In domains that deal with physical space and geometry, programmers need to track the coordinate systems that underpin a computation. We identify a class of geometry bugs that arise from confusing which coordinate system a vector belongs to. These bugs are not ruled out by current languages for vector-oriented computing, are difficult to check for at run time, and can generate subtly incorrect output that can be hard to test for. We introduce a type system and language that prevents geometry bugs by reflecting the coordinate system for each geometric object. A value's geometry type encodes its reference frame, the kind of geometric object (such as a point or a direction), and the coordinate representation (such as Cartesian or spherical coordinates). We show how these types can rule out geometrically incorrect operations, and we show how to use them to automatically generate correct-by-construction code to transform vectors between coordinate systems. We implement a language for graphics programming, Gator, that checks geometry types and compiles to OpenGL's shading language, GLSL. Using case studies, we demonstrate that Gator can raise the level of abstraction for shader programming and prevent common errors without inducing significant annotation overhead or performance cost.

Systems of transformations arise in many programming systems, such as in graphs of implicit type conversion functions. It is important to ensure that these diagrams commute: that composing any path of transformations from the same source to the same destination yields the same result. However, a straightforward approach to verifying commutativity must contend with cycles, and even so it runs in exponential time. Previous work has shown how to verify commutativity in the special case of acyclic diagrams in (| | 4 | | 2) time, but this is a batch algorithm: the entire diagram must be known ahead of time. We present an online algorithm that efficiently verifies that a commutative diagram remains commutative when adding a new edge. The new incremental algorithm runs in (| | 2 (| | + | |)) time. For the case when checking the equality of paths is expensive, we also present an optimization that runs in (| | 4) time but reduces to the minimum possible number of equality checks. We implement the algorithms and compare them to batch baselines, and we demonstrate their practical application in the compiler of a domain-specific language for geometry types. To study the algorithms' scalability to large diagrams, we apply them to discover discrepancies in currency conversion graphs.