Fdtxjr

Writing in a Christian voice

What is this high flying aircraft over Pennsylvania?

Highest stage count that are used one right after the other?

Why didn't Voldemort know what Grindelwald looked like?

Can a Knock spell open the door to Mordenkainen's Magnificent Mansion?

What do the positive and negative (+/-) transmit and receive pins mean on Ethernet cables?

Not hide and seek

Do I have to take mana from my deck or hand when tapping this card?

Is there a distance limit for minecart tracks?

What is the tangent at a sharp point on a curve?

What is the period/term used describe Giuseppe Arcimboldo's style of painting?

How do I lift the insulation blower into the attic?

Derivative of an interpolated function

What should be the ideal length of sentences in a blog post for ease of reading?

How can I, as DM, avoid the Conga Line of Death occurring when implementing some form of flanking rule?

Friend wants my recommendation but I don't want to give it to him

Recursively move files within sub directories

Is there a POSIX way to shutdown a UNIX machine?

Trouble reading roman numeral notation with flats

Did I make a mistake by ccing email to boss to others?

Connection Between Knot Theory and Number Theory

Why does a 97 / 92 key piano exist by Bosendorfer?

Mortal danger in mid-grade literature

Taking the numerator and the denominator

Weak Tangent Problem - Reconciling Two Approaches

Why is the derivative the tangent vector?Parametrized curve tangent to a lineDefinition of strong tangent.Weak tangent but not a strong tangentClosed Smooth Curve with Curvature 1 is a CircleWhat is the rigorous formal definition of the “limit position” of a line with variable parametersCan a “not strong tangent” trace admit a $C^infty$ curve parameterization?Weak tangent vs. Strong tangentWhen to use different formulas to find the slope of a tangent lineConfusion about a tangent line approaching an asymptote

0

$begingroup$

The problem below is given in Do Carmo's Differential Geometry of Curves and Surfaces.

My question is regarding part (a).

Let us first show $alpha:IrightarrowmathbbR^3$ has weak tangent at $t_0=0$.

The line determined by $alpha(h)$ and $alpha(0)$ is given by

$$ell(s)=alpha(0)+s(alpha(h)-alpha(0)).$$ That is,

$$ell(s)=(0,0)+s(h^3,h^2).$$

Taking the limit as $hrightarrow 0$,

$$lim_hrightarrow0ell(s)=(0,0)+s(0,0).$$

Now, I'm not sure what to make of this limit. The defintion of weak tangent requires $ell$ have a unique limit position. I suppose this is in fact a limit position, but it does not really define a line, just a point!

Now, I've conducted some further investigation and noticed an alternative more satisfactory solution exists.

Rather than consider the point-direction form of the particular line, consider its slope. Let $alpha(t)=(x(t),y(t))$

$$lim_hrightarrow0fracy(h)-y(0)x(h)-x(0)=lim_hrightarrow0frach^2h^3.$$

This limit is clearly undefined as the slope is either $+infty$ or $-infty$. This means the slope is vertical, so we have a vertical line at $t=0$. Thus, due to the existence of a limiting line, we conclude there is a weak tangent at $t=0$.

How do we reconcile the two answers? The two approaches are both equally sound (in my view anyway, I could be wrong). The second approach yields a nice limiting line, whereas the first yields a point.

I think, as the parameter $srightarrowinfty$, $(0,0)+s(0,0)$ is an indeterminant form due to the multiplication of an infinite quantity by 0, but I cannot think of how to resolve this issue analytically.

differential-geometry curves tangent-line

share|cite|improve this question

asked Mar 13 at 18:16

SaruSaru

1097

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  "Now, I'm not sure what to make of this limit." You're trying to take the limit of the points separately and draw the line between those. You run into something much like a $frac 00$ problem this way. Instead, try to express, for any $h$, the line between $alpha(0)$ and $alpha(h)$ as, for instance, $y=ax+b$. Then take the limit of that as $hto0$.
  $endgroup$
  – Arthur
  Mar 13 at 18:29
  
  

  
  
  
  

add a comment | 

0

$begingroup$

The problem below is given in Do Carmo's Differential Geometry of Curves and Surfaces.

My question is regarding part (a).

Let us first show $alpha:IrightarrowmathbbR^3$ has weak tangent at $t_0=0$.

The line determined by $alpha(h)$ and $alpha(0)$ is given by

$$ell(s)=alpha(0)+s(alpha(h)-alpha(0)).$$ That is,

$$ell(s)=(0,0)+s(h^3,h^2).$$

Taking the limit as $hrightarrow 0$,

$$lim_hrightarrow0ell(s)=(0,0)+s(0,0).$$

Now, I'm not sure what to make of this limit. The defintion of weak tangent requires $ell$ have a unique limit position. I suppose this is in fact a limit position, but it does not really define a line, just a point!

Now, I've conducted some further investigation and noticed an alternative more satisfactory solution exists.

Rather than consider the point-direction form of the particular line, consider its slope. Let $alpha(t)=(x(t),y(t))$

$$lim_hrightarrow0fracy(h)-y(0)x(h)-x(0)=lim_hrightarrow0frach^2h^3.$$

This limit is clearly undefined as the slope is either $+infty$ or $-infty$. This means the slope is vertical, so we have a vertical line at $t=0$. Thus, due to the existence of a limiting line, we conclude there is a weak tangent at $t=0$.

How do we reconcile the two answers? The two approaches are both equally sound (in my view anyway, I could be wrong). The second approach yields a nice limiting line, whereas the first yields a point.

I think, as the parameter $srightarrowinfty$, $(0,0)+s(0,0)$ is an indeterminant form due to the multiplication of an infinite quantity by 0, but I cannot think of how to resolve this issue analytically.

differential-geometry curves tangent-line

share|cite|improve this question

asked Mar 13 at 18:16

SaruSaru

1097

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  "Now, I'm not sure what to make of this limit." You're trying to take the limit of the points separately and draw the line between those. You run into something much like a $frac 00$ problem this way. Instead, try to express, for any $h$, the line between $alpha(0)$ and $alpha(h)$ as, for instance, $y=ax+b$. Then take the limit of that as $hto0$.
  $endgroup$
  – Arthur
  Mar 13 at 18:29
  
  

  
  
  
  

add a comment | 

$begingroup$

The problem below is given in Do Carmo's Differential Geometry of Curves and Surfaces.

My question is regarding part (a).

Let us first show $alpha:IrightarrowmathbbR^3$ has weak tangent at $t_0=0$.

The line determined by $alpha(h)$ and $alpha(0)$ is given by

$$ell(s)=alpha(0)+s(alpha(h)-alpha(0)).$$ That is,

$$ell(s)=(0,0)+s(h^3,h^2).$$

Taking the limit as $hrightarrow 0$,

$$lim_hrightarrow0ell(s)=(0,0)+s(0,0).$$

Now, I'm not sure what to make of this limit. The defintion of weak tangent requires $ell$ have a unique limit position. I suppose this is in fact a limit position, but it does not really define a line, just a point!

Now, I've conducted some further investigation and noticed an alternative more satisfactory solution exists.

Rather than consider the point-direction form of the particular line, consider its slope. Let $alpha(t)=(x(t),y(t))$

$$lim_hrightarrow0fracy(h)-y(0)x(h)-x(0)=lim_hrightarrow0frach^2h^3.$$

This limit is clearly undefined as the slope is either $+infty$ or $-infty$. This means the slope is vertical, so we have a vertical line at $t=0$. Thus, due to the existence of a limiting line, we conclude there is a weak tangent at $t=0$.

How do we reconcile the two answers? The two approaches are both equally sound (in my view anyway, I could be wrong). The second approach yields a nice limiting line, whereas the first yields a point.

I think, as the parameter $srightarrowinfty$, $(0,0)+s(0,0)$ is an indeterminant form due to the multiplication of an infinite quantity by 0, but I cannot think of how to resolve this issue analytically.

differential-geometry curves tangent-line

share|cite|improve this question

asked Mar 13 at 18:16

SaruSaru

1097

$endgroup$

share|cite|improve this question

asked Mar 13 at 18:16

SaruSaru

1097

share|cite|improve this question

asked Mar 13 at 18:16

SaruSaru

1097

asked Mar 13 at 18:16

SaruSaru

1097

asked Mar 13 at 18:16

SaruSaru

1097