Articles of geometry

High School Projectile Motion and Quadratics

High school students are learning about the basics of solving quadratics and trigonometric ratios, including trigonometric inverses. The eventual goal of their project is to be able to show a reasonable firing solution, given in initial angle $\theta$ and initial velocity $v_0$. Projectile motion is given by $$y=\left(\tan{\theta}\right)x-\left(\frac{g}{2v_0^2\cos^2{\theta}}\right)x^2$$ where $x$ and $y$ are horizontal and […]

Covering all the edges of a hypercube?

Consider an arbitrary $n$- dimensional hypercube: If we select $n – 1$ corners of that hypercube and highlight all $(n – 2)$ dimensional elements that originate from each of the corners is it possible to cover all the $(n – 2)$ dimensional faces of the cube? Also, is it ever possible to cover all the […]

Find number of Cuboids in a larger Cuboid

This is a question posed to my brother in Grade 5. What would be the general approach to solve- How many cuboids of dimensions $a*b*c$ are there in a cuboid of dimension $d*e*f$? My brothers approach: Just get the ratio $\frac{d*e*f}{a*b*c}$ and round it off. My approach: Check 1: Each of $d$, $e$, $f$ must […]

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 […]

Symmetries of a Pentagon.

I’m supposed to find the Cayley Table of the group of symmetries for a regular pentagon. But to find the Cayley table, I need to be able to figure out the symmetries of the pentagon. I can see 6 symmetries of a pentagon. The identity, 4 rotations, and 1 reflections on the y-axis. There should […]

The unit square stays path-connected when you delete a cycle-free countable family of open segments?

This question was inspired to me by Lukas Geyer’s recent question. A positive answer to this question would easily entail a positive answer to Lukas’ question also, and a negative answer would probably be informative as well. Let $T=]0,1[^2$ be the open unit square. Let $(S_k)_{k\geq 1}$ be a countable family of open segments in […]

Does the shortest path have no crossings?

Suppose $A_1,A_2,A_3,\ldots,A_n$ are distinct points in the plane. A closed path on those points is formed from a permutation of the points, with line segments drawn between successive points (and with a segment connecting the last point to the first, to make the path “closed”). For the six points pictured below on the left, the […]

Why is the identity map never equal to the product of an odd number of reflections?

Suppose I have an some plane and an identity mapping on the points of the plane. I see that the identity can be expressed as a product of an even number of reflections, since any reflection has itself as its own inverse. But why is it impossible to ever express the identity as the product […]

inflection point of cubic bezier with restrictions

Say you have this type of cubic Bézier curve: The 4 control points A,B,C,D have restrictions: A & B have the same Y-axis coordinate C & D have the same Y-axis coordinate B & C have the same x-axis coordinate My question is: How do you express the X-axis coordinate of the inflection point of […]

Does $d(x+u, y + v) \le d(x, y) + d(u,v)$ holds for every metric?

The title said it, I want to prove that $$ d(x+u, y + v) \le d(x, y) + d(u,v) $$ for every metric $d$. If the metric is induced by a norm, i.e. $d(x,y) := ||x-y||$, then this is easy. \begin{align*} d(x+u, y+v) & = ||x+u – (y+v)|| \\ & = ||x-y + u – […]