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Question Number 3695 by Rasheed Soomro last updated on 19/Dec/15
Can we say that  A line is a circle whose radius is ∞  Or  A circle with ∞ radius is a line  ?
$$\mathcal{C}{an}\:{we}\:{say}\:{that} \\ $$$$\mathcal{A}\:{line}\:{is}\:{a}\:{circle}\:{whose}\:{radius}\:{is}\:\infty \\ $$$$\mathcal{O}{r} \\ $$$${A}\:{circle}\:{with}\:\infty\:{radius}\:{is}\:{a}\:{line}\:\:? \\ $$
Commented by Filup last updated on 19/Dec/15
A line l^→ =∞ is like a vector  l^→  moves in one direction (1D).  l^→ ∈R^1     A circle with radus r^→ =∞is similar.  Except a circle will eventually create  tangents facing every possible direction  in what we can call a 2D plane.  r^→ ∈R^2     This is just what i think. I could be  wrong.
$${A}\:\mathrm{line}\:\overset{\rightarrow} {{l}}=\infty\:\mathrm{is}\:\mathrm{like}\:\mathrm{a}\:\mathrm{vector} \\ $$$$\overset{\rightarrow} {{l}}\:\mathrm{moves}\:\mathrm{in}\:\mathrm{one}\:\mathrm{direction}\:\left(\mathrm{1D}\right). \\ $$$$\overset{\rightarrow} {{l}}\in\mathbb{R}^{\mathrm{1}} \\ $$$$ \\ $$$$\mathrm{A}\:\mathrm{circle}\:\mathrm{with}\:\mathrm{radus}\:\overset{\rightarrow} {{r}}=\infty\mathrm{is}\:\mathrm{similar}. \\ $$$$\mathrm{Except}\:\mathrm{a}\:\mathrm{circle}\:\mathrm{will}\:\mathrm{eventually}\:\mathrm{create} \\ $$$$\mathrm{tangents}\:\mathrm{facing}\:\mathrm{every}\:\mathrm{possible}\:\mathrm{direction} \\ $$$$\mathrm{in}\:\mathrm{what}\:\mathrm{we}\:\mathrm{can}\:\mathrm{call}\:\mathrm{a}\:\mathrm{2D}\:\mathrm{plane}. \\ $$$$\overset{\rightarrow} {{r}}\in\mathbb{R}^{\mathrm{2}} \\ $$$$ \\ $$$$\mathrm{This}\:\mathrm{is}\:\mathrm{just}\:\mathrm{what}\:\mathrm{i}\:\mathrm{think}.\:\mathrm{I}\:\mathrm{could}\:\mathrm{be} \\ $$$$\mathrm{wrong}. \\ $$
Commented by Filup last updated on 19/Dec/15
Actually, a correction i want to point out.    r^→ ∉R^2   the circle c^→ ∈R^2 .  The radius is 1D. r^→ ∈R^1     A line only has one feature:  its length l^→ ∈R^1     A circle (c^→ ) has two features.  The radius and the circumference.  r^→ ∈R^1             c^→ ∈R^2   The circle itself lies within a 2D plane.  The radius is in a 1D plane
$${A}\mathrm{ctually},\:{a}\:{correction}\:{i}\:{want}\:{to}\:{point}\:{out}. \\ $$$$ \\ $$$$\overset{\rightarrow} {{r}}\notin\mathbb{R}^{\mathrm{2}} \\ $$$${the}\:\mathrm{circle}\:\overset{\rightarrow} {{c}}\in\mathbb{R}^{\mathrm{2}} . \\ $$$$\mathrm{The}\:\mathrm{radius}\:\mathrm{is}\:\mathrm{1D}.\:\overset{\rightarrow} {{r}}\in\mathbb{R}^{\mathrm{1}} \\ $$$$ \\ $$$${A}\:\mathrm{line}\:\mathrm{only}\:\mathrm{has}\:\mathrm{one}\:\mathrm{feature}: \\ $$$$\mathrm{its}\:\mathrm{length}\:\overset{\rightarrow} {{l}}\in\mathbb{R}^{\mathrm{1}} \\ $$$$ \\ $$$$\mathrm{A}\:\mathrm{circle}\:\left(\overset{\rightarrow} {{c}}\right)\:\mathrm{has}\:\mathrm{two}\:\mathrm{features}. \\ $$$$\mathrm{The}\:\mathrm{radius}\:\mathrm{and}\:\mathrm{the}\:\mathrm{circumference}. \\ $$$$\overset{\rightarrow} {{r}}\in\mathbb{R}^{\mathrm{1}} \:\:\:\:\:\:\:\:\:\:\:\:\overset{\rightarrow} {{c}}\in\mathbb{R}^{\mathrm{2}} \\ $$$$\mathrm{The}\:\mathrm{circle}\:\mathrm{itself}\:\mathrm{lies}\:\mathrm{within}\:\mathrm{a}\:\mathrm{2D}\:\mathrm{plane}. \\ $$$$\mathrm{The}\:\mathrm{radius}\:\mathrm{is}\:\mathrm{in}\:\mathrm{a}\:\mathrm{1D}\:\mathrm{plane} \\ $$
Commented by RasheedSindhi last updated on 19/Dec/15
Perhaps I can′t clear what I mean.  If an object revolves on a circular  path, the object is said to be    contineously changing direction.  Consider my ′change of direction′  somewhat like that.  Consider a circle path of moving  point which keeps changing  direction at every position.
$${Perhaps}\:{I}\:{can}'{t}\:{clear}\:{what}\:{I}\:{mean}. \\ $$$$\mathcal{I}{f}\:{an}\:{object}\:{revolves}\:{on}\:{a}\:{circular} \\ $$$${path},\:{the}\:{object}\:{is}\:{said}\:{to}\:{be}\:\: \\ $$$${contineously}\:{changing}\:{direction}. \\ $$$${Consider}\:{my}\:'{change}\:{of}\:{direction}' \\ $$$${somewhat}\:{like}\:{that}. \\ $$$${Consider}\:{a}\:{circle}\:{path}\:{of}\:{moving} \\ $$$${point}\:{which}\:{keeps}\:{changing} \\ $$$${direction}\:{at}\:{every}\:{position}. \\ $$
Commented by prakash jain last updated on 19/Dec/15
Infinite radius means r→∞.  What is the area of line? If circle is  treated as line.
$$\mathrm{Infinite}\:\mathrm{radius}\:\mathrm{means}\:{r}\rightarrow\infty. \\ $$$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{line}?\:\mathrm{If}\:\mathrm{circle}\:\mathrm{is} \\ $$$$\mathrm{treated}\:\mathrm{as}\:\mathrm{line}. \\ $$
Commented by Rasheed Soomro last updated on 19/Dec/15
Area of line (which is considered as circle) is  half plane. Half plane in which centre occurs  may be considered as interior of the ′circle′  and the other half plane is exterior and length  of line (of course ∞) as perimeter.    When I consider a circle of ∞ radius I see a line!  Its change of direction is 0 .As r gets greater, change  of direction of circle gets smaller and smaller.
$${Area}\:{of}\:{line}\:\left({which}\:{is}\:{considered}\:{as}\:{circle}\right)\:{is} \\ $$$$\boldsymbol{\mathrm{half}}\:\boldsymbol{\mathrm{plane}}.\:{Half}\:{plane}\:{in}\:{which}\:{centre}\:{occurs} \\ $$$${may}\:{be}\:{considered}\:{as}\:\boldsymbol{\mathrm{interior}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{the}}\:'\boldsymbol{\mathrm{circle}}' \\ $$$${and}\:{the}\:{other}\:{half}\:{plane}\:{is}\:\boldsymbol{\mathrm{exterior}}\:{and}\:{length} \\ $$$${of}\:{line}\:\left({of}\:{course}\:\infty\right)\:{as}\:\boldsymbol{\mathrm{perimeter}}. \\ $$$$ \\ $$$$\mathcal{W}{hen}\:\mathcal{I}\:{consider}\:{a}\:{circle}\:{of}\:\infty\:{radius}\:\mathcal{I}\:{see}\:{a}\:{line}! \\ $$$$\mathcal{I}{ts}\:{change}\:{of}\:{direction}\:{is}\:\mathrm{0}\:.\mathcal{A}{s}\:{r}\:{gets}\:{greater},\:{change} \\ $$$${of}\:{direction}\:{of}\:{circle}\:{gets}\:{smaller}\:{and}\:{smaller}. \\ $$
Commented by prakash jain last updated on 19/Dec/15
If by change of direction you mean angle  toward the center=((arc length)/(radius))  So total =(L/r)=2π and it is independent of radius  r→∞, Total change of direction is still 2π
$$\mathrm{If}\:\mathrm{by}\:\mathrm{change}\:\mathrm{of}\:\mathrm{direction}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{angle} \\ $$$$\mathrm{toward}\:\mathrm{the}\:\mathrm{center}=\frac{\mathrm{arc}\:\mathrm{length}}{\mathrm{radius}} \\ $$$$\mathrm{So}\:\mathrm{total}\:=\frac{\mathrm{L}}{{r}}=\mathrm{2}\pi\:\mathrm{and}\:\mathrm{it}\:\mathrm{is}\:\mathrm{independent}\:\mathrm{of}\:\mathrm{radius} \\ $$$${r}\rightarrow\infty,\:\mathrm{Total}\:\mathrm{change}\:\mathrm{of}\:\mathrm{direction}\:\mathrm{is}\:\mathrm{still}\:\mathrm{2}\pi \\ $$
Commented by RasheedSindhi last updated on 19/Dec/15
By change of direction I mean  change of angle of tangents.  Let the center of the  circle is  origin of the plane and the point  of x−axis through which the  circle passes is point A.  At A the tangent is perpendicular  to the x−axis.Let B is an other  point of circle. Tangent on B  will make angle (say) θ with  x−axis. I mean  by ′change of  direction′ was keeping change of the  value of θ.
$${By}\:{change}\:{of}\:{direction}\:{I}\:{mean} \\ $$$${change}\:{of}\:{angle}\:{of}\:{tangents}. \\ $$$${Let}\:{the}\:{center}\:{of}\:{the}\:\:{circle}\:{is} \\ $$$${origin}\:{of}\:{the}\:{plane}\:{and}\:{the}\:{point} \\ $$$${of}\:{x}−{axis}\:{through}\:{which}\:{the} \\ $$$${circle}\:{passes}\:{is}\:{point}\:{A}. \\ $$$${At}\:{A}\:{the}\:{tangent}\:{is}\:{perpendicular} \\ $$$${to}\:{the}\:{x}−{axis}.{Let}\:{B}\:{is}\:{an}\:{other} \\ $$$${point}\:{of}\:{circle}.\:{Tangent}\:{on}\:{B} \\ $$$${will}\:{make}\:{angle}\:\left({say}\right)\:\theta\:{with} \\ $$$${x}−{axis}.\:{I}\:{mean}\:\:{by}\:'{change}\:{of} \\ $$$${direction}'\:{was}\:{keeping}\:{change}\:{of}\:{the} \\ $$$${value}\:{of}\:\theta. \\ $$
Commented by prakash jain last updated on 19/Dec/15
Let AB be a arc of circle and ((length of arc AB)/r)=θ  then Angle between tangent at A and tangent at  B=θ.  So change in angle of tangent is same as the angle   subtended by arc at the center.
$$\mathrm{Let}\:\mathrm{AB}\:\mathrm{be}\:\mathrm{a}\:\mathrm{arc}\:\mathrm{of}\:\mathrm{circle}\:\mathrm{and}\:\frac{\mathrm{length}\:\mathrm{of}\:\mathrm{arc}\:\mathrm{AB}}{{r}}=\theta \\ $$$$\mathrm{then}\:\mathrm{Angle}\:\mathrm{between}\:\mathrm{tangent}\:\mathrm{at}\:\mathrm{A}\:\mathrm{and}\:\mathrm{tangent}\:\mathrm{at} \\ $$$$\mathrm{B}=\theta. \\ $$$$\mathrm{So}\:\mathrm{change}\:\mathrm{in}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{tangent}\:\mathrm{is}\:\mathrm{same}\:\mathrm{as}\:\mathrm{the}\:\mathrm{angle}\: \\ $$$$\mathrm{subtended}\:\mathrm{by}\:\mathrm{arc}\:\mathrm{at}\:\mathrm{the}\:\mathrm{center}. \\ $$
Commented by prakash jain last updated on 19/Dec/15
I might have been wrong in my earlier arguments.  But let us we have have circle centered at  origin (0,0)  x^2 +y^2 =r^2 ⇒(x^2 /r^2 )+(y^2 /r^2 )=1  lim_(r→∞)  what does the above equation become?  Now let us say center of the circle is at (r,0)  (x−r)^2 +y^2 =r^2   x^2 −2xr+y^2 =0  ((x^2 +y^2 )/r)−2x=0  as r→∞  equation reduced to x=0 (y−axis)  So you can say that the line y−axis is a circle  with center at (∞,0).
$$\mathrm{I}\:\mathrm{might}\:\mathrm{have}\:\mathrm{been}\:\mathrm{wrong}\:\mathrm{in}\:\mathrm{my}\:\mathrm{earlier}\:\mathrm{arguments}. \\ $$$$\mathrm{But}\:\mathrm{let}\:\mathrm{us}\:\mathrm{we}\:\mathrm{have}\:\mathrm{have}\:\mathrm{circle}\:\mathrm{centered}\:\mathrm{at} \\ $$$$\mathrm{origin}\:\left(\mathrm{0},\mathrm{0}\right) \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \Rightarrow\frac{{x}^{\mathrm{2}} }{{r}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{r}^{\mathrm{2}} }=\mathrm{1} \\ $$$$\underset{{r}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{what}\:\mathrm{does}\:\mathrm{the}\:\mathrm{above}\:\mathrm{equation}\:\mathrm{become}? \\ $$$$\mathrm{Now}\:\mathrm{let}\:\mathrm{us}\:\mathrm{say}\:\mathrm{center}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{is}\:\mathrm{at}\:\left({r},\mathrm{0}\right) \\ $$$$\left({x}−{r}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} ={r}^{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} −\mathrm{2}{xr}+{y}^{\mathrm{2}} =\mathrm{0} \\ $$$$\frac{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{{r}}−\mathrm{2}{x}=\mathrm{0} \\ $$$$\mathrm{as}\:{r}\rightarrow\infty \\ $$$$\mathrm{equation}\:\mathrm{reduced}\:\mathrm{to}\:{x}=\mathrm{0}\:\left({y}−{axis}\right) \\ $$$$\mathrm{So}\:\mathrm{you}\:\mathrm{can}\:{say}\:{that}\:{the}\:{line}\:{y}−{axis}\:\mathrm{is}\:\mathrm{a}\:\mathrm{circle} \\ $$$${with}\:{center}\:{at}\:\left(\infty,\mathrm{0}\right).\: \\ $$
Commented by Rasheed Soomro last updated on 19/Dec/15
That is a point to think!
$$\mathcal{T}{hat}\:{is}\:{a}\:{point}\:{to}\:{think}! \\ $$
Commented by prakash jain last updated on 19/Dec/15
Also with x=0 being the circle also makes  your earlier argument about half plane  being the area valid.  And all tangents are parallel.  You were correct!
$$\mathrm{Also}\:\mathrm{with}\:{x}=\mathrm{0}\:\mathrm{being}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{also}\:\mathrm{makes} \\ $$$$\mathrm{your}\:\mathrm{earlier}\:\mathrm{argument}\:\mathrm{about}\:\mathrm{half}\:\mathrm{plane} \\ $$$$\mathrm{being}\:\mathrm{the}\:\mathrm{area}\:\mathrm{valid}. \\ $$$$\mathrm{And}\:\mathrm{all}\:\mathrm{tangents}\:\mathrm{are}\:\mathrm{parallel}. \\ $$$$\mathrm{You}\:\mathrm{were}\:\mathrm{correct}! \\ $$
Commented by Rasheed Soomro last updated on 20/Dec/15
An other term in which we can clear  our idea is ′curveness′  As radius of circle gets greater  the curveness gets smaller.  For r=1, for curve of unit length  the curveness is obvious.But for  r=1000,it is not so obvious.  So we can say:  r→∞ ⇒ curveness→0  or curveness→straightness  or  circle→line
$${An}\:{other}\:{term}\:{in}\:{which}\:{we}\:{can}\:{clear} \\ $$$${our}\:{idea}\:{is}\:'{curveness}' \\ $$$${As}\:{radius}\:{of}\:{circle}\:{gets}\:{greater} \\ $$$${the}\:{curveness}\:{gets}\:{smaller}. \\ $$$${For}\:{r}=\mathrm{1},\:{for}\:{curve}\:{of}\:{unit}\:{length} \\ $$$${the}\:{curveness}\:{is}\:{obvious}.{But}\:{for} \\ $$$${r}=\mathrm{1000},{it}\:{is}\:{not}\:{so}\:{obvious}. \\ $$$${So}\:{we}\:{can}\:{say}: \\ $$$${r}\rightarrow\infty\:\Rightarrow\:{curveness}\rightarrow\mathrm{0} \\ $$$${or}\:{curveness}\rightarrow{straightness} \\ $$$${or} \\ $$$${circle}\rightarrow{line} \\ $$

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