As I’ve written about several times lately, the Smith chart is the image of a rectangular grid in the right half-plane under the function f(z) = (z − 1)/(z + 1). What would the image of a grid in the left half-plane look like? For starters, since f maps the right half-plane to the interior of the unit […] The post The anti-Smith chart first appeared on John D. Cook.| John D. Cook
The cross ratio of four points A, B, C, D is defined by where XY denotes the length of the line segment from X to Y. The idea of a cross ratio goes back at least as far as Pappus of Alexandria (c. 290 – c. 350 AD). Numerous theorems from geometry are stated in terms of the cross ratio. For example, the cross […] The post Cross ratio first appeared on John D. Cook.| John D. Cook
The inverse of the matrix is the matrix assuming ad − bc ≠ 0. Also, the inverse of the bilinear function (a.k.a. Möbius transformation) is the function again assuming ad − bc ≠ 0. The elementary takeaway is that here are two useful equations that are similar in appearance, so memorizing one makes it easy to memorize […] The post Inverting matrices and bilinear functions first appeared on John D. Cook.| John D. Cook
We will see that the 3D plot of \(x^2 + (y+zi)^2 = 1\) contains a circle and a hyperbola, where \(i\) is the imaginary number. Beyond the visuals, this helps us understand the (complex) eigenvalues of real matrices.| Toby Lam’s Blog