
Lissajous Figures 
For mathematical details concerning these figures
(and many even more amusing ones!)
see, for example, here. 
If the oscilloscope beam is deflected horizontally and
vertically by alternating voltages which are exactly in phase with
each other, the result on the screen is a straight line. 
If the amplitudes of the voltages
(and the x and y sensitivities of the oscilloscope) are equal the line
will be at 45° as shown here. 
For animations illustrating these situations, see In Phase, 45° out of Phase and
90° out of Phase 



If the phase of one of the varying voltages changes
the image on the screen goes through variously shaped
ellipses, finally becoming a circle, when the phase
difference is 90° (or π/2 rads)
as shown in the next three diagrams. 





Continue changing the phase of one of the voltages
(in the same sense) and the image cycles through the
same series of ellipses, circle and finally a straight
line again as shown in the next three diagrams. 





We now know that the two alternating voltages are in
antiphase or 180° (or π rads)
out of phase. 



