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

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)
The fixed wheel (ring gear) is smaller 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 smaller as 2:1 = 2 (in the pictures below is i = 4:3 = 1,333).
- The quantity of loops / cusps (sharp corners) depends on the dividend of the transmission ratio (in this case: i=4:3 and that means the number of loops or cusps is 4)
- Points outside the moving wheel creates curtate hypotrochoids (left)
- Points on the tread of the moving wheel creates common hypotrochoids (also named hypocycloids) (middle)
- Points inside he moving wheel creates prolate hypotrochoids (right)
Special case: The fixed wheel (ring gear) is exact twice as large as the moving wheel
- Points inside the moving wheel creates curtate hypotrochoids and this hypotrochoids are always ellipsis (links)
- Points on the tread of the moving wheel creates common hypotrochoids and this hypotrochoids are always straights, which are passed twice because the point generating the straight claps (mitte)
- Points outside of the moving wheel creates also curtate hypotrochoids based on the rule, that the same hypotrochoide can be created twice (with different diameters of the wheels) (right)
- In this special case with the transmission ratio i = 'Radius of the fixed wheel (ring gear)' / 'Radius des moving wheel' = 2:1 no loop will be generated. The number of cusps (sharp corners) is 2 (both ends of the straight, which will be generated by a claping point).

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

As examplte the classical apportionment does not consider

the variation of the number of Self-Intersection Points
die Variation of the number of changes of the center of curvature from one side to the other side of the hypotrochoid
(Changes of right hand bends and left hand bends)
the variation of the number of Self-Tangential Points
the variation of the number of approximate straight-line pattern
the variation of the number of self-intersection points

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

a transmission ratio of i > 2:1 and
a transmission ratio of i < 2:1.

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

by a pair of wheels with a transmission ratio 'i > 2:1' and also
by a pair of wheels with a transmission ratio 'i < 2:1'

- 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' = i_Z/i_N.

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 = i_Z2/i_N2 of the second pair of wheels can be calculated as follows:

The numeratori_Z2 remains unchanged (i_Z2=i_Z)
The denominator i_N2 = i_Z - i_N

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:

a₂ = i_N * (i_Z - i_N) / a

example of calculation (editable)

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 Ber�cksichtigung 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:

All points on a circle around the center of the rotating wheel generate the same curve (albeit inclined at a different angle)

Curves of points near the center of the circle (when 'i > 2:1')

Let us now consider a moving wheel and a fixed hollow wheel with a fixed ratio of radii,

The center of the rotating wheel creates a circle, i.e. a path with a constant radius of curvature
A point in the neighborhood of the center point creates a curve with a variable radius of curvature, but the center of curvature is always inside the curve. The track therefore corresponds to a racetrack with only right-hand bends - without straights and without left-hand bends.
If the point generating the hypotrochoid is placed further and further away from the center point, it will eventually pass through an (approximate) straight line (in this case there are 3 approximate straight lines that are passed through one after the other). However, the center of curvature does not yet change sides. The track therefore corresponds to a racetrack with right-hand bends and straight lines - but without left-hand bends

If you move the mouse over the middle image, a green circlular area v appears within the moving wheel.

It is a ring that is bounded on the outside by the points is bordered by a section of an approximated straight line.
This boundary is also called the BALL circle
The ring is bounded on the inside by the point that creates a circular curve. Because the point is identical to the center of the ring, the ring degenerates into a circular disk.
All points within the (degenerated) ring generate curved hypotrochids whose center of curvature is always on the same side, as the animated displacement of the point generating the hypotrochoid within the green ring illustrates.

Now the qualitative curve is described of all hypotrochoids whose generating points lie in the green ring bounded by the BALL circle is described. (visible when the cursor is positioned over the middle image.)
Surprising hypotrochoid curves are therefore excluded. Only the radius of curvature of the hypotrochid and its dimensions change when the distance of the point generating a hypotrochoid is varied. (See variations of the position of the point within the green circle when the cursor is positioned over the center image.)

The radius of the BALL circle is calculated as follows:

r_B = r _G² / (r_R-r_G)
mit r_B = Radius of the BALL circle
mit r_R = Radius of the fixed hollow wheel
mit r_G = Radius of the moving wheel

example of calculation (editable)

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

Now let's take a look at the yellow ring that remains of the rotating wheel after the green ring has been drawn in
( middle image - but only visible when the cursor is over the picture)

Points on the inner boundary of the yellow ring (the boundary is only visible in the middle image) create a path with an (approximate) straight line whose center of curvature does not change sides (left image)
Points inside the yellow ring create hypotrochoids whose center of curvature changes sides several times (column 2). The curve therefore corresponds to a racetrack with right and left curves, but no straight lines. The position of each (approximated) straight line in column 1 is therefore replaced by a left-hand curve and again by a right-hand curve, i.e. the center of curvature changes twice to the other side of the curve.
Is the point generating the hypotrochoid part of the tread of the wheel, then the "left hand bend" degenerate to a cusp (sharp corner). The tread of the wheel is identical with the Moving Centrode (which is degenerated to a circle) . Aside of the cusp, the center of curvature is all the time on the same side - inside the curve.

Now the qualitative shape of hypotrochoids is known for all points inside the green ring (visible when the cursor is positioned over the middle image) - i.e. within the Moving Centrode (within the moving wheel) and outside the BALL circle (and therefore outside the green ring).
Surprising hypotrochoids are therefore excluded. There are only differences in the depth of the 'waves' (middle image) and in the extend of the curves.

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

Let us have a look on the area, which is outside of the yellow ring. This plane is not colored. The boundary of this ring is the infinite. Sie bildet aber einen Ring, der bis ins Unendliche reicht.

Points of the Moving Centrode (Points of the tread of the moving wheel) create hypotrochoids with cusps (linkes Bild).

Points outside of the Moving Centrode (Points outside of the moving wheel) create curves with loops instead of cusps.
The center of curvature is again all the time on the same side of the curve. With other words: the curve is like a racetrack with left hand bend exclusively - no straight section of the road and no right hand bend.
Instead of the cusps there is the same number of loops and thus the same number of (additional) self-intersection points.

All hypotrochoids with a transmission ratio i > 2:1 have several loops. If the point generating the hypotrochoid is shifted outwards from the gear pole curve (from the outer edge of the rotating wheel) the loops will

intersect at some point (bullet point A image 3.),
or touch and intersect (bullet point B image 3. - for better recognition of the touching curve sections, 2 curves are colored green: curve sections of the same color touch each other, those of different colors intersect at the same point)

There are 3 cased to be distinguished (A, B.1 and B.2):

Moving the point that generates the hypotrochoid creates a multiple self-intersection point that is not superimposed by self-tangential points (bullet point A image 3),

The number of self-intersection points of the hypotrochoids does not change even if the generating point is moved further outwards (image 4 above). The (dotted) circle around the 'center of the moving wheel' and through the 'point that generates hypotrochoids with self-intersection points' (only visible in pictures 2 and 4 when the cursor is positioned over the pictures), separates the outer ring into two rings. However, all points of these two rings generate hypotrochoids with the same number of self-intersection points, self-touch points (0) and side switch of the center of curvature (0) , which is why the rings are not colored differently here. However, the hypotrochoids differ in the border of the center field:
It is formed either by concave

or by convex

curve sections (compare images 2 and 4).

Points that generate hypotrochoids with multiple self-intersection points always lie on a circle around the 'center of the rotating wheels' that passes through the 'center of the fixed hollow wheel'.

Points that generate curves with self-tangential points (Figure 3 below), are known as transition curve points. These points form a transition curve, which degenerates into a circle in the case of hypotrochoids.
(Moving the mouse over image 2 will display a blue circular transition curve outside the moving wheel, which forms the outer edges of an orange-colored ring.)

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=8), is greater by the 'number of loops' (Figure 2 to 4: count=4) than by points inside the transition curve (Figure 2: count=4).

1)

2)

3)

4)

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

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.)
so the 'count of self-intersection points' of the hypotrochoids generated by points outside the transition curve (figure 4: count=15), is greater by 'twice the number of loops' (figure 2 to 4: count=5) than by points inside the transition curve (figure 2: count=5).

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

In this case (B.2)

there is (at least) one more transition curve beyond the one already identified (similar to cases B.1 or B.2),
or the hypotrochoid has a self-intersection point with multiple intersections (similar to case A).

Now the qualitative shape of hypotrochoids is known for all points outside of the Moving Centrode (outside of the moving wheel) and (if existing) inside the Transition Curve [and thus inside the (first) orang ring]. (only visible, if the cursor is postioned over a figure 2 or 4 in the picture heading B.2.)
Surprising shapes of hypotrochoides are impossible. A variation exists only in the proportion of the loop compared with the remaining part of the curve (figure in the middle) (Bilder in Spalte 2 und 4) and in the extend of the curve.
(Please view the variation of the position of the point outside the yellow ring, if the cursor is over the pictures in column 2 or 4.)

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:

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).

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).

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).

1)

2)

3)

4)

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

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)

The browser have to be Javascript activated. This is not true for this session and this is the reason, why the calculation does not work.

Example: Radius of the fixed hollow wheel r_R = (i_Z) = radius of the moving wheel r_G = (i_N) =

Quantity of loops or rather cusps: n_S&S =
Number of revolutions of the center of the moving wheel per period: n_U =
Number of changes of the curvature to the other side of the trochoid, which are caused by points between BALL Curve and Moving Centrode: n_kMax =
(all other points creates hypotrochoids with the quantity: n_K = 0)
Radius of the BALL Circle: r_B =
Radius of the Moving Centrode: r_G =
Number of Transition Curves: n_Ümax =
Minimum quantity of Self-Intersection Points: n_Smin = Quantity of Self-Intersection Points of the Moving Centrode (of the moving wheel) =
Quantity of Self-Intersection Points between Moving Centrode and Transition Curve: n_S1 =
Maximum quantity of Self-Intersection Points: n_Smax = Quantity of Self-Intersection Points der =
Radius of the Transiton Curve r_T = 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:

eMail: V.Jaekel@t-online.de

Complete investigation of the shape diversity of Hypotrochoids / Hypocycloids

The classical apportionment

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

The fixed wheel (ring gear) is smaller than twice as large as the moving wheel

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.

Consideration about the derivation of the shape diversity of hypotrochoids

Double generation of trochoids

example of calculation (editable)

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

Curves of points near the center of the circle (when 'i > 2:1')

example of calculation (editable)

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')

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