the ninety-degree counterclockwise rotation of $u$ in the plane. Therefore your desired blue vector is in the direction $$\alpha v^{\perp} - \beta u^{\perp}.$$
To get the right magnitude, we scale by $\|\bar u\|/\|\bar w\|$: $$\frac{\sqrt{\|u\|^2+\alpha^2}}{\sqrt{\|v^{\perp} - \beta u^{\perp}\|^2 + \|u\|^2\|\|v\|^2}}\left(v^{\perp} - \beta u^{\perp}\right).$$
You asked for "numerical values." To be honest, I don't know that I have any more skill at plug-and-chug than you do, but if it's useful:
Let's try $u = (1,-.2)$ and $v = (.5, .8)$, which very roughly corresponds to your picture above. We expect $w$ to point left and up in the plane. Indeed using the formulas we get $$\alpha = 0.52266$$ $$\beta = -0.650518$$ Moreover $$u^{\perp} = (.2, 1),\quad v^{\perp} = (-.8, .5)$$ so that $$v^{\perp} - \beta u^{\perp} = (-0.415882, 0.847919).$$ Normalizing this direction using the formula above gives $$(-0.504623, 1.02885)$$ for the coordinates of the blue vector.