# Overview of Shape Variation of Epitrochoids / Epicycloids

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 changes of the center of curvature from one side to the other side of the Epitrochoid
(change from left to right turns)

### An Epitrochoid / Epicyloid will be created by a rotation of one wheel around an other one. In this overview all gear trains will be presented with a single-digit transformation ratio between 1:1 and 1:9 or rather 8:9.

The derivation of the complete investigation of the shape diversity of Epitrochoids with predetermined transformation ratio can be found here.
(Look at first the result of one transformation ratio of wheels and then the derivation or vice versa)

If the cursor is over one of the images, an animation starts or something similar.

• You can switch to an explanation page by clicking on an image. But this works only for the first rows currently.

Loops=1

approximate straight-line pattern:
0 oder 1

 detailed description transformation ratio: i=1:1 cycles U=1 intersection points: 0 oder 1

Loops=2

approximate straight-line pattern:
0 oder 2

 detailed description transformation ratio: i=2:1 cycles U=1 intersection points: 0,2 oder 4

Loops=3

approximate straight-line pattern:
0 oder 3

 detailed description transformation ratio: i=3:1 cycles U=1 intersection points: 0,3 oder 9

 detailed description transformation ratio: i=3:2 cycles U=2 intersection points: 3,6 oder 12

Loops=4

approximate straight-line pattern:
0 oder 4

 detailed description transformation ratio: i=4:1 cycles U=1 intersection points: 0,4,12 oder 16

 detailed description transformation ratio: i=4:3 cycles U=3 intersection points: 8,12,202 oder 24

Loops=5

approximate straight-line pattern:
0 oder 5

 detailed description transformation ratio: i=5:1 cycles U=1 intersection points: 0,5,15 oder 25

 detailed description transformation ratio: i=5:2 cycles U=2 intersection points: 5,10,20 oder 30

 detailed description transformation ratio: i=5:3 cycles U=3 intersection points: 10,15,25 oder 35

 detailed description transformation ratio: i=5:4 cycles U=4 intersection points: 15,20,30 oder 40

Loops=6

approximate straight-line pattern:
0 oder 6

 detailed description transformation ratio: i=6:1 cycles U=1 intersection points: 0,6,18,30 oder 36

 . transformation ratio: i=6:5 cycles U=5 intersection points: 24,30,42,54 oder 60

Loops=7

approximate straight-line pattern:
0 oder 7

 transformation ratio: i=7:1 cycles U=1 intersection points: 0,7,21,35 oder 49

 transformation ratio: i=7:2 cycles U=2 intersection points: 7,14,28,42 oder 56

 transformation ratio: i=7:3 cycles U=3 intersection points: 14,21,35,49 oder 63

 transformation ratio: i=7:4 cycles U=4 intersection points: 21,28,42,56 oder 70

 transformation ratio: i=7:5 cycles U=5 intersection points: 28,35,49,63 oder 77

 transformation ratio: i=7:6 cycles U=6 intersection points: 35,42,56,70 oder 84

Loops=8

approximate straight-line pattern:
0 oder 8

 transformation ratio: i=8:1 cycles U=1 intersection points: 0,8,24,40,56 oder 64

 transformation ratio: i=8:3 cycles U=3 intersection points: 16,24,40,56,72 oder 80

 transformation ratio: i=8:5 cycles U=5 intersection points: 32,40,56,72,88 oder 96

 transformation ratio: i=8:7 cycles U=7 intersection points: 48,56,72,104 oder 102

Loops=9

approximate straight-line pattern:
0 oder 9

 transformation ratio: i=9:1 cycles U=1 intersection points: 0,9,18,36,54,72 oder 90

 transformation ratio: i=9:2 cycles U=2 intersection points: 9,27,45,63 oder 81

 transformation ratio: i=9:4 cycles U=4 intersection points: 27,36,54,72,90 oder 108

 transformation ratio: i=9:5 cycles U=5 intersection points: 36,45,63,81,99 oder 117

 transformation ratio: i=9:7 cycles U=7 intersection points: 54,63,81,99,117 oder 135

 transformation ratio: i=9:8 cycles U=8 intersection points: 63,72,90,108,126 oder 144