Reparameterize the helix, $alpha(t) = (a*cos(t),a*sin(t),b*t)$ by arc length
$begingroup$
Reparameterize $alpha(t) = (a*cos(t),a*sin(t),b*t)$ by arc length. That is, give an equivalent parameterization $gamma$ of the helix such that $|gamma'(t)|=1$
If $gamma(t)= (a*cos(t/sqrt2a),a*sin(t/sqrt2a),t/sqrt2)$
$rightarrow$ $gamma'(t)= (sqrt2*cos(t/sqrt2a),sqrt2*sin(t/sqrt2a),1/sqrt2)$
$|gamma'(t)|= (sqrt2*cos(t/sqrt2))^2 + (sqrt2*sin(t/sqrt2))^2 + (1/sqrt2)^2$
= $1/2 + 1/2 = 1$
Is this correct? And to get from one parameterization to the other I would use a homeomorphism that sends $t rightarrow t/sqrt2$. Correct?
differential-geometry
edited Jan 12 at 16:24
the_fox
2,90031538
$endgroup$
add a comment |
$begingroup$
Reparameterize $alpha(t) = (a*cos(t),a*sin(t),b*t)$ by arc length. That is, give an equivalent parameterization $gamma$ of the helix such that $|gamma'(t)|=1$
If $gamma(t)= (a*cos(t/sqrt2a),a*sin(t/sqrt2a),t/sqrt2)$
$rightarrow$ $gamma'(t)= (sqrt2*cos(t/sqrt2a),sqrt2*sin(t/sqrt2a),1/sqrt2)$
$|gamma'(t)|= (sqrt2*cos(t/sqrt2))^2 + (sqrt2*sin(t/sqrt2))^2 + (1/sqrt2)^2$
= $1/2 + 1/2 = 1$
Is this correct? And to get from one parameterization to the other I would use a homeomorphism that sends $t rightarrow t/sqrt2$. Correct?
differential-geometry
edited Jan 12 at 16:24
the_fox
2,90031538
$endgroup$
$begingroup$
Unroll the cylinder on which the helix is wrapped. One revolution corresponds to a slanted line segment havibg one componeny of length $2pi a$ and a perpendicular component of length $2pi b$; while one revolution increments $t$ by $2pi$. Work out the length of the unwrapped cycle, compare with $t$ changing by $2pi$, and thence infer the relation between $s$ and $t$.
$endgroup$
– Oscar Lanzi
Jan 12 at 16:31
add a comment |
$begingroup$
Reparameterize $alpha(t) = (a*cos(t),a*sin(t),b*t)$ by arc length. That is, give an equivalent parameterization $gamma$ of the helix such that $|gamma'(t)|=1$
If $gamma(t)= (a*cos(t/sqrt2a),a*sin(t/sqrt2a),t/sqrt2)$
$rightarrow$ $gamma'(t)= (sqrt2*cos(t/sqrt2a),sqrt2*sin(t/sqrt2a),1/sqrt2)$
$|gamma'(t)|= (sqrt2*cos(t/sqrt2))^2 + (sqrt2*sin(t/sqrt2))^2 + (1/sqrt2)^2$
= $1/2 + 1/2 = 1$
Is this correct? And to get from one parameterization to the other I would use a homeomorphism that sends $t rightarrow t/sqrt2$. Correct?
differential-geometry
edited Jan 12 at 16:24
the_fox
2,90031538
$endgroup$
Reparameterize $alpha(t) = (a*cos(t),a*sin(t),b*t)$ by arc length. That is, give an equivalent parameterization $gamma$ of the helix such that $|gamma'(t)|=1$
If $gamma(t)= (a*cos(t/sqrt2a),a*sin(t/sqrt2a),t/sqrt2)$
$rightarrow$ $gamma'(t)= (sqrt2*cos(t/sqrt2a),sqrt2*sin(t/sqrt2a),1/sqrt2)$
$|gamma'(t)|= (sqrt2*cos(t/sqrt2))^2 + (sqrt2*sin(t/sqrt2))^2 + (1/sqrt2)^2$
= $1/2 + 1/2 = 1$
Is this correct? And to get from one parameterization to the other I would use a homeomorphism that sends $t rightarrow t/sqrt2$. Correct?
differential-geometry
differential-geometry
edited Jan 12 at 16:24
the_fox
2,90031538
edited Jan 12 at 16:24
the_fox
2,90031538
edited Jan 12 at 16:24
the_fox
2,90031538
edited Jan 12 at 16:24
the_fox
2,90031538
edited Jan 12 at 16:24
the_fox
2,90031538
2,90031538
asked Jan 12 at 16:01
user624065
$begingroup$
Unroll the cylinder on which the helix is wrapped. One revolution corresponds to a slanted line segment havibg one componeny of length $2pi a$ and a perpendicular component of length $2pi b$; while one revolution increments $t$ by $2pi$. Work out the length of the unwrapped cycle, compare with $t$ changing by $2pi$, and thence infer the relation between $s$ and $t$.
$endgroup$
– Oscar Lanzi
Jan 12 at 16:31
add a comment |
$begingroup$
Unroll the cylinder on which the helix is wrapped. One revolution corresponds to a slanted line segment havibg one componeny of length $2pi a$ and a perpendicular component of length $2pi b$; while one revolution increments $t$ by $2pi$. Work out the length of the unwrapped cycle, compare with $t$ changing by $2pi$, and thence infer the relation between $s$ and $t$.
$endgroup$
– Oscar Lanzi
Jan 12 at 16:31
$begingroup$
Unroll the cylinder on which the helix is wrapped. One revolution corresponds to a slanted line segment havibg one componeny of length $2pi a$ and a perpendicular component of length $2pi b$; while one revolution increments $t$ by $2pi$. Work out the length of the unwrapped cycle, compare with $t$ changing by $2pi$, and thence infer the relation between $s$ and $t$.
$endgroup$
– Oscar Lanzi
Jan 12 at 16:31
$begingroup$
Unroll the cylinder on which the helix is wrapped. One revolution corresponds to a slanted line segment havibg one componeny of length $2pi a$ and a perpendicular component of length $2pi b$; while one revolution increments $t$ by $2pi$. Work out the length of the unwrapped cycle, compare with $t$ changing by $2pi$, and thence infer the relation between $s$ and $t$.
$endgroup$
– Oscar Lanzi
Jan 12 at 16:31
add a comment |
1 Answer
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$begingroup$
No, the re-parameterization is $tto frac{t}{sqrt{2}a}$
if $b=a$; otherwise the re-parameterization is $tto frac{t}{|(a,b)|}$.
edited Jan 12 at 17:27
Namaste
1
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
$endgroup$
$begingroup$
Misread the formula, missing that small "$a$".
$endgroup$
– Oscar Lanzi
Jan 12 at 16:46
add a comment |
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1 Answer
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$begingroup$
No, the re-parameterization is $tto frac{t}{sqrt{2}a}$
if $b=a$; otherwise the re-parameterization is $tto frac{t}{|(a,b)|}$.
edited Jan 12 at 17:27
Namaste
1
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
$endgroup$
$begingroup$
Misread the formula, missing that small "$a$".
$endgroup$
– Oscar Lanzi
Jan 12 at 16:46
add a comment |
$begingroup$
No, the re-parameterization is $tto frac{t}{sqrt{2}a}$
if $b=a$; otherwise the re-parameterization is $tto frac{t}{|(a,b)|}$.
edited Jan 12 at 17:27
Namaste
1
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
$endgroup$
$begingroup$
Misread the formula, missing that small "$a$".
$endgroup$
– Oscar Lanzi
Jan 12 at 16:46
add a comment |
$begingroup$
No, the re-parameterization is $tto frac{t}{sqrt{2}a}$
if $b=a$; otherwise the re-parameterization is $tto frac{t}{|(a,b)|}$.
edited Jan 12 at 17:27
Namaste
1
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
$endgroup$
No, the re-parameterization is $tto frac{t}{sqrt{2}a}$
if $b=a$; otherwise the re-parameterization is $tto frac{t}{|(a,b)|}$.
edited Jan 12 at 17:27
Namaste
1
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
edited Jan 12 at 17:27
Namaste
1
edited Jan 12 at 17:27
Namaste
1
edited Jan 12 at 17:27
Namaste
1
1
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
answered Jan 12 at 16:28
Federico FalluccaFederico Fallucca
2,270210
2,270210
$begingroup$
Misread the formula, missing that small "$a$".
$endgroup$
– Oscar Lanzi
Jan 12 at 16:46
add a comment |
$begingroup$
Misread the formula, missing that small "$a$".
$endgroup$
– Oscar Lanzi
Jan 12 at 16:46
$begingroup$
Misread the formula, missing that small "$a$".
$endgroup$
– Oscar Lanzi
Jan 12 at 16:46
$begingroup$
Misread the formula, missing that small "$a$".
$endgroup$
– Oscar Lanzi
Jan 12 at 16:46
add a comment |
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$begingroup$
Unroll the cylinder on which the helix is wrapped. One revolution corresponds to a slanted line segment havibg one componeny of length $2pi a$ and a perpendicular component of length $2pi b$; while one revolution increments $t$ by $2pi$. Work out the length of the unwrapped cycle, compare with $t$ changing by $2pi$, and thence infer the relation between $s$ and $t$.
$endgroup$
– Oscar Lanzi
Jan 12 at 16:31