Parametrized curves

Here are the curves from the textbook, Section 13.1, Exercises 21-26.

#21

Note: The curve satisfies the equation $x^2 + z^2 = t^2 (\cos^2(t) + \sin^2(t)) = t^2 = y^2$. Therefore, the curve lies within the surface defined by $x^2 + z^2 = y^2$, which is a cone.

#22

Note: The curve lies within the surface defined by $x^2 + y^2 = 1$, which is a cylinder around the z-axis.

#23

#24

Note: Because of the trigonometric identity $\cos(2t) = \cos^2(t) - \sin^2(t)$, the curve lies within the surface defined by $z = x^2 - y^2$, which is a hyperbolic paraboloid.

Moreover, the curve also lies within the surface defined by $x^2 + y^2 = 1$, which is a cylinder around the z-axis. In fact, the curve is the intersection of those two surfaces.

#25

Note: The curve lies within the surface defined by $x^2 + y^2 = 1$, which is a cylinder around the z-axis.

#26

Note: The curve lies within the plane defined by $x + y = 1$.