Jerry Shaw
01-10-2011, 09:41 PM
http://upload.wikimedia.org/math/7/a/1/7a1e18cb71571553e1e5c38505278b6b.png
or equivalently
http://upload.wikimedia.org/math/9/8/4/984f1b5320ac279bf39cd24b37725081.png
It is common to assume that a counterclockwise motion results in a positive change of angle and a clockwise one will correspond to a negative change of angle. To accommodate this convention we introduced a minus sign in the formula above. (http://upload.wikimedia.org/math/6/c/e/6cebc296a06759c1e4467bdfb04af8d4.png)
Let (xc,yc) be the coordinates of the center of C2 in the absolute system of coordinates. Then R − r represents the radius of the trajectory of the center of the inner circle, and
http://upload.wikimedia.org/math/3/d/f/3df9b10cdd4264707d50892cfb0f693a.png
The coordinates of A in the new system are http://upload.wikimedia.org/math/9/3/5/935d47cefe061544d26574396740abac.png and they obey the regular law of circular motion (the angle of rotation in the relative system is http://upload.wikimedia.org/math/1/0/0/1003afc8969f92c0140a6a4363ed58dd.png):
http://upload.wikimedia.org/math/3/a/a/3aaf593a949f552ca05b6e751de6f220.png
In order to obtain the trajectory of A in the absolute (old) system of coordinates we add these two motions:
http://upload.wikimedia.org/math/b/9/0/b909a1c3b3413fe3f32823b6c598f9d1.png
where ρ is defined above.
Now we use the relation between t and http://upload.wikimedia.org/math/1/0/0/1003afc8969f92c0140a6a4363ed58dd.png as discussed above to obtain equations describing the trajectory of point A in terms of one parameter t:
http://upload.wikimedia.org/math/b/d/e/bde48b6aa37516a13086e97adde52811.png
(we used the fact that function sin is odd)
It is convenient to represent the equation above in terms the radius R of the largest circle and dimensionless parameters describing the structure of the spirograph. Namely, let
http://upload.wikimedia.org/math/b/6/2/b6271212607a4a059dff2bdf523629f4.png
and
http://upload.wikimedia.org/math/f/8/b/f8b1773ef6b83b3898cfe8d76fc761f2.png
The parameter http://upload.wikimedia.org/math/8/3/5/835031e0d0c6df286de7a62c28c4682a.png represents how far the point A is located from the center of the inner circle. At the same time, http://upload.wikimedia.org/math/f/b/f/fbfe5299be232bac4ffbb420f3171f4a.png represents how big the inner circle is with respect to the large one.
We observe that
http://upload.wikimedia.org/math/f/9/f/f9f6981e038f634e15fbea7e05b01c02.png
and therefore the trajectory equations take form of
http://upload.wikimedia.org/math/3/7/e/37e9dcfeaa81b3eef9317d21921eb918.png
or equivalently
http://upload.wikimedia.org/math/9/8/4/984f1b5320ac279bf39cd24b37725081.png
It is common to assume that a counterclockwise motion results in a positive change of angle and a clockwise one will correspond to a negative change of angle. To accommodate this convention we introduced a minus sign in the formula above. (http://upload.wikimedia.org/math/6/c/e/6cebc296a06759c1e4467bdfb04af8d4.png)
Let (xc,yc) be the coordinates of the center of C2 in the absolute system of coordinates. Then R − r represents the radius of the trajectory of the center of the inner circle, and
http://upload.wikimedia.org/math/3/d/f/3df9b10cdd4264707d50892cfb0f693a.png
The coordinates of A in the new system are http://upload.wikimedia.org/math/9/3/5/935d47cefe061544d26574396740abac.png and they obey the regular law of circular motion (the angle of rotation in the relative system is http://upload.wikimedia.org/math/1/0/0/1003afc8969f92c0140a6a4363ed58dd.png):
http://upload.wikimedia.org/math/3/a/a/3aaf593a949f552ca05b6e751de6f220.png
In order to obtain the trajectory of A in the absolute (old) system of coordinates we add these two motions:
http://upload.wikimedia.org/math/b/9/0/b909a1c3b3413fe3f32823b6c598f9d1.png
where ρ is defined above.
Now we use the relation between t and http://upload.wikimedia.org/math/1/0/0/1003afc8969f92c0140a6a4363ed58dd.png as discussed above to obtain equations describing the trajectory of point A in terms of one parameter t:
http://upload.wikimedia.org/math/b/d/e/bde48b6aa37516a13086e97adde52811.png
(we used the fact that function sin is odd)
It is convenient to represent the equation above in terms the radius R of the largest circle and dimensionless parameters describing the structure of the spirograph. Namely, let
http://upload.wikimedia.org/math/b/6/2/b6271212607a4a059dff2bdf523629f4.png
and
http://upload.wikimedia.org/math/f/8/b/f8b1773ef6b83b3898cfe8d76fc761f2.png
The parameter http://upload.wikimedia.org/math/8/3/5/835031e0d0c6df286de7a62c28c4682a.png represents how far the point A is located from the center of the inner circle. At the same time, http://upload.wikimedia.org/math/f/b/f/fbfe5299be232bac4ffbb420f3171f4a.png represents how big the inner circle is with respect to the large one.
We observe that
http://upload.wikimedia.org/math/f/9/f/f9f6981e038f634e15fbea7e05b01c02.png
and therefore the trajectory equations take form of
http://upload.wikimedia.org/math/3/7/e/37e9dcfeaa81b3eef9317d21921eb918.png