Question

You are given functions x[t], y[t],and z[t]. When you plot x[t], y[t], z[t] in threedimensions for 0 <= t <= 4, you get a curve. What would you plot in three dimensions to getthe tube consisting of the edges of all circles of radius 0.5 parallel to the xy - plane andcentered on the curve?What would you do to plot a sample of a horn consisting of edges of all circles parallel tothe xy-plane and centered on the curve?

Answer 1

I'm not even sure this makes sense, but something like\[(x-x(t))^2+(y-y(t))^2=\frac14\]should be a set of circles centered on whatever points (x(t),y(t)) you choose
z(t) can vary normally I suppose
a 'horn' should be something where the radius of the circles depend on z, so\[(x-x(t))^2+(y-y(t))^2=z(t)\]perhaps
I'm not really positive though... maybe someone can build on what I did

Answer 2

\[(x-x(t))^2+(y-y(t))^2=(z(t))^2\]I meant for the horn