I’ve been thinking a bit about interpolation recently. In animation programming, we often use interpolation as a way to ensure continuity when transitioning from one animation to another. The most basic type of interpolation is linear interpolation, given by $$ a +_s b = sa + (1-s)b\,, $$ for vectors \(a, b\) and weight \(s\). …
Author Archives: john
How big is a trillion, anyway?
2020 was a sad year for lovers of large numbers. On 9th March, we lost Richard K. Guy at the age of 103; barely a month later, John Horton Conway is no longer with us. These two mathematicians, between them, in their Book of Numbers (1996) outlined a system for naming all numbers strictly less …
‘Spaces’
In early 2017, the Natural History Museum in London removed from their grand entrance hall the skeleton of a diplodocus, which had been a fixture of the museum since 1905. Affectionately nicknamed ‘Dippy’, the skeleton had moved around between various museum halls, but had become a favourite of museum-goers during his tenure as the museum’s …
What’s so special about [0,1]?
It happens to everyone. First, you learn real analysis and suddenly you understand what it means to do rigorous mathematics. You learn the precise definition of what a continuous function is; you learn that if they are defined on a closed bounded interval then they are bounded and attain their bounds; you learn about things …
What’s the difference between an orientation and a rotation?
This question came up at work recently, and I was unable to find a really good answer online. For the purposes of this article, I will be using the terms orientation and rotation as they are used in computer graphics; this might not be the same definitions you are used to. If you search online …
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The topological closed graph theorem
There are a number of theorems in various settings that link continuity of a function \(f\colon X \to Y\) to closedness of that function’s graph – i.e., the set of all points \((x,y)\in X \times Y\) such that \(y = f(x)\). For example, we learn in functional analysis that if \(f \colon X \to Y\) …