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.