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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 anti-phase or 180 (or π rads) out of phase.
 
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