Show that a path lies on a quadric surface We have the path of a particle determined as $\vec{x}(t)=(a\cos(t), b\sin(t), ct)$, $a, b, c > 0$, $t$ is time. If I want to show that the path lies on a quadric surface, do I have to derive a definite equation of the quadric surface? Or can I go about it implicitly? I can show that the path is an oval: $(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$; and that $z=ct$, which implies that the particle "draws" a spiral oval as time goes. I'm, however, wondering if we can show by an explicit formula that the oval spiral lies on an oval surface. My guess is no.

The parametrization is sufficient to say that you are extruding an ellipse and on that prism you are going up a helical a path. The surface is an elliptic cylinder as distinguished from the circular cylinder.

It need not lie on a quadric surfaces _alone_. A rigid helix can be drawn on an infinite set of blown in or blown out surfaces.

Mathematical formulation may be possible, but involved. I for one like to avoid math if a physical situation can be imagined.

If a wire-frame model of the helix is made and somehow made to enclose space, dipped in a soap solution it spans a minimal surface. By putting in or taking our air into the inner space several surfaces of positive and negative Gauss curvature can be created as grooves or humps..

Unless a property of the surface is mentioned, a curve cannot determine a surface on which it mounts.