how to combine multiple equations?

Niek Sprakel shared this question 8 months ago
Answered

Hi.

Suppose you have two equations for two ellipses and you want to combine them, how would you do this?

For instance, when you try to approximate the shape of an egg with two ellipses, you want to combine the upper half of one ellipse and the lower half of the other ellipse.

https://i.imgur.com/rk0iTSt.png

Greetings and thanks in advance for any suggestions, Niek

Best Answer
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* Intersect them

* use Arc( <Ellipse>, <Point>, <Point> )

* put them in a list


(& post your .ggb file)

Comments (13)

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in not derivable way using arc of conic

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* Intersect them

* use Arc( <Ellipse>, <Point>, <Point> )

* put them in a list


(& post your .ggb file)

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Thanks!

My .ggb file was a bit of a mess since it had a lot of additional stuff unrelated to the question.

Often I find it's not easy to create a new ggb file based on existing ggb files, because somehow select-copy-paste operation fails between two ggb files that are open in two instances of the GeoGebra Geometry app and it's cumbersome to manually type over large expressions just to transfer them between two open ggb files.

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Perhaps you want use a parametric spline

https://www.geogebra.org/m/z8jpdcpr

Or Bézier curves

https://www.geogebra.org/m/sX7dmW6F

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Yes, I was kind of exploring to see if there is a natural or simple polar curve that matches up with the shape of an egg.

I can find one, but it's upside down (so I have to reflect it) and I'm still trying to figure out how to modify the equation of the curve so I no longer need that extra step to flip the curve upside down.

https://i.imgur.com/1fCRaHe.png

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If you post your .ggb file then we can help


Also you might like to try the FitImplicit() command if you're after a single formula

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Ok, here is the ggb file.

So I have the polar parametric curve which closely matches the shape of the egg, except that it's upside down, so I need an extra step to flip it upside down.

How would I modify the expression for the polar parametric curve so I don't need the extra step to reflect the curve?

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Try using this (cartesian) form

Curve((((v2 - 1) cos(2t - π) + 1) sin(t)^0.89)*cos(t), (((v2 - 1) cos(2t - π) + 1) sin(t)^0.89)*sin(t), t, 0, 2pi)

and now you can easily translate and reflect by just changing the x or y components

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Yes, that would be an option. But it seems to complicate the expression quite a bit. Also, I'm curious about how one would do it for the polar form. Basically I'm just kind of playing around with it to obtain some intuition for the relationship between simple functions like the sine and simple shapes, like circles, ellipses or egg shapes. Intuitively I'd think that there must be some elementary modifications of the sine function to obtain an ellipse or an egg shape instead of a circle within the framework of polar parametric curves.

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If you want to reflect/translate - I think it needs to be in cartesian form

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From what I recall it would make it a lot easier in cartesian form, but it's still possible to do it in polar form.

For instance in this ggb file I've used a polar parametric function to create circular curves corresponding to circles translated to evenly spaced locations along a bigger circle.

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Playing with Locus instead of equations..

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But ultimately I would like to end up with an equation or something I can use in other situations.

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