What is the easiest way to see that the path r _ : R →...

therightwomanwf

therightwomanwf

Answered

2022-07-03

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 ?

Answer & Explanation

Lana Schwartz

Lana Schwartz

Expert

2022-07-04Added 8 answers

Step 1
You 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

Palmosigx

Expert

2022-07-05Added 4 answers

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.

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