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 ycir 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 ye 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 ycir values:
ye = ycir * cos(α)
where cos(α) = √(1 - e²)
Similar to the ye values, the ycon 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:
ycon = ycir * cos(α)
where cos(α) = (1 - e²)1/4
In this case, the angle α is smaller, and the ellipse produced is of smaller eccentricity. To produce the ye values, the ycon values must be multiplied by an additional factor of cos(α):
ye = ycon * cos(α)
Data Points for One-quarter of an Ellipse
(a = 1)
xe | ycir | ye | ycon |
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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