Ellipse from 3 points
benpaulthurston shared this idea 7 years ago
I found these equations I call the Eliptical Pizza theorem because it looks like a pizza under perspective adjustments, I thought maybe they could be useful for a possible new feature of geogebra to calculate an ellipse from 3 points, once you calculate 2 other points using these equations you can then use the conic from 5 points to get the ellipse from three given points..http://benpaulthurstonblog....
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We won't be adding that, but you could make a custom tool:
What we want to add though is parametric conics:
f(t)=A+(B-A)*sin(t)+(C-A)*cos(t) will produce an ellipse with center A through B,C.
Wow that parametric conic function is interesting... Yes I see that that is better because with my method you would have to have some user interface function to tell the program which 3 points on the ellipse in the diagram it is you know, whereas the parametric conic you just have to find the center and 2 of the points and it also skips the step of finding the conic through 5 points... great thanks
This one I derived from the center and 2 point parameterization you mentioned for 3 points on the ellipse, C, B, D with segment between B and D being the shortest axis of the ellipse:f(t)=(1/2)*D+(1/2)*B+((1/2)*B-(1/2)*D)*sin(t)+(C-(1/2)*D-(1/2)*B)*cos(t)
Maybe this is off topic but I used the parameterization I derived from the one you mentioned and some sigmoidal functions to blend between them to make this:
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