Hypocycloid Equations required to form system of three rotating circles

In summary, the goal is to create a series of hypocycloids that resemble a spirograph. The outer circle remains stationary while the middle circle rotates along the inner edge of the outer circle and outer edge of the innermost circle. The innermost circle stays stationary but may rotate to accommodate the movement of the middle circle. A gear system may be needed to achieve this and the equations for a hypocycloid are x=a(cos(t) + k) and y=a(sin(t) + k), where 'k' is the ratio of the smaller circle's radius to the larger circle's radius. A provided program can be used as a reference for the desired outcome.
  • #1
Liam Semeniuk
4
0
Hi there,

I have never worked with Hypocycloids before so I'm unsure which equations I should be using; but I'll try and get across what I am trying to build. Essentially I am trying to create a series of hypocycloids that act in a similar manner to the "spirograph".

Goal: Three circles set within each other. The outer circle will stay stationary and not rotate. It should be static. The circle set within this circle will rotate along the inner edge of the outer circle and along the outer edge of the innermost circle. The innermost circle should stay stationary, but should rotate as required to accommodate the rotation and drift of the middle circle.

I have attached a diagram (not animated) of what the system should look like.

I have marked two points (Point A and Point B) that should not meet up until the circle has made five rotations around the circle (a total travel of 1800 degrees).

Is it likely that I will need a gear system to implement this?

Any help is appreciated! Thanks!


I have attached a program as well (in C#) that can be used to show what I am aiming for with the outer and middle circle. Set the form to the following:
A = 180
B = 130
C = 130
Fr/Rev = 20
 

Attachments

  • Hypocycloidal System.png
    Hypocycloidal System.png
    8.4 KB · Views: 438
  • howto_draw_hypotrochoid_animated.zip
    16.9 KB · Views: 135
Last edited:
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  • #2
RPM = 1This should give you an idea of what I am trying to achieve. If you are looking for equations, the equation for a hypocycloid is x=a(cos(t) + k) and y=a(sin(t) + k). The 'k' value is the ratio of radius of the smaller circle to the larger circle. You can use this equation to calculate the position of the inner circle at any given time. You may also need to add in a gear system to the program to ensure that the inner circle rotates in the opposite direction to the middle circle.
 

1. What is a hypocycloid equation?

A hypocycloid equation is a mathematical equation used to describe the shape of a hypocycloid, a curve traced by a fixed point on a smaller circle as it rolls around the inside of a larger circle.

2. How many circles are needed to form a hypocycloid system?

A hypocycloid system requires three circles: a large fixed circle, a smaller rolling circle, and a third circle that rotates around the center of the smaller circle.

3. What are the variables in a hypocycloid equation?

The variables in a hypocycloid equation are the radius of the smaller rolling circle (r), the radius of the larger fixed circle (R), and the distance between the centers of the two circles (d).

4. What is the significance of hypocycloid equations?

Hypocycloid equations have numerous applications in mathematics and engineering, such as in the design of gears and gear ratios, the study of planetary motion, and in creating aesthetically pleasing designs in art and architecture.

5. How do you solve a hypocycloid equation?

To solve a hypocycloid equation, you can use parametric equations or polar coordinates to calculate the x and y coordinates of points on the hypocycloid curve. Alternatively, you can use computer software or graphing calculators to plot the curve.

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