I'm an Engineering student. All of the sudden I need to know about "Family of Lines" which is a topic in "Projective Geometry". I've found the old book of Veblen & Young (and two other books) but I unfortunately this is barley explained in the last chapter of the book. This topic is very important for me and I know It'll take time and effort for me to grasp the gist of this topic but I feel I'm not dealing with it in the right way and the books are not helping. Can anyone help me through?
BTW, What is "variety of rank 2" or how 3 skew lines define a "Regulus"... I've got some questions!!!
The first question: rank of a variety is defined to be the degree of certain dual object. For a planar curve, the rank is defined to be the degree of its tangent line equation, which is the dual object of the curve in the plane. The general definition relies on the notion of dual relation between points and hyperplanes.
The second question: In Plucker line coordinates $$ (l,\bar l), l,\bar l\in\mathbb R^3, l\cdot \bar l=0, \|l\|^2+\|\bar l\|^2\neq 0. $$ the set of lines $(l,\bar l)$ that intersect the three given lines $(l_i,\bar l_i),i=1,2,3$, is given by three linear homogeneous equations $$ l\cdot \bar l_i+\bar l\cdot l_i=0, i=1,2,3. $$ and therefore, it defines a $2$-plane in the five dimensional projective space $\mathbb RP^5$. Notice that the set of all lines is not the whole of $\mathbb RP^5$, but is a quadric given by: $$ l\cdot \bar l=0. $$ which is called the Klein quadric. So the lines that intersect three given skew lines is represented by the intersection curve of the aforementioned $2$-plane with the Klein quadric, which is also quadratic by Bezout theorem, i.e., a $1$-parameter line series. Going back from line coordinates to point coordinates in $\mathbb RP^3$, the line series gives a ruled quadratic surface. The three lines form one of the two family of rulings on this quadratic surface.
Four skew lines have only $2$ lines that intersect all of them, and in turns defines a line congruence.
These parts of math education is lost from modern college program, and is a pain in the axx to self study. And you should not expect new applications. Its already been done. I will suggest you to simply walk away and pick a modern robotics topic such as planning and control, sensor fusion, machine learning.