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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

• 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)
3. ### 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' = 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 numeratoriZ2 remains unchanged (iZ2=iZ)
• The denominator iN2 = iZ - iN
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 = iN * (iZ - iN) / a

### example of calculation (editable)

 Example: Radius of the fixed hollow wheel: rR = iZ = Radius of the moving wheel: rG = iN = Transmission ratio of the source pair of wheels i = Transmission ratio of the alternative pair of wheels: i2 = iZ2 =      iN2 = distance 'a' of the source pair of wheels a = distance 'a' of the alternative pair of wheels: a2 =

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:

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

• Varying the distance of a point to the center of the rotating wheel is therefore sufficient to run through all curve shapes.

### 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,
1. The center of the rotating wheel creates a circle, i.e. a path with a constant radius of curvature
2. 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.
3. 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
 1) 2) 3)

### 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')

• 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)
1. 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)
2. 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.
3. 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.
 1) 2) 3)
• 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')

• 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.

# 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