Articles of geometry

Calippo, toothpaste and milk .. packing

This post was posing the question “The Calippo™ popsicle has a specific shape … Does this shape have an official name?”, and the accepted answer was that it is a Right Conoid which in fact is much resembling. However, a conoid is not developable and so cannot be economically made out of a cardboard sheet. […]

Confused about dimension of circle

I confused myself when thinking about the circle: It can be parameterised as $$ C(t) = (\cos t , \sin t)$$ for $t \in [0,2\pi)$. This makes it clear that the circle is one dimensional. But then the circle is also defined by $x,y$ such that $$ x^2 + y^2 = 1$$ If we try […]

How did Archimedes find the surface area of a sphere?

In school we are told that the surface area of a sphere is $4\pi$. Is it true that Archimedes found the surface area of a sphere using the Archimedes Hat-Box Theorem? Is there a simple proof for this theorem? Thank you. Added: Does that kind of projection as mentioned in the Archimedes Hat-Box Theorem preserve […]

A right angle at the focus of a hyperbola

$P$ is a point on a hyperbola. The tangent at $P$ cuts a directrix at point $Q$. Prove that $PQ$ subtends a right angle to the focus $F$ corresponding to the directrix. I have tried to use the general equation of the hyperbola and gradient method to show, but too many unknowns and I can’t […]

Area covered by Moving Circle?

Consider a situation where we have a point (x,y) moving on a 2-D plane. In fact, the point is function of time x=f(t),y=g(t). Centered around (x,y) is a circle of radius r? Obviously, we can visualize a circle moving in the 2-D plane. Compute the area covered by the circle from the the start of […]

Prove that the circle contains the polygon.

Given a convex polygon. The circle is constructed for every triple of consecutive vertices of the polygon.We get the n circles. Select the circle with the largest radius. Prove that the circle contains the polygon. My work so far: $n=3 -$ triangle – obviously. $n=4 -$ If $\angle B = \max \left\{A,B,C,D \right\}$ then $ABCD […]

Partition the points

Given an even number of points in general positions on the plane (that is, no three points co-linear), can you partition the points into pairs and connect the two points of each pair with a single straight line such that the straight lines do not overlap?

Missing solution in quadratic equation

This is a reformulation of this question to better fit this forum. I removed the mentioning of sage math and this question is now 100% math. Given, the following quadrilateral: I want to describe the position of point X in terms of a multiple $t$ of the vector $HG$. The points $H$ and $G$ themselves […]

If any x points are elected out of a unit square, then some two of them are no farther than how many units apart?

If 5 points are randomly positioned in a unit square, no two points can be greater than square root of 2 divided by 2 apart; divide up the unit square into four squares, and, based on the pigeonhole principle, five points (pigeons) fitting into four squares (holes) means that no two points can be greater […]

Spatial angles in higher dimensions

Wikipedia gives an excellent treatise about solid angles in 1-2-3-Dimensions. But how about n-D? I read once some notes from a seminar held during WWII in Switzerland, and one result concerned spatial angles in even dimensions (I have forgotten the reference), but I would like to have a similar general definition, likely computationally nice. So […]