Articles of circle

How to find the central angle of a circle?

My book states the following: Likewise, we can take a circular cone with base radius $r$ and slant height $l$, cut it along the dashed line in Figure 2: and flatten it to form a sector of a circle with radius $l$ and central angle $ \theta = \frac{2\pi r}{l} $. Why does $ \theta […]

How to maximize area of two circles inside a rectangle without overlapping?

Two circles have to be drawn inside a rectangle of dimensions $W\times H$ such that the area of both circles is to be as large as possible without overlapping. Let the radii of the circles be $r_1$ and $r_2$ How do I go about formulating the constraints? Thank you.

Help setting out a proof about the circle $x^{2} + y^{2} + 2gx + 2fy + c = 0$

16. Given that the circle $$x^{2} + y^{2} + 2gx + 2fy + c = 0$$ touches the $y$-axis, prove that $f^{2} = c$. So, because the circle touches the $y$-axis, we know that there is a solution to this equation where $x = 0$, so we can say: $y^{2} + 2fy + c = […]

Longest chord inside the intersection area of three circles

I am currently working on my masters thesis in computer science and I stumbled onto a geometry problem. My goal is to compute the length of the longest possible chord inside the intersection area of three circles. I know the following thing about the circles: their radius is r Construction: Assume that there is a […]

Meaning of the expression “orientation preserving” homeomorphism

The only time that I’ve heard the term “orientation-preserving map” was in Linear Algebra, but today I read the term orientation-preserving homeomorphism of the circle in the following context: If a homeomorphism $f$ of $S^1$ has a rational rotation number $r$ ($r=\lim_{n\to\infty} \frac{F^n(x)-x}{n}$ where $F$ is a “lift” of $f$ to $\mathbb{R}$) and if $f$ […]

Find sagitta of a cubic Bézier-described arc

I have a situation where I have an arc that was mangled (irrelevant: by c#’s GraphicsPath.AddArc() function). The original arc is guaranteed to be circular, and the new data I have describes the Bézier approximation for the arc instead. I’m not hugely up on Béziers, or complex geometry, so am hoping someone can help me! […]

How many points must the arc intersect?

Let’s say we have some number of points $\{x_i\}$, which lie on a circle. We wish to position an arc somewhere on the circle, which has an arc length equal to $\frac1c$ of the circle’s circumference, such that we minimise the number of the $\{x_i\}$ that lie on the arc. Can we always position the […]

Check whether $n$ disks intersect

I struggle with the following problem: Given $N$ disks $D_i = (x_i, y_i,r_i)$, calculate whether they ALL intersect. $D_1 \cap D_2 \cap \dots \cap D_N = \emptyset $ ? I do not care about the intersection area, just want to know whether they do or not. I know how to check whether two disks intersect. […]

Why the unit circle in $\mathbf{R^2}$ has one dimension?

When I was reading ‘Convex Optimization, Stephen Boyd’, I was wondering of following steps Consider the unit circle in $\mathbf{R^2}$, $i.e.$, $\{x\in\mathbf{R^2}|x^2_1+x^2_2=1\}$. Its affine hull is all of $\mathbf{R^2}$, so its affine dimension is two. By most definitions of dimension, however, the unit circle in $\mathbf{R^2}$ has dimension one. I understood the affine hull of […]

Technique for proving four given points to be concyclic?

While making my way through an exercise, I stalled on question 7: 7. Prove that the points $(9, 6)$, $(4, -4)$, $(1, -2)$, $(0, 0)$ are concyclic. The book does not provide any guidance on how to tackle such a question and I can only assume that the authors are assuming that anybody using their […]