September 15, 2010

More More More Trochoid Doodling

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7 comments:

sam said...

Hiya Laz sam again . Lets talk.

jehovajah@ntlworld.com
fractalforums.com

Studying your trochoids(now roulettes) for a reason.

sam said...

Think i have all your apps now man soo doodling away to learn the structure.

sam said...

This is so cool.

Using the trochoid in java script app.

So boxes are split into spin over radius over a kind of thickness. This third one affects the space in which the trochoid is drawn. It's kind of like a relative radius. So i am thinking it is a relative radius between the inner and outer circles?

The second box is the second circle, but the 3rd box is way out thereman! it looks like it controls the thickness around the trochoid line so the 4th box controls the ineer/outer circles for this circle pair.

So what i got so far is a circle pair which is the base pair then taking each point on a trochoid you can draw an orthogonal circle pair, that is orthogonal to the tangent to the base trochoid line. So it turns the line into spaghetti, man and you can bolognese it up like sauce dripping round the spaghetti strand!

And then --outstanding dude!!!- you can do it again for the new lines!

This is Lie group heaven Man, seriously radical topology.

Go to the top of the class!

sam said...

So i have looked as carefully as i can and i think the box is split spin rate then revolutions of the circumference then radius of the circle.

It seems that the second circle is orthogonal to the first,but its revolution of its circumference is used to ratio that of the first circle. This ratio the main one is used to determine the path of the point relative to the second circleand its radius. The image is animated in 3d.

Iam now going to find out what the 3rd circle does.

sam said...

The introduction of each new circle introduces a new orthogonal plane. The programme can draw a 4 cube, but stops there, however the circles continue to rotate in new "orthogonal planes.

So the spin is actually a spin in a different plane each time the revolution comparator compares with the previous circle and the radius adjust the radius of the new circle, but the circle is not drawn if the comparator is set to 0.

So very easily thr espacially simple lie group could be drawn if the programme can support the number of circles.
radial ratios produce hypo or epi trochoids

sam said...

The rolling first circle may be inside or outside the second, or "kissing", circle. If it is inside, the curve trace is called a hypotrochoid; if outside, an epitrochoid.

The tracing point may be inside the rolling first circle, outside it, or on the circle. If it is inside, then the curve trace is said to be curtate; if outside, it is prolate. If the point is on the circle, then the curve trace is an epicycloid or hypocycloid.

A trochoid is a closed curve, of finite length, precisely when the radius r of the rolling first circle is a rational multiple of the radius R of the second circle. I will use the convention that this ratio, which I will call the basic ratio, is positive if the two circles curve the same way at the point of kissing. Thus the curve is a hypotrochoid if the basic ratio is r:R R≥r; if R or r is negative or r greater than R, then the curve is an epitrochoid. The basic ratio is between the third items in the box.

The distance from the center of the rolling first circle to the tracing point I will call the arm, and the ratio of this to the radius of that first circle I will denote by A, and call the arm ratio. this is the second item in big numbers in the box. i called it the thickness before but now i see that it is the ratio of the arm to the first circles radius. The ratio is distributed between the second items in the boxes but relates to or controls the arm of the first circle in a pair of circles.

The first item in the box is the spin rate for that circle, but it spins orthonal to the preceding circles axis of spin. Once it gets beyond the 3 orthogonal axis system it moves into a generalised cooerdinate system to facilitate different spin planes.

sam said...

http://userpages.monmouth.com/~chenrich/Trochoids/Trochoids.html

This is a reference for some of the basics, and you can see i have modified it to suit.
The spin rate may in fact do the same job as the greatest common divisor if the ratio of the spin rates is adjuted to have an integer highest common divisor, i will look see.

This is great for my proportion and ratio project!