[alogo] Simplified absolute motion of the moon

[ Simplified] , because speeds, rotation speeds and paths, in more realistic models, are determined by solving numerically the differential equations of the [ three-body-motion-problem] , for bodies interacting through gravitation. Here the speeds and shape of the earth's path are set arbitrarily.

The earth T moves on an elliptical path f. The Moon X, moves also in an elliptical path m, whose one focus is
the earth. As X moves, the ellipse m rotates also about T. Thus it results a complex absolute movement of the
moon. We can trace the trajectory of X by attaching a pen to it. Part of the trajectory is shown (red).

Construction: (You see the names by clicking on the white space while holding down the SHIFT key)
1) First the ellipse f, through [Curves\Ellipse\Ellips(2 foci+pt)] tool. Foci at L, M, point at N.
2) Set a point P on the ellipse [Ctrl+E] to select the tool, click next to ellipse holding SHIFT-key down.
3) Create a motor, by clicking on the motor button on the upper-right toolbar.
4) Set moving point on f, starting at P: Right click on the motor, select [ Moving] , click at P, then click at the motor. This action creates a point T, initially identified with P. To distinguish from it, you
must move it a little further from P. Press the green button of the motor, then after a while press it again to stop.
5) Create a point R, with fixed relative position w.r. to the moving T. For this select from the context-menu
of the motor the item [Fixed] . Then click at T, then click at R.
6) Create a point S rotating about T, on the circle defined by the moving T(center) and R(pass). For this, select
from motor's context menu the item [Moving] and click at R and the motor.
7) S is initially identified with R, move it a little further by starting/stopping the motor, as in step-4.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

8) T and S will be used as foci of the moving ellipse m, whose another point is V. We want an ellipse rotating about T (one focus of m) but preserving its shape. For this we construct a triangle j and addapt it
by similarity to the segment TS: Construct triangle j. Select the tool [Shape-Tools\Polygon tools\Polygon Compasses], click on one side of j and then at T and S. This defines m(TSV).
8) Using the same tool as in step-1, click three times, respectively, next to the points T,S,V. This sets the ellipse m.
9) Start/Stop the motor to see how nice the ellipse moves.
10) Now stop the movement and doing the same as in step-2, put a point W on the ellipse m.
11) Create a moving point point X on m, doing the same as in step-4. Start/stop the motor to see the movement.
12) Attach a pen to the point X. Start the motor. While the system moves, press the F2 key to trace the trajectory of X.

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