A guide to geometric algebra and interval arithmetic — from blades and rotors through to a working Ruby implementation and conformal-space worked examples.
An interval \([a, b] \subset \mathbb{R}\) with \(a \leq b\) represents a quantity whose precise value is unknown but known to lie within those bounds. The arithmetic of intervals was systematically developed by…
In \(\mathcal{G}(\mathbb{R}^2)\), the basis vectors \(\mathbf{e}_1\) and \(\mathbf{e}_2\) satisfy \(\mathbf{e}_1^2 = \mathbf{e}_2^2 = 1\) and \(\mathbf{e}_1\mathbf{e}_2 = -\mathbf{e}_2\mathbf{e}_1\). Define the…
Classical interval arithmetic defines division by an interval not containing zero as:
Constructive Solid Geometry (CSG) builds complex shapes from primitive solids (spheres, planes, cylinders) combined by Boolean operations: union, intersection, and difference. Ray tracing a CSG scene requires…
Consider the coupled linear system:
The implementation follows three guiding principles drawn from the mathematical structure itself:
Testing an interval library differs from testing ordinary numerical code in one fundamental respect: the correct answer is a set , not a number. A test that checks result == 3.14159 is asking the wrong question.…
CGA, developed in modern form by Hestenes, Li, and Rockwood [1] , adds two extra basis vectors \(\mathbf{e}_+\) and \(\mathbf{e}_-\) to the Euclidean basis, with metric: