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

Points inside the moving wheel creates curtate epitrochoids (left)
Points on tread of the wheel creates common epitrochoids (middle)
Points outside the moving wheel creates prolate epitrochoids (right)
The number of loops / cusps (sharp corners) depends on the numerator of the transmission ratio i = 'Radius of the fixed wheel' / 'Radius of the moving wheel' (in this case: i=4:3 and that means the number of loops or cusps is 4)

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::

All points of a circle around the pivot (center) of the moving wheel creates the same curve (with an other angle relative to the ground)

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

Let us review a moving wheel and a fixed wheel without modifyiing the ratio of the radii,
(first and second row of figures)

The center of the moving wheel creates a circle, with other words a curve with a constant radius of curvature
A point in the neighborhood of the center creates a curve with varying radius of curvature, but the center of curvature is inside of the curve allways. With other words: the curve is like a racetrack with right hand bend exclusively - no straight section of the road and no left hand bend.
If the point generating the curve is moving away and away from the pivot then in a special situation the point will create an approximate straight-line (2 straight lines in blue indicate the position of the approximate straight-line). The center of curvature does not switch to the other side of the curve, but this situation is nearby. With other words: the curve is identical with a racetrack with right hand bends and straight sections of the road - but without any left hand bend.

If you move with the mouse obove an figure in the first row of pictures then an additional circle-shaped surface will overlay inside the moving wheel.

The circle-shaped surface is a ring. Outside the ring is bordered by points, which creates epitrochoids with approximate straight-lines.
This border is called the BALL Circle
The inner border of the ring is the point, which creates a circle. Because this point is identical with the center of the ring, the ring will degenerate into a circular disc.
All points in the degenerated ring create curtate epitrochoids with centers of curvature, which are inside the ring all the time.

Now the qualitative shape of epitrochoids is known for all points inside the green ring, which is bordered by the BALL circle. (only visible, if the cursor is over a picture in the first row)
Surprising shapes of epitrochoides are impossible. Variation of the distance between the point generating an epitrochoid and the pivot (center) of the moving wheel modifies only the radius of curvature and the extend of the epitrochoid (Please view the variation of the position of the point inside the green circle, if the cursor is over the picture below.)

In the pictures of the second row the green circle is unfortunately so small, that it is behind the pivot.

radius of the BALL circle

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

example of calculation (editable)

Curves of points inside the tread of the moving wheel

Let us have a look on the yellow ring, which remains of the moving wheel after plotting the green ring
(first and second row of figures - In the pictures of the second row the green circle is unfortunately so small, that it is behind the pivot.)

points on the inner edge of the yellow ring (visible in the first row, if the cursor if over an figure) creates a curve with an approximate straight-line, but no switch of the center of curvature to the other side.
Points inside the yellow ring create epitrochids with switching centers of curvature to the other side several times. With other words: the curve is identical with a racetrack with right hand bends and left hand bends - but without any straight sections of the road. The approximate straight-line is replaced by switching from left hand bend to right hand bend and back - thus the center of curvature switches to the other side of the curve twice.
Is the point generating the epitrochoid 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 epitrochoids is known for all points in the yellow ring (only visible, if the cursor is over a picture in the first row) - thus inside the Moving Centrode (inside the moving wheel) and outside of the BALL circle (and therefore outside of the green ring).
Surprising shapes of epitrochoides are impossible. A variation exists only in the high of the 'waves' (figure in the middle) and the extend of the curves. (Please view the variation of the position of the point inside the yellow ring, if the cursor is over the picture below.)

Curves of points outside the tread of the moving wheel

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.
(first and second row of figures)

Points of the Moving Centrode (Points of the tread of the moving wheel) create epitrochoids with cusps.
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 right hand bend exclusively - no straight section of the road and no left hand bend.
Instead of the cusps there is the same number of loops and thus the same number of (additional) self-intersection points.
Exists only one loop, then moving the generating point from the Moving Centrode (tread of the moving wheel) away do not modify the qualitative shape of the epitrochoid. (first row). The curves differs only in the extend and in the proportion of the loop compared with the remaining part of the curve
Exists multiple loops on the contrary, the loops will tangent each other in a special situation, if the generating point will be moved from the Moving Centrode (the tread of the moving wheel) away.
Points generating curves with self-tangential points are called Transition Curve Points. This points are part of the Transition Curve, which degenerates to a Transition Circle for all epitrochoids.
(When the cursor is over an figure in the second row of pictures, a blue circular Transition Curve will be added as border of an orange ring. In column 3 the 3 self-tangential points are highlighted, if the cursor is over the picutre.)

In the pictures of the second row the green circle is unfortunately so small, that it is behind the pivot.

Now the qualitative shape of epitrochoids 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 over a picture in the second row)
Surprising shapes of epitrochoides are impossible. A variation exists only in the proportion of the loop compared with the remaining part of the curve (figure in the middle) 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 below.)

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!

Transition curves (shown in blue in the picture below) always have a larger radius than the Moving Centrode and the radius is at maximum as large as the distance between the centers of the two wheels. The picture shows an example with 3 transition curves
Let us now look at the area that is limited by an inner (and possibly also an outer) transition curve

Points on the smaller transition curve generate epitrochoids with self-tangential points (column 1).

Points outside the smaller transition curve create paths with additional self-intersection points per loop - compared to points located on or inside the smaller transition curve..

Under normal circumstances, the number per loop increases by 2 self-intersection points (column 2)

This results in 6 additional self-intersection points in row 1, making a total of 9 self-intersection points,

This results in 8 additional self-intersection points in row 2, making a total of 12 self-intersection points

In the special case that the transition curve passes through the center of the fixed wheel, the number per loop increases by only one self-intersection point. (column 4)

This special case always occurs when the numerator of the transmission ratio is even.
In this special case, all self-tangential points are located on top of each other (column 3), and in this point there are also self-intersection points formed by neighboring loops.
In this example, 2 self-tangential points and four self-intersection are located on top of each other.
The reason for the reduced increment of the number of self-intersection points is that in this case there are not neighboring loops touching each other, but opposite loops in pairs and therefore half as many as the number of loops.
With four neighboring loops, there are only 2 self-tangential points.
(See column 3, in which two opposite loops turn green when the cursor is positioned over the image. The other two touching loops remain red).
- A comparison with column 3 therefore results in 4 additional self-intersection points in column 4 / row 2, i.e. a total of 16 self-intersection points

The number of transition curves is dependent on the numerator of the transmission ratio

The number of transition curves is independent on the denominator of the transmission ratio
The number of transition curves rises by increasing the numerator to the next even number and
the number of transitions curves is identical with the half of the even numerator (numerator/2).
The number of transition curves for an odd numerator is: (numerator-1)/2

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)

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 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 epitrochoids 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_Tmax =
Minimum quantity of Self-Intersection Points: n_Smin = 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: n_S1 =
Maximum quantity of Self-Intersection Points: n_Smax = Quantity of Self-Intersection Points outside of the =
Radius of the Transiton Curve r_T = 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:

eMail: V.Jaekel@t-online.de