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mun
 Post subject: two curves touch
PostPosted: Thu, 8 Mar 2012 18:45:46 UTC 
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when did two curves touch each other.
if there slopes are equal at their point of intersection then is it sufficient?

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Shadow
 Post subject: Re: two curves touch
PostPosted: Thu, 8 Mar 2012 21:02:39 UTC 
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mun wrote:
when did two curves touch each other.
if there slopes are equal at their point of intersection then is it sufficient?


Huh? What do you mean by "touch", because usually that means they are just intersecting. If you're demanding tangency, you need to make sure the tangent spaces at the point are the same.

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Shadow
 Post subject: Re: two curves touch
PostPosted: Thu, 8 Mar 2012 21:19:54 UTC 
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Shadow wrote:
mun wrote:
when did two curves touch each other.
if there slopes are equal at their point of intersection then is it sufficient?


Huh? What do you mean by "touch", because usually that means they are just intersecting. If you're demanding tangency, you need to make sure the tangent spaces at the point are the same.


where "the same" means that considering them as naturally embedded in the space where you have your curves are equal.

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outermeasure
 Post subject: Re: two curves touch
PostPosted: Fri, 9 Mar 2012 08:16:47 UTC 
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Shadow wrote:
Shadow wrote:
mun wrote:
when did two curves touch each other.
if there slopes are equal at their point of intersection then is it sufficient?


Huh? What do you mean by "touch", because usually that means they are just intersecting. If you're demanding tangency, you need to make sure the tangent spaces at the point are the same.


where "the same" means that considering them as naturally embedded in the space where you have your curves are equal.


where we are assuming the curves are differentiable at the common point, blahblahblah.

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\begin{aligned} Spin(1)&=O(1)=\mathbb{Z}/2&\quad&\text{and}\\ Spin(2)&=U(1)=SO(2)&&\text{are obvious}\\ Spin(3)&=Sp(1)=SU(2)&&\text{by }q\mapsto(\mathop{\mathrm{Im}}\mathbb{H}\ni p\mapsto qp\bar{q})\\ Spin(4)&=Sp(1)\times Sp(1)&&\text{by }(q_1,q_2)\mapsto(\mathbb{H}\ni p\mapsto q_1p\bar{q_2})\\ Spin(5)&=Sp(2)&&\text{by }\mathbb{HP}^1\cong S^4_{round}\hookrightarrow\mathbb{R}^5\\ Spin(6)&=SU(4)&&\text{by the irrep }\Lambda_+\mathbb{C}^4 \end{aligned}
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Shadow
 Post subject: Re: two curves touch
PostPosted: Fri, 9 Mar 2012 08:36:13 UTC 
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outermeasure wrote:
Shadow wrote:
Shadow wrote:
mun wrote:
when did two curves touch each other.
if there slopes are equal at their point of intersection then is it sufficient?


Huh? What do you mean by "touch", because usually that means they are just intersecting. If you're demanding tangency, you need to make sure the tangent spaces at the point are the same.


where "the same" means that considering them as naturally embedded in the space where you have your curves are equal.


where we are assuming the curves are differentiable at the common point, blahblahblah.


:)

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