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Re: Intersection of 2D Surfaces in 3D
*To*: mathgroup at smc.vnet.net
*Subject*: [mg86899] Re: [mg86892] Intersection of 2D Surfaces in 3D
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Wed, 26 Mar 2008 04:47:15 -0500 (EST)
*References*: <200803250617.BAA10628@smc.vnet.net> <46E32647-0187-4DF7-B920-62BBF4E39C61@mimuw.edu.pl> <C3A823D9-8CDB-4137-A4E5-C8BC6FAF59BF@mimuw.edu.pl>
On 25 Mar 2008, at 08:53, Andrzej Kozlowski wrote:
>
> On 25 Mar 2008, at 08:11, Andrzej Kozlowski wrote:
>>
>> On 25 Mar 2008, at 07:17, Narasimham wrote:
>>> Following is an example (slightly altered) given in intersection
>>> of 2-
>>> D curves with one real root.
>>>
>>> c1 = {x - (t^2 - 1), y - (s^3 + s - 4) };
>>> c2 = {x - (s^2 + s + 5), y - (t^2 + 7 t - 2) };
>>>
>>> It uses NSolve[Join[c1, c2], {x, y}, {s, t}] for supplying real
>>> roots
>>> of 2D curves in 2D itself.
>>>
>>> Next, how to generalize further to Solve and find real intersection
>>> curves of two parameter surfaces in 3-D by extending the same
>>> Mathematica Join procedure?
>>>
>>> And how to Show the one parameter 3D space curve of intersection so
>>> obtained ? The following attempt of course fails.
>>>
>>> c3 = {x - (t^2 - 1), y - (s^3 + s - 4), z - (t + s)};
>>> c4 = {x - (s^2 + s + 5), y - (t^2 + 7 t - 2),z -( t + s^2/2)};
>>> NSolve[Join[c3, c4], {x, y, z}, {t,s}];
>>>
>>> FindRoot also was not successful.
>>>
>>> Regards,
>>> Narasimham
>>>
>>>
>>
>>
>> Your two surfaces do not interesect:
>>
>> In[58]:= GroebnerBasis[Join[c3, c4], {x, y, s, t}]
>> Out[58]= {1}
>>
>> So what do you mena by "fails"? What would constitute a "success"
>> here?
>>
>> Andrzek Kozlowski
>
>
> Actually, it should have been
>
> In[60]:= GroebnerBasis[Join[c3, c4], {x, y, z}, {s, t}]
> Out[60]= {1}
>
> But they still do not intersect.
>
> Andrzej
I was too quick in replying both times. What this shows is not that
the two surfaces do not intersect but only that you can's
simultaneusly choose values of t and s which give a point of
intersection. This is a rather different matter. If we want to find a
point of intersection rather than a point of intersection which
corresponds to the same values of the parameters, we have to treat the
parameters as independent. One way to proceed is as follows:
c3 = {x - (t^2 - 1), y - (s^3 + s - 4), z - (t + s)};
c4 = {x - (s^2 + s + 5), y - (t^2 + 7 t - 2), z - (t + s^2/2)};
g1 = GroebnerBasis[c3, {x, y, z}, {t, s}];
g2 = GroebnerBasis[c4, {x, y, z}, {t, s}];
g = GroebnerBasis[Join[g1, g2], {x, y, z},
MonomialOrder -> EliminationOrder];
{x^4 - 8*z*x^3 - 48*x^3 + 24*z^2*x^2 - 8*y*x^2 +
284*z*x^2 + 736*x^2 - 32*z^3*x - 560*z^2*x +
176*y*x + 32*y*z*x - 2848*z*x - 4068*x + 16*z^4 +
368*z^3 + 16*y^2 - 32*y*z^2 + 2756*z^2 - 564*y -
368*y*z + 7360*z + 7483, z^6 - 3*x*z^4 - z^4 -
2*y*z^3 - 8*z^3 + 3*x^2*z^2 + 6*x*z^2 + 4*z^2 -
24*x*z - 6*x*y*z - 8*y*z - 32*z - x^3 - 5*x^2 +
y^2 - 8*x + 8*y + 12}
Now,
sols = NSolve[g == 0, {x, y, z}, WorkingPrecision -> 100];
During evaluation of In[11]:= NSolve::"infsolns" : "Infinite solution
set has dimension at least 1 . Returning intersection of solutions
with (28373 x)/30371-(85901 y)/60742+z == 1.
I have supressed the solutions but we can check one of them:
In[7]:= Chop[g1 /. First[sols]]
Out[7]= {0}
In[8]:= Chop[g2 /. First[sols]]
Out[8]= {0}
So we certianly do have solutions, even an infinte set of solutions,
but none of them correspond to the same values of s and t. I am not
sure if that is what you wanted or not.
Andrzej Kozlowski
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