Camera calibration With OpenCV - tutorial

i am reading the very interesting tutorial https://docs.opencv.org/3.4.2/d4/d94/... and i find it difficult to understand the equations for tangential distortion

xdistorted=x+[2p1xy+p2(r^2+2x^2)]

ydistorted=y+[p1(r^2+2y^2)+2p2xy]

specifically i don't understand the terms p2(r^2+2x^2) and p1(r^2+2y^2) which seems to me should have always the same sign no matter what sign of p1 or p2, in all 4 areas of the Cartesian plane

could you please provide a link with more detailed explanations of these equations (and would be wonderful also with some special case example)?

also i find some difficulties in the camera matrix equations

i would write

((

in case of central misalignment between optics and sensor we keep the spatial Cartesian reference on the optics and consider the optical center on the sensor (cx,cy) with a misaligned image cartesian refernce

x(camerapixel)-cx(pixel)=fx(pixel)X(mm)/Z(mm) y(camerapixel)-cy(pixel)=fy(pixel)Y(mm)/Z(mm) Z=Z

where fx and fy are proportional to the focal lengths and ideally measure a pixel/mm normalized conversion at Z(mm)=1

))

but i am not sure how to precisely relate this to those equations