There are a number of ways that you could find a horizontally displaced object. You could for example just look.
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It is called the displacement in the horizontal direction.
20 degrees is a measure of angular displacement. This cannot be converted to lateral (horizontal) displacement.
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On the horizontal axis you would probably plot the time. On the vertical axis you could plot displacement, velocity or acceleration.
They are the axes. Usually horizontal = x-axis, vertical = y-axis. But that need not always apply. In a displacement-time or speed-time graph, for example, the horizontal axis = t-axis (for time).