Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have a curve given by this equation: $$c(t) = (t\cos t,t \sin t,t)$$

I need to prove that this curve lies on a cone, and draw that cone and curve in Sage.

I've read somewhere that I could prove it by taking some random point and then put it into equation. But is there a more general proof, formal way of proving it lies on a cone?

How should I prove it (the easiest way?)

share|improve this question
5  
It's obvious that $z^2=x^2+y^2$ and that's the equation of a (double) cone. –  kiwi May 20 '12 at 23:42
1  
@kiwi Can you add that as answer so that this question gets an answer? Thanks! –  user17762 May 21 '12 at 0:22
add comment

1 Answer

In this case since you have your parametrized curve given by $\gamma(t) = (t\cos{t}, t\sin{t}, t)$, one way of trying to find an algebraic equation satisfied by any point on your curve is writing $x = t\cos{t}$, $y = t\sin{t}$ and $z = t$. Now since you have sine and cosine, you have to think about the relations that these functions satisfy.

The easiest one, which you obviously know, is $\cos^2{t} + \sin^2{t} = 1$. Now in terms of $x, y, z$ you see that

$$\cos{t} = \frac{x}{z} \quad \text{and} \quad \sin{t} = \frac{y}{z}$$

Then this tells you that

$$\left( \frac{x}{z} \right )^2 + \left( \frac{y}{z} \right )^2 = 1$$

and this implies that $\ x^2 + y^2 = z^2 \ $ for any point $(x, y, z) = (t\cos{t}, t\sin{t}, t) \ $ on the curve $\gamma$.

So this tells you that the curve $\gamma$ is contained in the surface of equation $x^2 + y^2 = z^2$, which is the equation of a cone as already pointed out in the comments.

Finally since you needed to draw the curve in SAGE, the following is a plot of the curve and the cone. You can see how the curve wraps around the cone.

enter image description here

In case you want it, the SAGE code I used is the following.

var('x, y, z, t')

implicit_plot3d(x^2 + y^2 - z^2, (x, -15, 15), (y, -15, 15), (z, -15, 15), opacity = 0.5, color = 'blue', aspect_ratio = 1) + parametric_plot3d([t*cos(t), t*sin(t), t], (t, -5*pi, 5*pi), aspect_ratio = 1, thickness = 5, color = 'red')

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.