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# The derivation of the 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 Einteilung einer eben bewegten Ebene in Felder mit qualitativ gleichen Koppelpunktbahnen unter besonderer Berücksichtigung der Übergangskurve. All required equations are concentrated in table 4.1. But for most people the phenomenological reflection should be enough:

• 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)

• Therefore the variation of the distance between the point and the pivot (center) of the moving wheel is sufficiently to create all shapes.

### 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 icons)
1. The center of the moving wheel creates a circle, with other words a curve with a constant radius of curvature
2. 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.
3. 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.
 1) 2) 3)

### 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 icons - In the pictures of the second row the green circle is unfortunately so small, that it is behind the pivot.)
1. points on the inner edge of the yellow ring (visible in the first row, if the cursor if over an icon) creates a curve with an approximate straight-line, but no switch of the center of curvature to the other side.
2. 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.
3. 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.
 1) 2) 3)

### 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 icons)
1. Points of the Moving Centrode (Points of the tread of the moving wheel) create epitrochoids with cusps.
2. 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.

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 int the dimension and in the proportion of the loop compared with the remaining part of the curve

3. 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 icon 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.)
 1) 2) 3)

### Curves of points outside the (first) Transition Curve

 1) 2) 3) 4)

# Calculation of the quantity of self-intersection points of an Epitrochoid or rather an Epicyloid (editable)

 Example: Radius of the fixed 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 epitrochoids with the quantity: nK = 0) Radius of the BALL Circle: rB = Radius of the Moving Centrode: rG = Number of Transition Curves: nTmax = Minimum quantity of Self-Intersection Points: nSmin = 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: nS1 = Maximum quantity of Self-Intersection Points: nSmax = Quantity of Self-Intersection Points outside of the = Radius of the Transiton Curve rT = Distance 'a' between - a point which creates a Epitrochoide with Self-Tangential Point - and the 'pivot (center) of the moving wheel': A Multi-Self-Intersection of the Epitrochoide without any additional Self-Tangential Points exists
Switch to animation of epitrochoids