Finite element method approximation

[Mechanics_student]

In the picture above, we have on the left a figure representing spatial element, and on the right a figure representing reference element.

The type of element used in the method here is a linear quadrilateral element.

What I understand is moving or mapping from the reference element domain ##\Omega^R## where it seems that the quadrilateral element looks square; I don't understand why it is a square, to the spatial element domain ##\Omega^e##, we use with such mapping the basis functions.

I think the basis functions ##N## is what is giving us the quadrilateral shape?

For approximating spatial coordinates, ##x_i## at ##\Omega^e## is approximated by the summation over the nodal values ##x_i^k##, (here ##k## symbolizes node and ##i## symbolizes direction), I'm not sure if ##x_i## will be the average coordinate of the individual element since we're summing over the coordinates of each node.

Clarification is appreciated. Thank you.