A guide to geometric algebra and interval arithmetic — from blades and rotors through to a working Ruby implementation, conformal-space worked examples, and projective GA for graphics.
Here is a question that sounds too simple to be worth asking: what does the number 3 mean? In one sense nothing could be more obvious — it's three. But put it on a line, and "3" starts doing at least three…
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:
Mathematics keeps inventing new kinds of geometric object. First vectors. Then complex numbers. Then quaternions. Then the blades and multivectors of Geometric Algebra developed across the rest of this series.…
Chapter 9 established what a blade is: the outer product of a set of vectors, encoding an oriented, extended geometric entity. This chapter is the workshop, following the construction Dorst, Fontijne, and…
A graphics programmer's toolbox holds several separate mathematical systems. Matrices carry transformations. The dot and cross products measure angles and find normals. Quaternions handle rotation, dual…
Chapter 9 gave us blades to describe shapes and rotors to move them. A rotor moves a shape through an angle. Let that angle run through a full turn and the shape leaves a trail, which is a new shape one dimension…
Chapter 8 represents a sphere as a single blade, moved by a sandwich product. Chapter 11 represents a plane, a line, and a point the same way, moved by a motor. Both chapters raise the same question in …
The chapters of this series build one line of argument. The sources below reach the same ground from other directions, and several let you move things on screen instead of reading about them. They're grouped by what…