Finding point on a curve that is d units apart from given point in direction of increasing arc length
$begingroup$
Suppose we are given a differentiable curve ${bf r} : mathbb{R}
to mathbb{R}^3$ with ${bf r}(t) = (x(t),y(t),z(t)). $ Find the point
on the curve that is at a distance of $d$ units along the curve from
the point $(a,b,c)$ in the direction of increasing arc length.
Generalize.
Attempt:
Suppose that point occurs at some time $T$: that is the point is $(x(T),y(T),z(T))$. The arc-length parametrization is given by
$$ s(T) = intlimits_t^T || {bf r'(t) } || mathrm{d} t$$
where $t$ is the found by solving the system
begin{array}
d x(t) = a \
y(t) = b \
z(t) = c \
end{array}
I want $s(T) = d$ (given ) and this should give an explicit value for $T$. I dont see any way to express the point since the system cannot always be solved, so I guess this only works for certain $t$. Is this a correct reasoning?
calculus
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
$endgroup$
add a comment |
$begingroup$
Suppose we are given a differentiable curve ${bf r} : mathbb{R}
to mathbb{R}^3$ with ${bf r}(t) = (x(t),y(t),z(t)). $ Find the point
on the curve that is at a distance of $d$ units along the curve from
the point $(a,b,c)$ in the direction of increasing arc length.
Generalize.
Attempt:
Suppose that point occurs at some time $T$: that is the point is $(x(T),y(T),z(T))$. The arc-length parametrization is given by
$$ s(T) = intlimits_t^T || {bf r'(t) } || mathrm{d} t$$
where $t$ is the found by solving the system
begin{array}
d x(t) = a \
y(t) = b \
z(t) = c \
end{array}
I want $s(T) = d$ (given ) and this should give an explicit value for $T$. I dont see any way to express the point since the system cannot always be solved, so I guess this only works for certain $t$. Is this a correct reasoning?
calculus
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
$endgroup$
$begingroup$
That should be $Bbb R to Bbb R^3$, not $Bbb R^3 to Bbb R$. Also, what does "in the direction of increasing arc length" mean?
$endgroup$
– TonyK
Jan 7 at 13:20
add a comment |
$begingroup$
Suppose we are given a differentiable curve ${bf r} : mathbb{R}
to mathbb{R}^3$ with ${bf r}(t) = (x(t),y(t),z(t)). $ Find the point
on the curve that is at a distance of $d$ units along the curve from
the point $(a,b,c)$ in the direction of increasing arc length.
Generalize.
Attempt:
Suppose that point occurs at some time $T$: that is the point is $(x(T),y(T),z(T))$. The arc-length parametrization is given by
$$ s(T) = intlimits_t^T || {bf r'(t) } || mathrm{d} t$$
where $t$ is the found by solving the system
begin{array}
d x(t) = a \
y(t) = b \
z(t) = c \
end{array}
I want $s(T) = d$ (given ) and this should give an explicit value for $T$. I dont see any way to express the point since the system cannot always be solved, so I guess this only works for certain $t$. Is this a correct reasoning?
calculus
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
$endgroup$
Suppose we are given a differentiable curve ${bf r} : mathbb{R}
to mathbb{R}^3$ with ${bf r}(t) = (x(t),y(t),z(t)). $ Find the point
on the curve that is at a distance of $d$ units along the curve from
the point $(a,b,c)$ in the direction of increasing arc length.
Generalize.
Attempt:
Suppose that point occurs at some time $T$: that is the point is $(x(T),y(T),z(T))$. The arc-length parametrization is given by
$$ s(T) = intlimits_t^T || {bf r'(t) } || mathrm{d} t$$
where $t$ is the found by solving the system
begin{array}
d x(t) = a \
y(t) = b \
z(t) = c \
end{array}
I want $s(T) = d$ (given ) and this should give an explicit value for $T$. I dont see any way to express the point since the system cannot always be solved, so I guess this only works for certain $t$. Is this a correct reasoning?
calculus
calculus
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
edited Jan 7 at 13:21
Jimmy Sabater
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
asked Jan 7 at 13:16
Jimmy SabaterJimmy Sabater
2,225319
2,225319
$begingroup$
That should be $Bbb R to Bbb R^3$, not $Bbb R^3 to Bbb R$. Also, what does "in the direction of increasing arc length" mean?
$endgroup$
– TonyK
Jan 7 at 13:20
add a comment |
$begingroup$
That should be $Bbb R to Bbb R^3$, not $Bbb R^3 to Bbb R$. Also, what does "in the direction of increasing arc length" mean?
$endgroup$
– TonyK
Jan 7 at 13:20
$begingroup$
That should be $Bbb R to Bbb R^3$, not $Bbb R^3 to Bbb R$. Also, what does "in the direction of increasing arc length" mean?
$endgroup$
– TonyK
Jan 7 at 13:20
$begingroup$
That should be $Bbb R to Bbb R^3$, not $Bbb R^3 to Bbb R$. Also, what does "in the direction of increasing arc length" mean?
$endgroup$
– TonyK
Jan 7 at 13:20
add a comment |
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$begingroup$
That should be $Bbb R to Bbb R^3$, not $Bbb R^3 to Bbb R$. Also, what does "in the direction of increasing arc length" mean?
$endgroup$
– TonyK
Jan 7 at 13:20