tangent to ellipse

jaaan.k shared this idea 6 years ago

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I need in my geometrical optics construction to build tangents to ellipses, would it be possible to extend accordingly the functionality of GeoGebra? A tangent to an ellipse through a point, and tangents to two ellipses?

Thank you for the very useful software!

Jaan

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Best Answer

mathmagic ● 6 years ago

https://www.geogebra.org/m/Yr9Jfinu

Comments (12)

jaaan.k ● 6 years ago

Thanks (when I first tried, it didn't work for some reason)! Though, I'd need also tangents to a pair of ellipses and preferably also tangents to ellipse with a fixed direction (e.g. perpendicular to a line). More specifically, the task is here: http://www.ioc.ee/~kalda/ip....

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Michael Borcherds ● 6 years ago

tangents to ellipse with a fixed direction

I think this command is the first step to doing that

https://wiki.geogebra.org/e...

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jaaan.k ● 6 years ago

Yes, with polar I can construct a tangent to an ellipse through a point. I can't see how I can use it to draw a line which is tangent to two arbitrarily oriented (but not nested) and arbitrarily sized ellipses. Neither can I see how to use this step to draw a tangent to an ellipse which is also perpendicular to another given line. Bounding box doesn't seem to help, neither ...

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Michael Borcherds ● 6 years ago

Rotate the ellipse so that the tangent line will be horizontal

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Michael Borcherds ● 6 years ago

Maybe a better method: rotate the ellipse and find the bounding box:

https://math.stackexchange....

https://www.geogebra.org/m/uxAeCJFx

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Michael Borcherds ● 6 years ago

tangents to a pair of ellipses

https://math.stackexchange....

Not sure what the answer means exactly though

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mathmagic ● 6 years ago ●

https://www.geogebra.org/m/Yr9Jfinu

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jaaan.k ● 6 years ago

That's great! If I want to do this by myself with arbitrarily drawn ellipses from scratch, do I need to enter manually the calculations of the polar conjugate conic section using lists?

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mathmagic ● 6 years ago

you can to move the points or download the applet and redefine the conics with an algebraic expressions in the algebra window

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jaaan.k ● 6 years ago

Yes, I already used this way, I was just thinking that perhaps there are other ways which I am not aware of. Either way, thank you very much!

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mathmagic ● 6 years ago

you can create an inputbox for each conic also

I edited my work in this way

https://www.geogebra.org/m/Yr9Jfinu

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Michael Borcherds ● 6 years ago

You can make a tool from mathmagic's file, see attached. Then you get a new command:

TangentEllipses[h, k]

Files: EllipseTangents...

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