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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 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'
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:
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'.
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).