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The derivation of the complete investigation of the shape diversity of Epitrochoids / Epicycloids

An Epitrochoid is a curve (in the picture colored in red). Epitrochoids are created while a moving wheel (colored in yellow) is scrolling around a fixed wheel (colored in grey) The point generating the curve can reside inside the yellow wheel, part of the tread of the wheel or outside of the wheel (like in the picture).

The animation starts, if the Cursor is over the picture

The classical apportionment


It is not possible to get an impression about the derivation of the shape diversity by the classical apportionment.

To determine the complete shape diversity some phenomenological inspection of an epitrochoid / epicyloid is required:

The theory for the phenomenological inspection is described in chapter 4 of the book Einteilung einer eben bewegten Ebene in Felder mit qualitativ gleichen Koppelpunktbahnen unter besonderer Berücksichtigung der Übergangskurve. All required equations are concentrated in table 4.1. But for most people the phenomenological reflection should be enough:

Curves of points closed to the pivot (center) of the moving wheel

                 
 
1)

 
 
2)

 
 
3)

Curves of points inside the tread of the moving wheel

                 
 
1)

 
 
2)

 
 
3)

Curves of points outside the tread of the moving wheel

                 
 
1)

 
 
2)

 
 
3)

Curves of points outside the (first) Transition Curve

                 
 
1)

 
 
2)

 
 
3)
 
 
4)

Calculation of the quantity of self-intersection points of an Epitrochoid or rather an Epicyloid (editable)

 
Example: Radius of the fixed wheel rR = (iZ) = radius of the moving wheel rG = (iN) =     

Quantity of loops or rather cusps: nS&S =
Number of revolutions of the center of the moving wheel per period: nU =
Number of changes of the curvature to the other side of the trochoid, which are caused by points between BALL Curve and Moving Centrode: nkMax =
(all other points creates epitrochoids with the quantity: nK = 0)
Radius of the BALL Circle: rB =
Radius of the Moving Centrode: rG =
Number of Transition Curves: nTmax =
Minimum quantity of Self-Intersection Points: nSmin = Quantity of Self-Intersection Points inside of the Moving Centrode (of the moving wheel) =
Quantity of Self-Intersection Points between Moving Centrode and Transition Curve: nS1 =
Maximum quantity of Self-Intersection Points: nSmax = Quantity of Self-Intersection Points outside of the =
Radius of the Transiton Curve rT = Distance 'a' between - a point which creates a Epitrochoide with Self-Tangential Point - and the 'pivot (center) of the moving wheel':

A Multi-Self-Intersection of the Epitrochoide without any additional Self-Tangential Points exists
Switch to animation of epitrochoids

Recapitulation of the links of this page:

 

© Volker Jaekel, October 25th 2015

eMail: V.Jaekel@t-online.de