What is the mathematical description of vanishing line in art?

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In drawing tutorials on how to give a drawing the '3d -effect', the concept of vanishing line is brought up. I don't exactly understand it, but it seems so that all parallel lines on the object that we draw intersect at a point on the vanishing line.

Could a more detailed description of what is going on with the vanishing line be given using projective geometry concepts in mathematics? My main doubts is how exactly is it that the vanishing lines allow us to give the 3D effect to the drawings we make. Like what mathematical idea is underlying it?