Tangents to a cubic

jtico shared this question 7 years ago

Answered

Hi.

In the file, the cubic was created with the command Cubic[A,B,C,3]. The theory says that the circumcenter (D) is on the curve. If you try to draw the tangent to the curve at the point D, you get a set of lines. On a mac OSX, GG 4.2.3.0 Java: 1.6.0_37.

jtico

https://ggbm.at/559313 17fe666fcd7edb704182f32a231af340

Files: ct.png

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Comments (6)

Michael Borcherds ● 7 years ago

If you try to draw the tangent to the curve at the point D, you get a set of lines

What were you expecting?

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jtico ● 7 years ago

What were you expecting?

I think this issue was discussed before. When a point is on the cubic, using the tangent tool, the result I expect is _the_ tangent to the curve at that point (only one line) as the example below. In fact if you move the points A, B or C, some times you get only one tangent. b6b8ce8e3f2dc2d96c932f725a016a99

Files: ct2.png

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Michael Borcherds ● 7 years ago

D isn't a point fixed on the cubic (in the sense of Point[cubic]) - it could move off. (and GeoGebra can't tell that it in fact can't without doing some very hard maths which would slow everything down... and wouldn't work in all circumstances anyway...)

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jtico ● 7 years ago

Hi.

I'm not sure that I understand your answer. As a user, I feel that this is an inconsistent behaviour of GG. From your answer, it looks like it's not easy to change this, so here is a workaround. Just hide the unwanted tangents.

jtico

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Zbynek ● 7 years ago

Hi,

what we could consider doing is

* detect that D belongs to d symbolically -- as Mike says, this is a performance killer

* detect that D belongs to d numerically / probabilistically -- won't be very stable

* make sure that at least Tangent[ClosestPoint[d,D],d] works -- I guess we can do that

Cheers,

Zbynek

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jtico ● 7 years ago

Hi Zbynek. Thanks for your answer. Now I understand that as developers you are concerned about stability.

I was wondering. What about Distance[Point,Object]?. It looks like this is a robust numerical test for deciding if a point is on a curve or not.

Cheers, jtico.

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