Find curve satisfying conditions.
Given a point:
$$ overline{x} = (x_1,x_2)$$
And a curve:
$$ a_1(t) = (a_{11}(t), a_{12}(t)) $$
I am asked to find the function
$$ a_2(t) = (a_{21}(t), a_{22}(t))$$
Such that:
$$ (a_2(t)-overline{x})cdot(a_2(t)-overline{x})=c_1tag1$$
$$ (a_2(t)-a_1(t))cdot(a_2(t)-a_1(t))=c_2tag2$$
If we define $$F:R^3 mapsto R, F(overline{x},t) = (overline{x}-a_1(t))cdot (overline{x}-a_1(t))tag3$$
$$ h(t) = (a_{21}(t), a_{22}(t), t) tag4$$
Since $ F circ h(t) = c_2 $ using the chain rule:
$$ (a_1(t) cdot a'_1(t)) + (a_2(t) cdot a_2'(t))- (a_1'(t)cdot a_2(t) )- (a'_2(t)cdot a_1(t)) = 0tag5$$
From here I suppose that $$ a_1(t) = (c_1) ^2(sin(k(t)) + x_1, cos(k(t)) + x_2)tag6$$
And try t solve for $k(t)$.
I dont have idea how to solve the problem, I need a hint or something because I dont think I am going in the right diretion.
calculus real-analysis geometry multivariable-calculus differential-geometry
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
asked Nov 29 at 3:58
cer
405
add a comment |
Given a point:
$$ overline{x} = (x_1,x_2)$$
And a curve:
$$ a_1(t) = (a_{11}(t), a_{12}(t)) $$
I am asked to find the function
$$ a_2(t) = (a_{21}(t), a_{22}(t))$$
Such that:
$$ (a_2(t)-overline{x})cdot(a_2(t)-overline{x})=c_1tag1$$
$$ (a_2(t)-a_1(t))cdot(a_2(t)-a_1(t))=c_2tag2$$
If we define $$F:R^3 mapsto R, F(overline{x},t) = (overline{x}-a_1(t))cdot (overline{x}-a_1(t))tag3$$
$$ h(t) = (a_{21}(t), a_{22}(t), t) tag4$$
Since $ F circ h(t) = c_2 $ using the chain rule:
$$ (a_1(t) cdot a'_1(t)) + (a_2(t) cdot a_2'(t))- (a_1'(t)cdot a_2(t) )- (a'_2(t)cdot a_1(t)) = 0tag5$$
From here I suppose that $$ a_1(t) = (c_1) ^2(sin(k(t)) + x_1, cos(k(t)) + x_2)tag6$$
And try t solve for $k(t)$.
I dont have idea how to solve the problem, I need a hint or something because I dont think I am going in the right diretion.
calculus real-analysis geometry multivariable-calculus differential-geometry
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
asked Nov 29 at 3:58
cer
405
I took the liberty of numbering your equations so that they could be referred to. Now, for any given $t$ you may just find $a_2(t)$ from (1) and (2). It's pure planimetry, nothing more. You don't need any calculus at all.
– Ivan Neretin
Nov 29 at 9:57
add a comment |
Given a point:
$$ overline{x} = (x_1,x_2)$$
And a curve:
$$ a_1(t) = (a_{11}(t), a_{12}(t)) $$
I am asked to find the function
$$ a_2(t) = (a_{21}(t), a_{22}(t))$$
Such that:
$$ (a_2(t)-overline{x})cdot(a_2(t)-overline{x})=c_1tag1$$
$$ (a_2(t)-a_1(t))cdot(a_2(t)-a_1(t))=c_2tag2$$
If we define $$F:R^3 mapsto R, F(overline{x},t) = (overline{x}-a_1(t))cdot (overline{x}-a_1(t))tag3$$
$$ h(t) = (a_{21}(t), a_{22}(t), t) tag4$$
Since $ F circ h(t) = c_2 $ using the chain rule:
$$ (a_1(t) cdot a'_1(t)) + (a_2(t) cdot a_2'(t))- (a_1'(t)cdot a_2(t) )- (a'_2(t)cdot a_1(t)) = 0tag5$$
From here I suppose that $$ a_1(t) = (c_1) ^2(sin(k(t)) + x_1, cos(k(t)) + x_2)tag6$$
And try t solve for $k(t)$.
I dont have idea how to solve the problem, I need a hint or something because I dont think I am going in the right diretion.
calculus real-analysis geometry multivariable-calculus differential-geometry
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
asked Nov 29 at 3:58
cer
405
Given a point:
$$ overline{x} = (x_1,x_2)$$
And a curve:
$$ a_1(t) = (a_{11}(t), a_{12}(t)) $$
I am asked to find the function
$$ a_2(t) = (a_{21}(t), a_{22}(t))$$
Such that:
$$ (a_2(t)-overline{x})cdot(a_2(t)-overline{x})=c_1tag1$$
$$ (a_2(t)-a_1(t))cdot(a_2(t)-a_1(t))=c_2tag2$$
If we define $$F:R^3 mapsto R, F(overline{x},t) = (overline{x}-a_1(t))cdot (overline{x}-a_1(t))tag3$$
$$ h(t) = (a_{21}(t), a_{22}(t), t) tag4$$
Since $ F circ h(t) = c_2 $ using the chain rule:
$$ (a_1(t) cdot a'_1(t)) + (a_2(t) cdot a_2'(t))- (a_1'(t)cdot a_2(t) )- (a'_2(t)cdot a_1(t)) = 0tag5$$
From here I suppose that $$ a_1(t) = (c_1) ^2(sin(k(t)) + x_1, cos(k(t)) + x_2)tag6$$
And try t solve for $k(t)$.
I dont have idea how to solve the problem, I need a hint or something because I dont think I am going in the right diretion.
calculus real-analysis geometry multivariable-calculus differential-geometry
calculus real-analysis geometry multivariable-calculus differential-geometry
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
asked Nov 29 at 3:58
cer
405
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
asked Nov 29 at 3:58
cer
405
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
edited Nov 29 at 9:51
Ivan Neretin
8,79021535
8,79021535
asked Nov 29 at 3:58
cer
405
asked Nov 29 at 3:58
cer
405
asked Nov 29 at 3:58
cer
405
405
I took the liberty of numbering your equations so that they could be referred to. Now, for any given $t$ you may just find $a_2(t)$ from (1) and (2). It's pure planimetry, nothing more. You don't need any calculus at all.
– Ivan Neretin
Nov 29 at 9:57
add a comment |
I took the liberty of numbering your equations so that they could be referred to. Now, for any given $t$ you may just find $a_2(t)$ from (1) and (2). It's pure planimetry, nothing more. You don't need any calculus at all.
– Ivan Neretin
Nov 29 at 9:57
I took the liberty of numbering your equations so that they could be referred to. Now, for any given $t$ you may just find $a_2(t)$ from (1) and (2). It's pure planimetry, nothing more. You don't need any calculus at all.
– Ivan Neretin
Nov 29 at 9:57
I took the liberty of numbering your equations so that they could be referred to. Now, for any given $t$ you may just find $a_2(t)$ from (1) and (2). It's pure planimetry, nothing more. You don't need any calculus at all.
– Ivan Neretin
Nov 29 at 9:57
add a comment |
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I took the liberty of numbering your equations so that they could be referred to. Now, for any given $t$ you may just find $a_2(t)$ from (1) and (2). It's pure planimetry, nothing more. You don't need any calculus at all.
– Ivan Neretin
Nov 29 at 9:57