Abstract: The paper uses, for the first time, the graphic method to systematically express the
existence of three forms of line pencils in the projective plane. The line pencils show a certain
pattern in the formation of second-order curve: the two pencils of lines with infinite points as the
base form a hyperbola; the two pencils of lines with both finite and infinite points as the base form
a parabola; the two pencils of lines with both finite points as the base form a hyperbola, a parabola
or an ellipse upon rotating any one of them. These conclusions lay a theoretical foundation for the
realization of the transition between the second-order curve.

Key words: projective plane, the curve of second order, degenerate type