How do you use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces $x^{2} + y^{2} = 25$ $x^2+y^2=25$ and $z = 3 x^{2}$ $z=3x^2$ ?

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How do you use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces $x^{2} + y^{2} = 25$ $x^2+y^2=25$ and $z = 3 x^{2}$ $z=3x^2$ ?

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Answer 1 · 2015-03-08T03:14:11

Use the following parametrization for the curve $s$ $s$ generated by the intersection:

$s (t) = (x (t), y (t), z (t)), t \in [0, 2 π)$ $s(t)=(x(t), y(t), z(t)), t in [0, 2pi)$
$x = 5 cos (t)$ $x = 5cos(t)$
$y = 5 sin (t)$ $y = 5sin(t)$
$z = 75 {cos}^{2} (t)$ $z=75cos^2(t)$

Note that $s(t): RR -> RR^3$ is a vector valued function of a real variable.

To reach this result, consider the curves that these equations define on certain planes.

The equation $x^2+y^2=25$ defines a circle of radius $5$ centered on the $z$ -axis on the planes $z=c_1$ , where $c_1 in RR$ is any constant.
The equation $z=3x^2$ defines a parabola on any plane $y=c_2$ , where $c_2 in RR$ is another constant.
The surfaces are, therefore, those obtained by translating the circle along the $z$ -axis and the parabola along the $y$ -axis.

To obtain a parametrization for the intersection curve $s$ , we must find equations for $x$ , $y$ and $z$ as functions of $t$ that obey both equations given in the problem.

Consider the standard parametrization for a circle $C$ of radius $r$ (it's easy to see that this parametrization fulfils the condition $x^2+y^2=r^2$ ):

$C(t)=(rcos(t), rsin(t)), t in [0,2pi)$

Checking the first equation, we get $r=5$ and

$x = 5cos(t)$
$y = 5sin(t)$

Now, we already have an expression for $x(t)$ . So, in order to obey the second condition, we make:

$z=3x^3=3(5cos(t))^2=75cos^2(t)$

And we have the parametrization $s(t)=(x(t), y(t), z(t)), t in [0, 2pi)$ for $s$ .

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How do you use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces $x^{2} + y^{2} = 25$ $x^2+y^2=25$ and $z = 3 x^{2}$ $z=3x^2$ ?

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How do you use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces x^2+y^2=25x2+y2=25x^2+y^2=25 and z=3x^2z=3x2z=3x^2?

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How do you use cos(t) and sin(t), with positive coefficients, to parametrize the intersection of the surfaces $x^{2} + y^{2} = 25$ $x^2+y^2=25$ and $z = 3 x^{2}$ $z=3x^2$ ?