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Complete investigation of the shape diversity of Hypotrochoids / Hypocycloids

An Hypotrochoid is a curve (in the picture colored in red). Hypotrochoids are created while a moving wheel (colored in yellow) is scrolling inside a fixed wheel (ring gear) (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

  1. The fixed wheel (ring gear) is larger than twice as large as the moving wheel

    • The transmission ratio i = 'Radius of the fixed wheel (ring gear)' / 'Radius des moving wheel' have to be larger as 2:1 = 2 . (in the pictures below is i = 4:1 = 4)
    • The quantity of loops / cusps (sharp corners) depends on the dividend of the transmission ratio (in this case: i=4:1, and that means the number of loops or cusps is 4)
    • Points inside the moving wheel creates curtate hypotrochoids (left)
    • Points on the tread of the moving wheel creates common hypotrochoids (also named hypocycloids) (middle)
    • Points outside the moving wheel creates prolate hypotrochoids (right)
  2. The fixed wheel (ring gear) is smaller than twice as large as the moving wheel

  3. Special case: The fixed wheel (ring gear) is exact twice as large as the moving wheel

The classical apportionment is not sufficient, to describe or rather calculate the shape diversity of hypotrochoids.

As examplte the classical apportionment does not consider

Consideration about the derivation of the shape diversity of hypotrochoids

As described in the chapter about the classical apportionment of the shape diversity of hypotrochoids, it has to be divided between hypotrochoids with

As indicated in the 2 image series each with 3 pictures with 'i > 2:1' or rather 'i < 2:1' (see above), identical hyportrochoids can be generated

- indeed with an other scale. This finding is called Double Generation of a hypotrochoid. Die zweifache Erzeugung erlaubt es, sich hier (vorerst) auf einen Fall der Erzeugung zu konzentrieren, nämich auf das Übersetzungsverhältnis 'i > 2:1'

Double generation of trochoids

Each trochoid can be generated by two different transmission ratios i = 'Radius of the fixed wheel' / 'Radius of the moving wheel' = iZ/iN.
The animation
"switching between the pair of wheels that generate the hypotrochoid"
starts when the cursor is over the image

Therefore, the transmission ratio i2 = iZ2/iN2 of the second pair of wheels can be calculated as follows:

The distance a2 between the 'point generating the trochoid' and the 'center of the moving wheel' of the second generating pair of wheels can be determined from the distance a of the original gear as follows:

However, in order for the second hypotrochoid to be the same size as the first, the scale of the second pair of wheels must be adjusted. The equations listed here do not take this scale into account because they work with the transmission ratio and not with the wheel dimensions.

In order to be able to determine the variety of forms, some phenomenological observations on the development of a hypotrochoid/hypocyloid are necessary:

If you want to know the details, you have to read chapter 4 of my dissertation Einteilung einer eben bewegten Ebene in Felder mit qualitativ gleichen Koppelpunktbahnen unter besonderer Berücksichtigung der Übergangskurve (German only). 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 near the center of the circle (when 'i > 2:1')


example of calculation (editable)

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

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

Radius of the BALL circle: rB =

Paths of points within the boundary of the rotating wheel (when 'i > 2:1')


Curves of points outside the tread of the moving wheel (when 'i > 2:1')

Curves of points outside the (first) Transition Curve (when 'i > 2:1')

Hypotrochoids generated from points outside a transition curve have aways more self-intersection points than those generated from points inside. The number of self-intersection points differs by twice the number of loops.

The case descriptions are repeated similarly to A, B.1 and B.2. However, the inner boundary is not the Moving Centrode (yellow ring) but the last determined transition curve (orange ring). For clarity, the correct terms and designations for these cases are U, V.1 and V.2.

If the point that generates a trochoid is shifted outward from the transition curve, it is necessary to distinguish between cases U, V.1 and V.2:

  1. Moving the point that generates the hypotrochoid (Figure 2) creates a multiple self-intersection point that are not overlaid by self-tangential points. (Figure 3),
    The number of self-intersection points, self-touch points (0) and side changes of the center of curvature(0) of the hypotrochoid remains constant even when the generating point is moved outward. (Please refer to Figures 2 und 4 above).

    The hypotrochoids inside and outside the dotted circle differ only in the border of the middle field. This border is formed by either (barely recognizable)
    concave or convex curve sections (as shown in Figures 2 and 4).

  2. Moving the point that creates the hypotrochoid (Figure 2 below) results in a self-tangetial point (Figure 3 below), The corresponding transition curve in blue is visualized in figures 2 and 4 (when the mouse is over either image).

    1. If the transition curve passes through the 'center of the hollow wheel', (Figure 2 or 4, visible when the cursor is positioned over the image.) the number of loops must be in this case even.

      The 'number of self-intersection points' of the hypotrochoids generated by points outside the transition curve (Figure 4: count=24), is greater by the 'number of loops' (Figure 2 to 4: count=6) than by points inside the transition curve (Figure 2: count=18).

      • In the pictures, the green ring is unfortunately so small that it almost disappears in the center of the rotating wheel

    2. If the radius of the transition curve is smaller than a circle around the 'center of the moving wheel' and through the 'center of the hollow wheel', (smaller than the dotted circle in Figure 2 below, visible when the cursor is positioned over the image) this transition curve is not the final division into rings to determine the variety of shapes.
      • In the pictures, the green ring is unfortunately so small that it almost disappears in the center of the rotating wheel

      In case (V.2)

      • the hypotrochoid still has a multiple self-intersection point (similar to case U).
      • or there is (at least) one additional transition curve outside the one already found (similar to cases V.1 or V.2)

      The U, V.1 and V.2 cases must be reevaluated until either U or V.1 are reached.

Calculation of the quantity of self-intersection points of a Hypotrochoid or rather an Hypocyloid (editable)

Example: Radius of the fixed hollow 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 hypotrochoids with the quantity: nK = 0)
Radius of the BALL Circle: rB =
Radius of the Moving Centrode: rG =
Number of Transition Curves: nÜmax =
Minimum quantity of Self-Intersection Points: nSmin = Quantity of Self-Intersection Points 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 der =
Radius of the Transiton Curve rT = distance 'a' between - a point which creates a hypotrochoid with Self-Tangential Point and - the 'pivot (center) of the moving wheel':

Eine Hypotrochoide mit einem Mehrfachselbstschnittpunkt, der kein Selbstberührungspunkt ist, existiert



Recapitulation of the links of this page:


© Volker Jäkel, February 7th 2024