0
$\begingroup$

I would like to have a function that gives the area of a rectangle at a certain theta of the spiral r=theta. The height of the rectangle is the y value of the point on the spiral and the base of the rectangle is the x value of the point on the spiral. I know the area would then just be xy (positive if its above the x axis and negative if below the x axis) for each point (x,y) on the curve. I know that r=theta can be written in cartesian coordinates as tan(sqrt(x^2+y^2)) = y/x, but I don't know if I can solve this equation for x or y.

For example, given F(5 * sqrt(2)) = 25:

blue square, green spiral on graph

The red line is the the angle on the spiral, and the blue lines are the high and base of the rectangle.

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ Although this is an interesting question, it is not research level, so, despite @JonathanLove's nice answer and comment, it will probably be closed. For future questions at this level, try our sister site MSE. $\endgroup$
    – LSpice
    Oct 9, 2021 at 1:40

1 Answer 1

Reset to default
1
$\begingroup$

A parametric form for this spiral is given by $x=\theta\cos\theta$ and $y=\theta\sin\theta$. So the area of the rectangle is $\theta^2\sin\theta\cos\theta=\frac{\theta^2}{2}\sin(2\theta)$.

Improve this answer
$\endgroup$
1
  • 2
    $\begingroup$ Also worth pointing out: the example you shared is not quite right. The correct value is $F(5\sqrt{2})=24.9996914\ldots$. The reason your picture looks the way it does is because $5\sqrt{2}$ is a very close approximation to $\frac{9\pi}{4}$. $\endgroup$
    – Jonathan Love
    Oct 9, 2021 at 1:19

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