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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 (available only in German) Einteilung einer eben bewegten Ebene in Felder mit qualitativ gleichen Koppelpunktbahnen unter besonderer Berücksichtigung der Übergangskurve. All the necessary equations are summarized in Table 4.1 (English version) For all others, the following phenomenological considerations should be sufficient::

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




example of calculation (editable)

Example: Radius of the fixed wheel: rR = (iZ) =

Radius of the moving wheel: rG = (iN) =          

Radius of the BALL circle: rB =

Curves of points inside the tread of the moving wheel




Curves of points outside the tread of the moving wheel




Curves of points outside the first Transition Curve

The following statements apply not only to the second transition curve, but also to all other transition curves!
Now the qualitative shape of epitrochoids is known for all points outside of the first Transition Curve..

All epitrochoids between 2 transition curves have the same qualitative shape. The same applies to all epitrochoids outside the outermost transition curve. There are only differences in the size of the loops and the extent of the curves.

The count of self-intersection points, self-tangential points (=0), cusps (=0), approximate straight-line (=0) and switched centers of curvature to the other side (=0) therefore remains constant for all epitrochoids with a generating point outside the largest transition curve.

With one exception: With odd transmission ratios of the generating wheels, the radius of the transition curve is smaller than the distance between the axles of the wheels. A special case occurs with these pairs of wheels if the distance of the point generating the trochoid is equal to the distance between the axes of the wheels. In this case, multiple self-intersection points coincide in the axis of the stationary wheel.

The epitrochoids whose generating point are located outside the transition curves, but have either a smaller or larger distance to the center of the stationary wheel than the axis of the moving wheel, differ qualitatively only in the outline of the central field:
It is formed either by concave or convex curve sections.

As the surface boundaries formed by concaves are always very small, a comparison can almost only be made in an animation.

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 Epitrochoid with Self-Tangential Point - and the 'pivot (center) of the moving wheel':

A Multi-Self-Intersection of the Epitrochoid without any additional Self-Tangential Points exists


Recapitulation of the links of this page:


© Volker Jaekel, February 7th 2024