Re: Intersection of 2D Surfaces in 3D

• To: mathgroup at smc.vnet.net
• Subject: [mg86933] Re: [mg86892] Intersection of 2D Surfaces in 3D
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 26 Mar 2008 04:56:15 -0500 (EST)
• References: <200803250617.BAA10628@smc.vnet.net> <46E32647-0187-4DF7-B920-62BBF4E39C61@mimuw.edu.pl>

```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

```

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