Articles of geometry

Involute of a circle – what is the separation distance?

It seems like a simple enough question. For the involute of a circle, what is the separation distance between successive turns? Is this derivation correct? Parametric formula for the y-coordinate: $ y = r(Sin(\theta) – \theta Cos(\theta)) $ Differentiating: $ \frac{dy}{d\theta} = r \theta Sin(\theta) $ Which has roots at $ \theta = \pi n, […]

How to calculate volume given by inequalities?

I need to find the volume of the 3d space that is given by the following conditions: \begin{array}{c} 0 < x_1 < 1\\ 0 < x_2 < 1\\ 0 < x_3 < 1\\ x_1 + x_2 + x_3 < a. \end{array} I also need to solve this problem for the $n$-dimensional space. Could anybody, please, […]

Triangle problem related to finding an area

Given a triangle $\triangle ABC$ . Points $P, Q, R$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively. $\overline{AP}$ bisects $\overline{BQ}$ at point $X$, $\overline{BQ}$ bisects $\overline{CR}$ at point $Y$, and $\overline{CR}$ bisects $\overline{AP}$ at point $Z$. Find the area of the triangle $\triangle XYZ$ in function area of $\triangle ABC$

Boundedness condition of Minkowski's Theorem

Statement: “Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if S is closed, then it suffices to take $Volume(S) \geq 2^ndet(L)$.” I am wondering if the condition for boundedness can be relaxed. […]

Trying to understand the limit of regular polygons: circle vs apeirogon (vs infinigon?)

In the definition of regular polygon at the Wikipedia, there is this statement about the limit of a n-gon: “In the limit, a sequence of regular polygons with an increasing number of sides becomes a circle, if the perimeter is fixed, or a regular apeirogon, if the edge length is fixed.” My question is: I […]

a question about differential geometry, how to prove that $|k(t_{0})|>{1\over |\alpha(t_{0})|}$

Let $\alpha:(a,b)\to\Bbb R^2$ be a regular parametrized plane curve. Assume that there exits $t_{0}$, $a<t_{0}<b$, such that the distance $|\alpha(t)|$ from the origin to the trace of $\alpha$ will be a maximum at $t_{0}$. Prove that the curvature $k$ of $\alpha$ at $t_{0}$ satisfies $|k(t_{0})|>{1\over |\alpha(t_{0})|}$. My thoughts: Without loss of generalization, I let t […]

Finding points on ellipse

I have ellipse in 2D. I want to compute fixed number of points on this ellipse with constant angular separation between those points. My first idea was to generate line equations from center of the ellipse and then solve equations of these lines with ellipse equation. But it’s not efficient computationally. Second idea is to […]

Eccentered Circles – determine space between circle at a given location

I need to figure out a way of calculating the dimensions x and y as shown on the attached image. I know the angles (in the example the inner circle is broken into 6 – 60 degree angles). I also know the diameters of both circles. I have tried a bunch of approaches but I […]

Relation between mean width and diameter

Question: Let $A$ be a compact set in $\mathbb R^n$. Is it always true that $\text{mean-width}(A) \ge C \cdot \text{diam}(A)$ for some constant $C$ depending only on the dimension? If not, is it true assuming $A$ is convex? Background / definitions: Given a compact set $A\subset\mathbb R^n$ and a direction $v\in\mathcal{S}^{n-1}$, the width $b(A, v)$ […]

Computing fundamental forms of implicit surface

This question already has an answer here: About the second fundamental form 1 answer