Show that the curve defined by γ ( s ) := ( a cos ⁡ ( s a ) , a sin ⁡ ( s a ) ) Is on a circle with the radius a and the center (0,0),also show that γ ( s ) has been parameterized by its arc length.I know that the parametric equation of a circle with radius a and center ( x 0 , y 0 ) is : x = x 0 + a cos ⁡ ( t ) y = y 0 + a sin ⁡ ( t ) If we denote the parametric equation of a circle with γ ( t ) = ( x 0 + a cos ⁡ ( t ) , y 0 + a sin ⁡ ( t ) ) ,then we have: d γ ( t ) d t = ( − a sin ⁡ ( t ) , a cos ⁡ ( t ) ) ‖ d γ ( t ) d t ‖ = a 2 = aSo the arc length is : s = ∫ 0 t ‖ d γ ( τ ) d t ‖ d τ = a ∫ 0 t d τ Which implies s = a t.And if we parameterize the circle by its arc length then: γ ( s ) = ( a cos ⁡ ( s a ) , a sin ⁡ ( s a ) ) Which is the given parametrization.But I have not shown that the center is at the origin.

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Show that the curve defined by  γ  (  s  )  :=      (    a    cos    ⁡          (              s        a            )        ,    a    sin    ⁡          (              s        a            )        )  Is on a circle with the radius   a and the center (0,0),also show that   γ  (  s  ) has been parameterized by its arc length.I know that the parametric equation of   a circle with radius a and center   (      x    0    ,      y    0    ) is :  x  =      x          0        +  a  cos  ⁡      (    t    )    y  =      y          0        +  a  sin  ⁡      (    t    )  If we denote the parametric equation of a circle with   γ  (  t  )  =      (          x              0              +    a    cos    ⁡          (      t      )        ,          y              0              +    a    sin    ⁡          (      t      )        )  ,then we have:            d      γ      (      t      )              d      t        =      (    −    a    sin    ⁡          (      t      )        ,    a    cos    ⁡          (      t      )        )        ‖                  d        γ        (        t        )                    d        t              ‖    =            a              2              =  aSo the arc length is :  s  =      ∫          0              t            ‖                  d        γ        (                  τ                )                    d        t              ‖    d      τ    =  a      ∫          0              t          d      τ  Which implies   s  =  a  t.And if we parameterize the circle by its arc length then:  γ  (  s  )  =      (    a    cos    ⁡          (              s        a            )        ,    a    sin    ⁡          (              s        a            )        )  Which is the given parametrization.But I have not shown that the center is at the origin.

Tristin Case · Accepted Answer

You are given the parametrization in the form   x  (  s  )  =  a  cos  ⁡      (          s      a        )    =  0  +  a  cos  ⁡      (          s      a        )  ,   y  (  s  )  =  a  sin  ⁡      (          s      a        )    =  0  +  a  sin  ⁡      (          s      a        )  . According to what you say you know about parametrizing circles, that   x  (  t  )  =      x    0    +  a  cos  ⁡  (  t  ) and   y  (  t  )  =      y    0    +  a  sin  ⁡  (  t  ) parametrizes a circle centered at   (      x    0    ,      y    0    ), then you should be able to conclude that   s  ↦            (        x  (  s  )  ,  y  (  s  )            )       parametrizes a circle centered at   (      x    0    ,      y    0    )  =  (  0  ,  0  ) of radius   a.

aggierabz2006zw · Accepted Answer

Note that   x  (  s  )  =  a  cos  ⁡      s    a   and   y  (  s  )  =  a  sin  ⁡      s    a  , which leads to      x    2    +      y    2    =      a    2        (          cos      2        ⁡          s      a        +          sin      2        ⁡          s      a        )    =      a    2  i.e. a circle of the radius a and center at origin.Also note   x  =  a  sin  ⁡  θ and   y  =  a  sin  ⁡  θ, which implies that the polar angle   θ is given by   θ  =      s    a  , or   s  =  a  θ. Thus, s is the arc length corresponding to the circle sector angle   θ and   γ  (  s  ) is parameterized in arc length.

Show that the curve defined by γ ( s ) := ( a c

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