Intersecting a circle with a cubic locus curve, mostly

ET66044 shared this problem 2 months ago
Not a Problem

I created two applets where I need to intersect a circle (circumscribed to triangle ABC there) with a cubic locus curve (orange color - created by me as the locus of the intersection points of all tangents from two points P and Q to all parabolas inscribed in triangle ABC).

If a locus curve is (functionally) just as with any other curve offered by GeoGebra (lines and conics, at least), then I should be able to use the "Intersect" command/Icon in order to obtain the intersection points - but when I try nothing happens!

Or is it that the curves needing to be intersected have to be of degrees one or two only (hence cubics won't be dealt with) ?

Further, I saw no "value" (i.e., equation) displayed for the cubic created: should GeoGebra not be able to assess that no transcendental function had been used, hence the resulting locus should be an algebraic curve, whose equation should then be determined, even when the degree would not be discernible ?

https://www.geogebra.org/m/f2fvhcpk

Same problem when I created quartics: no equation is displayed:

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

Going back to the cubics and their intersection with the circumcircle ABC: I had to "fake" their intersection, by attaching a point to one of the curves, then manually sliding that point along its curve path until the "intersection position" (visually) appears to be attained. In the second picture involving cubics, the intersection point F lies quite far from its correct intersection point position, between the cubic and the circle. I have manually corrected that (more than once!), but after I save it all, point F is still astray! Fortunately, I noticed that the "Refresh" button on the applet would bring point F in a correct position, but ... why should it not have been saved correctly there in the first place ? I had to put a message for users/viewers to first refresh the image ...

Finally: should GeoGebra not offer, along with its "Conic through five points" tool, the dual tool too ? That would be: "Conic tangent to five lines". There is quite a bit of work constructing it. I know I could simply create this tool myself, for my own use, but ... past attempts have shown that design flaws caused some of my tool to fail, at least in certain configurations (choice of initial data).

Thanks.

Comments (19)

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1

Have you tried LocusEquation() to get the equation if the locus?

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Hello,

Thank you for your promptness.

I did not know about it, but tried it now: the ellipses there were returning a quadratic equation, but the orange curve (which should be a cubic) returned "undefined" in both applets. I think I read somewhere in the help info that this command would only work up to conics. Too bad. Thanks again,

E.T. :)

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Hello,

Sorry, I forgot this: my main concern was if there was a way to intersect the (yellow) cubic curve with the circumcircle ABC. I had to "fake it", by placing a point on one of the curves only, then manually (visually) adjusting it to best representing an intersection point.

The big problem is, this goes out of alignment even for mysterious reasons, such as ... running the "Locus" command (which clearly "jolts" the picture, in the background).

And as I said the first time, I don't understand why, after manually choosing its correct position while editing the applet, upon saving it, the new preferred position is not recorded.

Thanks again,

E.T.

P.S.

This is harder than I thought: I went back to editing the applet, just to check something in its history: the "Locus" commands are all gone, although yesterday they had (counter-intuitively ?) been added to it, as latest steps - and I could see no way to remove them! And to top it all, GeoGebra seems again "willing" to remember correctly where I wanted point F to be saved ...

One question remains, though: is there a way to intersect cubics and conics ? If I understand this at all, if Groebner bases are used to determine equations/intersections, I don't see why higher degree algebraic curves could not be dealt with (in terms of intersecting them, or computing their equations).

Finally: I'm embarrassed to ask, but where is the "self-help" page where I could read about commands, functionality etc. ? Yes, there are tutorials, but one sometimes needs to find very specific pieces of information, if they are available. For instance, the info on the Locus command appears on a site which does not seem to belong to GeoGebra's ...

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1

LocusEquation will work for higher powers - please post your.ggb file where it's not working

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1

Hello,

I don't really understand how GeoGebra works: I don't think I was able to ever see anything from the ggb files I download. They open as "blanks", it's just the GeoGebra application that opens, when I click on my downloads. In any case, here is the download you requested. The png version works fine.

I've tried the "Locus" command again (for the orange cubic in the picture), with the same result: "undefined".

Sorry for giving you so much trouble. Thanks.

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1

Can you make the construction without using Polar()?


LocusEquation() only works with "simple" geometry

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1

(try File -> Open to load your files)

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Sir,

The construction involves a lot of polar lines, it's extremely painful to construct polars „from scratch”, because it gets even longer and would produce an even more crowded picture I already struggle to remember who is what, there.

The application I have downloaded („GeoGebra classic”) does not display in its menu the usual options („File”, among them - so I don't have that), it only shows the icons of various geometric operations.

And I'm terrible in the IT department ... Why would I try to „load” a file, when I would actually want to have/download a copy of the .ggb file(s) I've created, as it/they appear on the GeoGebra website - I don't understand.

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1

OK, if you don't want to try that then you can try putting some points on the locus and using Conic(list of 5 points) or ImplicitCurve( <List of Points> )

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Sir, its easier to assume I don't understand your directions. I know next to nothing about IT stuff. It's not that "I don't want to try" this or that. For instance, I said my GeoGebra application does not seem to have - believe it or not! - a "File/Open/etc." anywhere on its menus. Or that I do not understand why you would direct me to something called "Upload", when I actually want to download the file in question.

Equally, now I don't see how putting points and the orange locus curve and constructing a conic through it would help me with intersecting my circumcircle ABC with said orange locus. I did it though, I added 5 points on the cubic, then I constructed the conic through them. If I try to use the intersection command (I mean, icon), it would do nothing - in particular, it would not show/construct the 6th intersection point, otherwise visible in the picture.


Please keep in mind I know nothing about IT/GeoGebra in particular, so I can't understand most directions (because they don't seem to be in full detail. Until then, please remember I don't know how to download any applet here onto my computer (to send it to you here, as attachment).


If this is too much, I could give up - and thank you for your effort. The construction history now shows the conic being constructed, but no sign afterwards of my trying to intersect this conic with the yellow cubic.

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algunas ideas sobre su construccion

a) hay algún error interno del archivo GG pues el punto A no está bien definido. compare el valor de A con el numero x(A) despues de escribir x(A) en la linea de entrada. los valores son diferentes y si crea MM=A el punto MM se dibujara a la izquierda de A. creo que se cometió algún error en un nombramiento extraño

/wNanRbsukaZbQAAAABJRU5ErkJggg==


b) una vez corregido esto observar que locus debe ser hecho de la forma más simple posible. como ejemplo es mejor

A,B,C puntos libres y c=circle(A,B,C)

que c=circle(O,G) y A=point(c) B=point(c) C=point(c)

es muy posible que tanto punto sobre caminos produzca un locus no definido por no ser puntos libres. es mejor procurar que solo halla un punto en un camino para definir un locus

observa tu primer locus y su ecuacion realizado así en el adjunto

Files: foro.ggb
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1

Hello,

Thank you for your assistance - I'll have to look at your attached file first. Until then, let me say two things:

- I did not create anything by coding, it's all done by clicking on command icons. Therefore I can't see how anything mistaken/illegal could have been created. Otherwise, I cannot follow your comments above.

- I need the center of that circumcircle, because the only way I could (as accurately as possible) create a parabola tangent to the sides of a triangle is to first create the (pole-polar) dual conic to that parabola (the ellipse visible there), and make sure it passes through the circumcenter, making the dual of this curve be a parabola. In other words, I need to include the center O of this circle among the defining points of the ellipse which is supposed to be the dual of the desired parabola, before dualizing the ellipse to finally obtain that parabola.

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1

Did you look at mathmagic's file foro.ggb?


If you put the points on the grid then you get a nicer equation, see foro2.ggb

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@mathmagic ET6604's uploaded file Cubic controls intersection of Ellipse-Loci.ggb looks fine to me

<command name="Point">
<input a0="c"/>
<output a0="Α"/>
</command>

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see definition of red MM point

indeed a problem with A

I get the attached text. comments in red

and the list of points is the image

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@mathmagic: I have MM=A (and MM isn't used) so what's the problem with that?

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hey ET66044

I think the logic steps are: create A,B,C then the circle then the center of circle

you get the same elements with different freedom

you can see the code of ggb file opening it with 7zip or winzip then editing geogebra.xml

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OK, thanks, I'll try. I had little time since last time, I unfortunately haven't check the sent files yet (nor do I expect it to go fast, since I'm a beginner).

Thanks again,

E.T.

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hey Michael

in the file there is two points named A. see the jpg attached in above post

/wdjP2ecBOds3AAAAABJRU5ErkJggg==

it does not matter. the file can be fixed

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