# Geometry

## Fine Points on Polygons and Polyhedra

Last time, looking at degenerate polygons, I mentioned some other issues pertaining to the definition of a polygon. Let’s take the opportunity to look at them. This post supplements what was said previously in What is a Polyhedron … Really?

## Degenerate Polygons

We’ve been looking at degenerate figures, starting with the most interesting case, degenerate conic sections. But other things can also be degenerate, so we should take a look at some of these, which perhaps arise even more often. We’ll examine triangles that aren’t triangles, rectangles that aren’t rectangles, and bigger polygons – or smaller polygons! …

## Degenerate Conics II: Are Their Parameters Meaningful?

Last time we looked at what a degenerate conic section is, and how it relates on one hand to actual cones, and on the other to the general equation of the conic. Here we’ll look at the parameters of conic sections (focus, directrix, axes, and especially eccentricity) and how they apply to degenerate cases. Does …

## Degenerate Conics I: Mystery of the Missing Case

Degenerate cases are instances of a concept that are just on the edge of fitting its definition. They occur when we stretch a definition to its limits, at which point some of the original properties remain, but others break. We’ll start here with common instances of the phenomenon, in conic sections, pursuing the elusive case …

## A Tunnel Through the Earth

I have a very short problem this week: How deep will you go if you dig a straight tunnel through the earth, how long will it be, and what angle do you have to start at?

## A Hole in a Cube

(A new question of the week) Here is an interesting little question about how drilling a hole affects volume and surface area. We’ll have one answer, and several explanations.

## A Triangle Problem: With Trig, and Without

(A new question of the week) When you are given a problem about a triangle, there can be many ways to approach it: pure geometry, trigonometry, and analytic geometry come to mind. When the context doesn’t dictate a method (as turns out to be true here), you just have to try what feels right to …

## Average Distance Between Two Sets of Points

(A new question of the week) Here we have a different kind of question than usual: A conjecture about distances between points, with a request for confirmation. Normally we like to just give hints to help a student figure something out; this was a request for a theorem that ought to exist, and trying to …

## Application of Vectors: Airplane in the Wind

A recent question about the resultant velocity of an airplane illustrates different ways to make a diagram showing the bearings of air velocity and wind velocity, and to work out angles without getting too dizzy.

## Distance from a Point to a Line in Space

(A new question of the week) Among many interesting recent questions we have one about vectors and equations in three dimensions. We’ll see four different ways to find the distance from a point to a line, proving two formulas and catching some of the errors one can make along the way. We’ll also see a …