| To simplify analysis I changed tactics a little: defined the angle between each following phases (As you remember, in the examples above I defined only starting and ending phases and the animation was created by program automatically). I took the wheel with 16 spoke again, but in this case – regular figure (As you noticed, the distances between animations above weren’t equal). The angle between each spoke equals to 22.5 degrees (360/16=22.5). I took one sector, i.e. two neighboring spokes encircled with arc, (the length of arc is L), and divided this sector into two parts by bisector. The length L=2Пr/n, where n is a number of spokes n=natural number. Let’s take any point on the circle (for convenience – the point, where radius is touching the circle: green point-a on the diagram D). Let’s first consider the variant, when lagging (i.e. the length of arc- L1 from the point a to the next, following location of the same point a in next phase is less than L, i.e. less than arc between neighboring spokes. Let’s divide this particular task into three parts: when
1. L1<L/2,
2. L1=L/2,
3. L1>L/2;
In any case when L1<L/2, (angle between two following phases is within the range between 0 and 11.25 degrees) revolving is perceived as clockwise (variant A on the diagram), i.e. any following location of the first spoke in next phase falls left-side from the bisector (in our particular case the lagging equals to 5 degrees, i.e. is less than 11.25 degrees – between the first spoke and bisector – variant A on the diagram).
When L/2< L1<L, revolving is perceived as counterclockwise, i.e. any following location of the first spoke in next phase falls right-side from the bisector (in our particular case the lagging equals to 17 degrees, i.e. is less than 22.5 degrees – between the first and second spoke, right-side of the bisector - variant B1 on the diagram).
When L1= L/2, revolving is perceived in both directions equally simultaneously, as any following location of the first spoke in next phase falls on the bisector.
Let’s consider as well the variants, when lagging is more than L (L1> L).
So, the final formula can be:
If lagging: K*L+ L1, where k is whole number and L1< L/2, clockwise - Variant A
If lagging: K*L+ L1, where k is whole number and L1> L/2, counterclockwise -Variants B1 and B2.
If lagging: K*L+ L1, where k is whole number and L1= L/2, in both directions, i.e. revolving direction depends on your inclination and on your eye - Variant C.
The following position of the first spoke is marked as thick black on the nether part of diagram, where separate sectors are drawn. |