Editing Dual number (section)

===Cycles===
Given two dual numbers {{mvar|p}} and {{mvar|q}}, they determine the set of {{mvar|z}} such that the difference in slopes ("Galilean angle") between the lines from {{mvar|z}} to {{mvar|p}} and {{mvar|q}} is constant. This set is a '''cycle''' in the dual number plane; since the equation setting the difference in slopes of the lines to a constant is a [[quadratic equation]] in the real part of {{mvar|z}}, a cycle is a [[parabola]]. The "cyclic rotation" of the dual number plane occurs as a motion of [[#Projective line|its projective line]]. According to [[Isaak Yaglom]],<ref name="yaglom"/>{{rp|92–93}} the cycle {{math|''Z'' {{=}} {''z'' : ''y'' {{=}} ''αx''<sup>2</sup><nowiki>}</nowiki>}} is invariant under the composition of the shear
:<math>x_1 = x ,\quad  y_1 = vx + y </math>

with the [[translation (geometry)|translation]]
:<math>x' = x_1 = \frac{v}{2a}  ,\quad  y' = y_1 + \frac{v^2}{4a}. </math>