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Arc length of nowhere differentiable arcs.
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Does there exist an arc (something homeomorphic to the interval $[0, 1]$) that is nowhere differentiable? If so, how does one define the arc length function along such an arc? Can a topological arc have infinite length?
general-topology analysis
asked 57 mins ago
gorzardfu
428211
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Does there exist an arc (something homeomorphic to the interval $[0, 1]$) that is nowhere differentiable? If so, how does one define the arc length function along such an arc? Can a topological arc have infinite length?
general-topology analysis
asked 57 mins ago
gorzardfu
428211
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Does there exist an arc (something homeomorphic to the interval $[0, 1]$) that is nowhere differentiable? If so, how does one define the arc length function along such an arc? Can a topological arc have infinite length?
general-topology analysis
asked 57 mins ago
gorzardfu
428211
Does there exist an arc (something homeomorphic to the interval $[0, 1]$) that is nowhere differentiable? If so, how does one define the arc length function along such an arc? Can a topological arc have infinite length?
general-topology analysis
general-topology analysis
asked 57 mins ago
gorzardfu
428211
asked 57 mins ago
gorzardfu
428211
edited 46 mins ago
asked 57 mins ago
gorzardfu
428211
asked 57 mins ago
gorzardfu
428211
asked 57 mins ago
gorzardfu
428211
428211
add a comment |Â
add a comment |Â
1 Answer
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You're looking for the notion of a rectifiable curve. It's possible to talk about the length of a general curve by approximating it with line segments. As those line segments in the approximation get smaller, their total length will increase. If it doesn't go to infinity, then it approaches a value, defined to be the arc length of the curve, and the curve is said to be rectifiable. You can show that if the curve is differentiable, this process will give the same result for the arc length as the more familiar
$$int sqrt(x')^2+(y')^2,dt.$$
(See https://en.wikipedia.org/wiki/Arc_length for more details.)
As to the question of whether a nowhere differentiable curve can be rectifiable or not, using this definition one can show that a nowhere differentiable curve is never rectifiable (see e.g. this question).
answered 34 mins ago
Carmeister
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
You're looking for the notion of a rectifiable curve. It's possible to talk about the length of a general curve by approximating it with line segments. As those line segments in the approximation get smaller, their total length will increase. If it doesn't go to infinity, then it approaches a value, defined to be the arc length of the curve, and the curve is said to be rectifiable. You can show that if the curve is differentiable, this process will give the same result for the arc length as the more familiar
$$int sqrt(x')^2+(y')^2,dt.$$
(See https://en.wikipedia.org/wiki/Arc_length for more details.)
As to the question of whether a nowhere differentiable curve can be rectifiable or not, using this definition one can show that a nowhere differentiable curve is never rectifiable (see e.g. this question).
answered 34 mins ago
Carmeister
2,3691720
add a comment |Â
up vote
4
down vote
You're looking for the notion of a rectifiable curve. It's possible to talk about the length of a general curve by approximating it with line segments. As those line segments in the approximation get smaller, their total length will increase. If it doesn't go to infinity, then it approaches a value, defined to be the arc length of the curve, and the curve is said to be rectifiable. You can show that if the curve is differentiable, this process will give the same result for the arc length as the more familiar
$$int sqrt(x')^2+(y')^2,dt.$$
(See https://en.wikipedia.org/wiki/Arc_length for more details.)
As to the question of whether a nowhere differentiable curve can be rectifiable or not, using this definition one can show that a nowhere differentiable curve is never rectifiable (see e.g. this question).
answered 34 mins ago
Carmeister
2,3691720
add a comment |Â
up vote
4
down vote
up vote
4
down vote
You're looking for the notion of a rectifiable curve. It's possible to talk about the length of a general curve by approximating it with line segments. As those line segments in the approximation get smaller, their total length will increase. If it doesn't go to infinity, then it approaches a value, defined to be the arc length of the curve, and the curve is said to be rectifiable. You can show that if the curve is differentiable, this process will give the same result for the arc length as the more familiar
$$int sqrt(x')^2+(y')^2,dt.$$
(See https://en.wikipedia.org/wiki/Arc_length for more details.)
As to the question of whether a nowhere differentiable curve can be rectifiable or not, using this definition one can show that a nowhere differentiable curve is never rectifiable (see e.g. this question).
answered 34 mins ago
Carmeister
2,3691720
You're looking for the notion of a rectifiable curve. It's possible to talk about the length of a general curve by approximating it with line segments. As those line segments in the approximation get smaller, their total length will increase. If it doesn't go to infinity, then it approaches a value, defined to be the arc length of the curve, and the curve is said to be rectifiable. You can show that if the curve is differentiable, this process will give the same result for the arc length as the more familiar
$$int sqrt(x')^2+(y')^2,dt.$$
(See https://en.wikipedia.org/wiki/Arc_length for more details.)
As to the question of whether a nowhere differentiable curve can be rectifiable or not, using this definition one can show that a nowhere differentiable curve is never rectifiable (see e.g. this question).
answered 34 mins ago
Carmeister
2,3691720
answered 34 mins ago
Carmeister
2,3691720
answered 34 mins ago
Carmeister
2,3691720
answered 34 mins ago
Carmeister
2,3691720
2,3691720
add a comment |Â
add a comment |Â
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