START - starts the curve, a second or third curve can be added to the image

CLEAR - clears the screen and rescales the new picture

"slow <----> fast" regulates the velocity of the drawing

"time of graph ... years" counts the earth's revolutions during the drawing process.

The words "years" and "days" refer to earth years and days.

Venus' geocentric path displays a fivefold symmetry, due to the approximate commensurability of 13:8 of the two revolution times. That means during 8 (earth) years Venus revolves approximately 13 times around the sun. Assume we start our observation when Venus and earth are next to each other. Since Venus revolves faster than the earth it will have to pass the earth four more times during the next 8 years (= 8 earth revolutions). Each time earth and Venus "meet" a loop in the geocentric venus path arises due to the overtaking. Since the speed of both of them is almost regular (the excentricity is quite small and enters only quadratically in the ellipse equation) the pattern will display a five fold symmetry.

Note however that none of the mentioned symmetries here is exact, they are due to approximations of the ratios of revolution times which in reality are by no means whole numbers.

Mercury displays a threefold symmetry, due to the approximate commensurability of 1:4 of the two revolution times. A closer view reveals a ratio of 7:29 resulting in a 22-fold symmetry (the degree of rotational symmetry is always given by the difference of the two fractional components of the ratio of the two revolutions).

Mars', trajectory reveals at first sight a onefold symmetry (revolution ratio w.r.t. earth 1:2). A closer view reveals a seven fold symmetry, due to the closer approximation of the ratio by 8:15 and a still closer view results in a 37-fold symmetry, due still closer approximation of 42:79.

Jupiter's trajectory forms an eleven-fold symmetry (revolution ratio Jupiter - Earth 1:12). A closer view reveals a 76-fold symmetry, due to the better description of the ratio by 7:83.

Pressing CLEAR adapts the virtual size of the applet to the planets path. So if you want an overall image of the solar system you have to CLEAR and start with the outermost neptune before graphing the other planets. A reasonable picture is feasable for all outer planets Neptune to Mars or for the inner planets Mars to Mercury.

As an orientation its always helpful to add the sun's orbit to the graph. This is included in the second choice "geocentric path + sun"

Note that the cusps of the envelopes do not occur at the passages of "backward motion" of the planet but at the opposition. To elucidate this behaviour of the two graphical representations a third option has been introduced "geocentric path + connection earth_planet" where both are plotted at the same time. If you add the fourth option "heliocentric path" you will notice, that each connection line starts exactly on the heliocentric path curve, as it should.

Finally the geocentric path is generated graphically by translating each connection line to the center of the panel, the observer on the earth.

Jupiter - Earth form an interval close to 3 octaves + a fifth,

Mars - Earth form a major seventh,

Earth - Venus form a minor sixth and

Earth - Mercury form an interval of roughly two octaves.

For ways to represent these musical intervals graphically see: curves of Lissajous. This may offer another quite interesting view on our planets motions.

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Last modified: Feb. 2010