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Living a quiet life in Coquitlam, B.C.

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Location: Coquitlam, British Columbia, Canada

Saturday, December 08, 2012

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

 

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