What is the easiest way to see that the path r _ : R → R 3 : t ↦ ( sin t , cos t , cos t )traces out an ellipse in the plane y = z ?

Question

What is the easiest way to see that the path      r    _    :      R    →            R        3    :  t  ↦  (  sin    t  ,  cos    t  ,  cos    t  )traces out an ellipse in the plane   y  =  z ?

Lana Schwartz · Accepted Answer

Step 1You are almost there. The rotation by   π      /    4 will yield the path  (      x    ′    ,      y    ′    ,      z    ′    )  =  (  sin  ⁡  t  ,      2    cos  ⁡  t  ,  0  )in the new coordinates. How do you eliminate t? Use      sin    2    ⁡  t  +      cos    2    ⁡  t  =  1      x          ′      2        +            y              ′        2              2    =  1

Palmosigx · Accepted Answer

Your path is also contsined in       x    2    +      y    2    =  1 , which is a cylinder. So your path is in the intersection of this cylinder and the plane   y  =  z . Geometrically, this is an ellipse, since the plane is not paralel to the axis of the cyclinder, so it cuts all generatrices.

What is the easiest way to see that the path <munder> r _ </munde

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