Epitrochoid / Epicycloid: Shape Variation for a transmission ratio of 4:1

The overview of shape variation is based on

• the number of loops and cusps (sharp corners)
• the number of approximate straight-line pattern
• the number of self-tangential points
• the number of self-intersection points
• the number of alterations of the center of curvature from one side to the other side of the Epitrochoid
  (change from left to right turns)

The circles are known, which borders the rings. They are called

• the BALL Points are forming a circle. Epritochoids created by BALL Points pass through an approximate straight-line pattern
  (In this case, the BALL Circle is smaller than the icon of the pivot, so that it is not visible)
• the Moving Centrode is identical with the tread of wheel. Its points pass through Epritochoids with cusps.
• the Transition Curve (Uebergangskurve so called by by R. Mueller) with points passing through self-tangential points of Epitrochoids. The Number of Transition Curves varies between 0, 1 and a number greater than 1.
  (In this case the number is equal 2)

The epitrochoid is created by a point, and if this point moves from the inner edge of the ring on the ring-shaped surface, then the shape of the epitrochoid changes. The same applies when switching from the ring-shaped surface to the outer edge. If the generating point varies its location within the ring-shaped surface, then the epitrochoid itself varies, but there is no qualitative modification of the epitrochoid (the verbal description does not change).

The following table describes alternately the shape of an epitrochoid

1. if the generating point is part of the inner edge of a ring
2. if the generating point is part of the ring-shaped surface

• If the point is located on the outer edge of a ring, it is coinstantaneous the inner edge of the next larger ring. Thus the description 2 follows the description 1 on the ring next size up

In the middle of all rings is a disc whose center point is the inner edge of a "disc-shaped ring." The table begins with this center point.

---

	Number of loops 4 transformation ratio i=4:1 Number of cycles 1	Remarks about the point, which creates the epitrochoid Remark about the shape of the epitochoid
The point (which created the epitrochoid) is on the edge of a ring approximate straight-line patterns 0 cusps 0 self-intersection points 0 self-tangential points 0 number of alterations of the center of curvature 0		Point is identical with the center of the wheel The epitrochoid is a circle
The point (which created the epitrochoid) is on the ring-shaped surface approximate straight-line patterns 0 cusps 0 self-intersection points 0 self-tangential points 0 number of alterations of the center of curvature 0		Point resides on the ring-shaped surface between the center of the wheel and the BALL Circle (BALL Curve) The center of curvature does not alternate to the other site of the curve. (In this case the BALL Circle is smaller as the icon of the pivot. As result the shape of this ring is not visible)
The point (which created the epitrochoid) resides on the edge of a ring approximate straight-line patterns 4 cusps 0 self-intersection points 0 self-tangential points 0 number of alterations of the center of curvature 0		The point is a part of the BALL Circle (BALL Curve) The BALL Circle resides always between the pivot and the outer edge of the moving wheel The radius of the BALL Circle is 0.2 for this transmission ratio (multiplied by the radius of the moving wheel)
The point (which created the epitrochoid) is on the ring-shaped surface approximate straight-line patterns 0 cusps 0 self-intersection points 0 self-tangential points 0 number of alterations of the center of curvature 8		Point resides between the BALL Circle (BALL Curve) and the Moving Centrode (In this case, the BALL Circle is smaller than the icon of the pivot, so that the outer edge of this ring is not visible)
The point (which created the epitrochoid) is on the edge of a ring approximate straight-line patterns 0 cusps 4 self-intersection points 0 self-tangential points 0 number of alterations of the center of curvature 0		Point is part of the Moving Centrode The Moving Centrode is identical with the tread of the moving wheel. The radius of the circular Moving Centrode is 1.0 (multiplied by the radius of the moving wheel)
The point (which created the epitrochoid) is on the ring-shaped surface approximate straight-line patterns 0 cusps 0 self-intersection points 4 self-tangential points 0 number of alterations of the center of curvature 0		Point is outside of the Moving Centrode (outside of the tread of the wheel) Point resists between Moving Centrode and first Transition Curve.
The point (which created the epitrochoid) is on the edge of a ring approximate straight-line patterns 0 cusps 0 self-intersection points 4 self-tangential points 4 number of alterations of the center of curvature 0		Point is part of the Transition Curve 1. (of 2 Transition Curves) The circular Transition Curve resides outside the tread of the moving wheel The radius of the Transition Curve is smaller or identical with the distance between the pivot of the moving wheel and the pivot of the fixed wheel The radius of the Transition Curve for this transmission ratio is 4.0 (multiplied by the radius of the moving wheel) In this general case the near-by loops are in contact (and not the opposite loops) .
The point (which created the epitrochoid) is on the ring-shaped surface approximate straight-line patterns 0 cusps 0 self-intersection points 12 self-tangential points 0 number of alterations of the center of curvature 0		Point resids outside of the Transition Curve 1 (of 2 Transition Curves) The variaton of the distance between the point generating the epitrochoid and the pivot of the moving wheel does not modify the qualitative characteristic of the shape of the epitrochoid as long a the point generating the epitrochoid is positioned between Transition Curve1. and the Transition Curve 2.
The point (which created the epitrochoid) is on the edge of a ring approximate straight-line patterns 0 cusps 0 self-intersection points 12 self-tangential points 2 number of alterations of the center of curvature 0		Point is part of the Transition Curve 2. (of 2 Transition Curves) The circular Transition Curve resides outside the tread of the moving wheel The radius of the Transition Curve is smaller or identical with the distance between the pivot of the moving wheel and the pivot of the fixed wheel The radius of the Transition Curve for this transmission ratio is 5.0 (multiplied by the radius of the moving wheel) The radius of the Transition Curve is identical with the distance of both wheels: In this special case only the opposite loops are in contact and not the near-by loops
The point (which created the epitrochoid) is on the ring-shaped surface approximate straight-line patterns 0 cusps 0 self-intersection points 16 self-tangential points 0 number of alterations of the center of curvature 0		Point resids outside of the Transition Curve 2 (of 2 Transition Curves) Any stretching of the distance between the point generating the epitrochoid and the pivot of the moving wheel do not change the qualitative characteristic of the shape of the epitrochoid as long as the point generating the epitrochoid resides outside of the largest Transition Curve.

Recapitulation of the links of this page:

• The derivation of the complete investigation of the shape diversity of Epitrochoids
• Overview of Shape Variation of Epitrochoids
• Animation of trochoids

© Volker Jaekel, January third 2024

eMail: V.Jaekel@t-online.de