### Ellipse Sample Datapoints

This post presents sample data points for one-quarter of an ellipse that is created in two ways.

Values under the column labeled y

_{cir}are y-values for circular

motion (i.e. -

*e*= 0), to provide data against which the elliptical data can be compared.

Values under the column labeled y

_{e}are y-values calculated according to the standard equation for an ellipse, given a and

*e*:

(x² / a²) + (y² / b²) = 1

These numbers are the same y-values described by an ellipse that is created by (i) taking a cut through a right cylinder at an angle, or (ii) projecting a circle onto an inclined plane.

Note that these values can be produced from the y

_{cir}values:

y

_{e}= y

_{cir}* cos(α)

where cos(α) = √(1 -

*e*²)

Similar to the y

_{e}values, the y

_{con}numbers are created by (i) taking a cut through a right cylinder at an angle, or (ii) projecting a circle onto an inclined plane. (Both methods produce the same results.)

However, in this case, the angle at which the cut is taken through the right cylinder is slightly different:

y

_{con}= y

_{cir}* cos(α)

where cos(α) = (1 -

*e*²)

^{1/4}

In this case, the angle α is smaller, and the ellipse produced is of smaller eccentricity. To produce the y

_{e}values, the y

_{con}values must be multiplied by an additional factor of cos(α):

y

_{e}= y

_{con}* cos(α)

### Data Points for One-quarter of an Ellipse

(a = 1)

x_{e} |
y_{cir} |
y_{e} |
y_{con} |

e = N/A |
e = 0 |
e = 0.786151377746 |
e = 0.618033988747 |

0 | 1 | 0.618033988764 | 0.786151377746 |

0.05 | 0.998749217772 | 0.617260962835 | 0.785168073588 |

0.1 | 0.994987437107 | 0.614936054525 | 0.782210744536 |

0.15 | 0.988685996664 | 0.611041550153 | 0.777256858449 |

0.20 | 0.979795897113 | 0.605547166467 | 0.770267894439 |

0.25 | 0.96824583655 | 0.598408836468 | 0.761187798416 |

0.30 | 0.953939201417 | 0.58956684969 | 0.749940617493 |

0.35 | 0.93674969976 | 0.578943153416 | 0.736427067083 |

0.40 | 0.916515138991 | 0.566437507113 | 0.720519639256 |

0.45 | 0.893028554975 | 0.551921999911 | 0.702055628873 |

0.50 | 0.866025403784 | 0.535233134672 | 0.68082706436 |

0.55 | 0.835164654425 | 0.516160142649 | 0.656565843733 |

0.60 | 0.8 | 0.494427191011 | 0.628921102208 |

0.65 | 0.759934207679 | 0.46966516957 | 0.597423324374 |

0.70 | 0.714142842854 | 0.441364549716 | 0.561424379827 |

0.75 | 0.661437827766 | 0.408791059014 | 0.519990259601 |

0.80 | 0.6 | 0.370820393258 | 0.471690826656 |

0.85 | 0.526782687643 | 0.325569605656 | 0.414130935671 |

0.90 | 0.435889894354 | 0.26939477007 | 0.342675440998 |

0.95 | 0.31224989992 | 0.192981051139 | 0.245475689028 |

1.00 | 0 | 0 | 0 |

Labels: ellipse, ellipse coordinates, elliptical section

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