1

Is there a way to find $\min$ and $\max$ values of $x$, $y$, and $z$ coordinates of a circle in $\mathbb{R}^3$? The radius of, center of, and plane containing the circle are given (the plane is given by a normal vector and a point).

inavda
  • 827
michalt38
  • 137
  • I don't have the answer, but the first thing to note is that if you can solve the problem for a unit circle centered at the origin, you will be able to solve it for a circle of any radius and center by translating and scaling it accordingly. – inavda Oct 09 '20 at 20:46
  • Actually, I do have an answer, but it depends on what tools you are allowed to use. Do you know/can you use calculus for this problem? – inavda Oct 09 '20 at 20:48
  • 1
    @inavda I will have to code it but I can use any mathematical tools to solve it – michalt38 Oct 09 '20 at 20:52
  • Sounds good. I posted my answer. – inavda Oct 09 '20 at 21:02

1 Answers1

1

As noted in the comments, we will solve this question for a unit circle centered at the origin.

Using the technique presented in this answer, we will find a set of parametric equations describing the circle.

If $\mathbf{v}$ is the unit normal vector to the plane containing the center, we find a vector in the plane by solving $\mathbf{a} \cdot \mathbf{v} = 0$ for $\mathbf{a}$. Then we can take the cross product to get a vector in the plane that is orthogonal to $\mathbf{a}$: $$\mathbf{b} = \mathbf{a} \times \mathbf{v}$$

Make sure at this point that $\mathbf{a}$ and $\mathbf{b}$ are properly normalized so that they both have length 1 (otherwise we would get an ellipse).

Then we can represent the circle using the following parametric equations:

$$x(\theta) = \cos(\theta)a_1 + \sin(\theta)b_1 \,,$$ $$y(\theta) = \cos(\theta)a_2 + \sin(\theta)b_2 \,,$$ $$z(\theta) = \cos(\theta)a_3 + \sin(\theta)b_3 \,,$$ where $\mathbf{a} = (a_1,a_2,a_3)$ and $\mathbf{b} = (b_1,b_2,b_3)$.

Since the functions are differentiable, we know that their maxima and minima will be attained at points where their derivatives are zero. We take the derivatives:

$$x'(\theta) = -\sin(\theta)a_1 + \cos(\theta)b_1 \,,$$ $$y'(\theta) = -\sin(\theta)a_2 + \cos(\theta)b_2 \,,$$ $$z'(\theta) = -\sin(\theta)a_3 + \cos(\theta)b_3 \,.$$

The max and min for the $x$-coordinate will occur when $\tan(\theta) = \frac{b_1}{a_1}$.

The max and min for the $y$-coordinate will occur when $\tan(\theta) = \frac{b_2}{a_2}$.

The max and min for the $z$-coordinate will occur when $\tan(\theta) = \frac{b_3}{a_3}$.

inavda
  • 827