I'm reading about homogeneous coordinates from a computer vision textbook. I'm not sure what the author means by "instead of the four degrees that a 3D line truly has." I think in the first part, when the author writes, "A disadvantage of the endpoint representation for 3D lines is that it has too many degrees of freedom," the author means that the endpoints of a 3D line segment can be written as (x0, y0, z0) and (x1, y1, z1). Where do the four degrees come from?
The space of lines in 3-space is 4-dimensional. A direction vector for the line is a point on the unit sphere (up to sign, unless the line is oriented). We then specify a unique point on the line by choosing a point on the plane orthogonal to the direction vector. Thus, 2 parameters for a point on the sphere and 2 more parameters to specify the unique point.