1+ week, 6+ day ago  (165+ words) Tim F left a comment on my guitar pick post saying the image was a "squircle-ish analog for an isosceles triangle." That made me wonder what a more direct analog of the squircle might be for a triangle. A squircle…...

2+ week, 5+ day ago  (221+ words) A couple days ago I wrote a post about turning a trick into a technique, finding another use for a clever way to construct simple, accurate approximations. I used as my example approximating the Bessel function J(x) with (1 + cos(x))/2. I…...

2+ mon, 1+ week ago  (294+ words) In the previous two posts, we looked at why Mathematica and Sym Py did not simplify sinh(arccosh(x)) to "(x" " 1) as one might expect. After understanding why sinh(arccosh(x)) doesn't simplify nicely, it's natural to ask why sin(arccos(x)) does…...

2+ mon, 3+ week ago  (258+ words) Here is an identity that doesn't look correct but it is. For real'x and'y, I found the identity in [1]. The author's proof is short. First of all, Taking square roots completes the proof. Now note that the statement at the…...

2+ mon, 3+ week ago  (291+ words) I ran across an old article [1] that gave a sort of multiplication table for trig functions and inverse trig functions. Here's my version of the table. I made a few changes from the original. First, I used La Te X,…...

5+ mon, 2+ week ago  (376+ words) This post is a side quest in the series on navigating by the stars. It expands on a footnote in the previous post. There are six pieces of information associated with a spherical triangle: three sides and three angles. I…...

5+ mon, 3+ week ago  (236+ words) In spherical geometry, the interior angles of a triangle add up to more than ". And in fact you can determine the area of a spherical triangle by how much the angle sum exceeds ". On a sphere of radius 1, the area…...

5+ mon, 3+ week ago  (352+ words) Let " be the upper half plane, the set of complex real numbers with positive imaginary part. When we measure distances the way we've discussed in the last couple posts, the geometry of " is hyperbolic. What is a circle of radius…...

5+ mon, 3+ week ago  (329+ words) The previous post described a metric for the Poincar" upper half plane. The development is geometrical rather than analytical. There are also analytical formulas for the metric, at least four that I've seen. It's not at all obvious that the…...

5+ mon, 3+ week ago  (305+ words) One common model of the hyperbolic plane is the Poincar" upper half plane ". This is the set of points in the complex plane with positive imaginary part. Straight lines are either vertical, a set of points with constant imaginary part,…...