Learn Zig Series (#160) - 2D Transform Matrices
Published on HivePostify by @scipio · Sat Sep 05 2026
Learn Zig Series (#160) - 2D Transform Matrices
What will I learn? - Why every shape we have drawn so far is nailed to fixed pixel coordinates, and why hand-editing vertices to move it is the wrong answer; - Homogeneous coordinates: the small trick of giving a 2D point a third coordinate of 1 so that translation, rotation and scale all become the same thing -- a matrix multiply; - A tiny Mat3 affine-transform type written from scratch in Zig, with identity, translation, scaling, rotation, matrix product, and point application; - Why order matters -- rotate-then-translate is a different world from translate-then-rotate -- and how composing matrices lets you say exactly which one you mean; - How Zig's comptime, fixed-size arrays and !void keep the fast path allocation-free while making the batch path's one real failure mode honest; - Testing strategies that actually catch transform bugs: identity round-trips, inverse round-trips, and composition-order assertions; - Where this machinery lives in the real world (scene graphs, SVG transform, camera/view matrices) and how C, Rust and Go write the identical arithmetic.
Requirements - A working modern computer running macOS, Windows or Ubuntu; - An installed Zig 0.14+ distribution (download from ziglang.org) -- the code here is written and tested against Zig 0.16; - The Framebuffer, Point and the clipping setPixel/getPixel from episode 156, plus the fillPolygon scanline fill we built last episode -- today we feed transformed points into that same fill; - Comfort with f32 math, @floatFromInt/@intFromFloat, and the comptime-duck-typing we have leaned on since episode 13; - The ambition to learn Zig programming.
Difficulty - Advanced
Curriculum (of the Learn Zig Series): - [Zig Programming Tutorial - ep001 - Intro](https://hive.blog/programming/@scipio/zig-programming-tutoroial-ep001-intro) - [Learn Zig Series (#2) - Hello Zig, Variables and Types](https://hive.blog/hive-196387/@scipio/learn-zig-series-2-hello-zig-variables-and-types) - [Learn Zig Series (#3) - Functions and Control Flow](https://hive.blog/hive-196387/@scipio/learn-zig-series-3-functions-and-control-flow) - [Learn Zig Series (#4) - Error Handling (Zig's Best Feature)](https://hive.blog/hive-196387/@scipio/learn-zig-series-4-error-handling-zigs-best-feature) - [Learn Zig Series (#5) - Arrays, Slices, and Strings](https://hive.blog/hive-196387/@scipio/learn-zig-series-5-arrays-slices-and-strings) - [Learn Zig Series (#6) - Structs, Enums, and Tagged Unions](https://hive.blog/hive-196387/@scipio/learn-zig-series-6-structs-enums-and-tagged-unions) - [Learn Zig Series (#7) - Memory Management and Allocators](https://hive.blog/hive-196387/@scipio/learn-zig-series-7-memory-management-and-allocators) - [Learn Zig Series (#8) - Pointers and Memory Layout](https://hive.blog/hive-196387/@scipio/learn-zig-series-8-pointers-and-memory-layout) - [Learn Zig Series (#9) - Comptime (Zig's Superpower)](https://hive.blog/hive-196387/@scipio/learn-zig-series-9-comptime-zigs-superpower) - [Learn Zig Series (#10) - Project Structure, Modules, and File I/O](https://hive.blog/hive-196387/@scipio/learn-zig-series-10-project-structure-modules-and-file-io) - [Learn Zig Series (#11) - Mini Project: Building a Step Sequencer](https://hive.blog/hive-196387/@scipio/learn-zig-series-11-mini-project-building-a-step-sequencer) - [Learn Zig Series (#12) - Testing and Test-Driven Development](https://hive.blog/hive-196387/@scipio/learn-zig-series-12-testing-and-test-driven-development) - [Learn Zig Series (#13) - Interfaces via Type Erasure](https://hive.blog/hive-196387/@scipio/learn-zig-series-13-interfaces-via-type-erasure) - [Learn Zig Series (#14) - Generics with Comptime Parameters](https://hive.blog/hive-196387/@scipio/learn-zig-series-14-generics-with-comptime-parameters) - [Learn Zig Series (#15) - The Build System (build.zig)](https://hive.blog/hive-196387/@scipio/learn-zig-series-15-the-build-system-buildzig) - [Learn Zig Series (#16) - Sentinel-Terminated Types and C Strings](https://hive.blog/hive-196387/@scipio/learn-zig-series-16-sentinel-terminated-types-and-c-strings) - [Learn Zig Series (#17) - Packed Structs and Bit Manipulation](https://hive.blog/hive-196387/@scipio/learn-zig-series-17-packed-structs-and-bit-manipulation) - [Learn Zig Series (#18b) - Addendum: Async Returns in Zig 0.16](https://hive.blog/hive-196387/@scipio/learn-zig-series-18b-addendum-async-returns-in-zig-016) - [Learn Zig Series (#19) - SIMD with @Vector](https://hive.blog/hive-196387/@scipio/learn-zig-series-19-simd-with-vector) - [Learn Zig Series (#20) - Working with JSON](https://hive.blog/hive-196387/@scipio/learn-zig-series-20-working-with-json) - [Learn Zig Series (#21) - Networking and TCP Sockets](https://hive.blog/hive-196387/@scipio/learn-zig-series-21-networking-and-tcp-sockets) - [Learn Zig Series (#22) - Hash Maps and Data Structures](https://hive.blog/hive-196387/@scipio/learn-zig-series-22-hash-maps-and-data-structures) - [Learn Zig Series (#23) - Iterators and Lazy Evaluation](https://hive.blog/hive-196387/@scipio/learn-zig-series-23-iterators-and-lazy-evaluation) - [Learn Zig Series (#24) - Logging, Formatting, and Debug Output](https://hive.blog/hive-196387/@scipio/learn-zig-series-24-logging-formatting-and-debug-output) - [Learn Zig Series (#25) - Mini Project: HTTP Status Checker](https://hive.blog/hive-196387/@scipio/learn-zig-series-25-mini-project-http-status-checker) - [Learn Zig Series (#26) - Writing a Custom Allocator](https://hive.blog/hive-196387/@scipio/learn-zig-series-26-writing-a-custom-allocator) - [Learn Zig Series (#27) - C Interop: Calling C from Zig](https://hive.blog/hive-196387/@scipio/learn-zig-series-27-c-interop-calling-c-from-zig) - [Learn Zig Series (#28) - C Interop: Exposing Zig to C](https://hive.blog/hive-196387/@scipio/learn-zig-series-28-c-interop-exposing-zig-to-c) - [Learn Zig Series (#29) - Inline Assembly and Low-Level Control](https://hive.blog/hive-196387/@scipio/learn-zig-series-29-inline-assembly-and-low-level-control) - [Learn Zig Series (#30) - Thread Safety and Atomics](https://hive.blog/hive-196387/@scipio/learn-zig-series-30-thread-safety-and-atomics) - [Learn Zig Series (#31) - Memory-Mapped I/O and Files](https://hive.blog/hive-196387/@scipio/learn-zig-series-31-memory-mapped-io-and-files) - [Learn Zig Series (#32) - Compile-Time Reflection with @typeInfo](https://hive.blog/hive-196387/@scipio/learn-zig-series-32-compile-time-reflection-with-typeinfo) - [Learn Zig Series (#33) - Building a State Machine with Tagged Unions](https://hive.blog/hive-196387/@scipio/learn-zig-series-33-building-a-state-machine-with-tagged-unions) - [Learn Zig Series (#34) - Performance Profiling and Optimization](https://hive.blog/hive-196387/@scipio/learn-zig-series-34-performance-profiling-and-optimization) - [Learn Zig Series (#35) - Cross-Compilation and Target Triples](https://hive.blog/hive-196387/@scipio/learn-zig-series-35-cross-compilation-and-target-triples) - [Learn Zig Series (#36) - Mini Project: CLI Task Runner](https://hive.blog/hive-196387/@scipio/learn-zig-series-36-mini-project-cli-task-runner) - [Learn Zig Series (#37) - Markdown to HTML: Tokenizer and Lexer](https://hive.blog/hive-196387/@scipio/learn-zig-series-37-markdown-to-html-tokenizer-and-lexer) - [Learn Zig Series (#38) - Markdown to HTML: Parser and AST](https://hive.blog/hive-196387/@scipio/learn-zig-series-38-markdown-to-html-parser-and-ast) - [Learn Zig Series (#39) - Markdown to HTML: Renderer and CLI](https://hive.blog/hive-196387/@scipio/learn-zig-series-39-markdown-to-html-renderer-and-cli) - [Learn Zig Series (#40) - Key-Value Store: In-Memory Store](https://hive.blog/hive-196387/@scipio/learn-zig-series-40-key-value-store-in-memory-store) - [Learn Zig Series (#41) - Key-Value Store: Write-Ahead Log](https://hive.blog/hive-196387/@scipio/learn-zig-series-41-key-value-store-write-ahead-log) - [Learn Zig Series (#42) - Key-Value Store: TCP Server](https://hive.blog/hive-196387/@scipio/learn-zig-series-42-key-value-store-tcp-server) - [Learn Zig Series (#43) - Key-Value Store: Client Library and Benchmarks](https://hive.blog/hive-196387/@scipio/learn-zig-series-43-key-value-store-client-library-and-benchmarks) - [Learn Zig Series (#44) - Image Tool: Reading and Writing PPM/BMP](https://hive.blog/hive-196387/@scipio/learn-zig-series-44-image-tool-reading-and-writing-ppmbmp) - [Learn Zig Series (#45) - Image Tool: Pixel Operations](https://hive.blog/hive-196387/@scipio/learn-zig-series-45-image-tool-pixel-operations) - [Learn Zig Series (#46) - Image Tool: CLI Pipeline](https://hive.blog/hive-196387/@scipio/learn-zig-series-46-image-tool-cli-pipeline) - [Learn Zig Series (#47) - Build a Shell: Parsing Commands](https://hive.blog/hive-196387/@scipio/learn-zig-series-47-build-a-shell-parsing-commands) - [Learn Zig Series (#48) - Build a Shell: Process Spawning](https://hive.blog/hive-196387/@scipio/learn-zig-series-48-build-a-shell-process-spawning) - [Learn Zig Series (#49) - Build a Shell: Built-in Commands](https://hive.blog/hive-196387/@scipio/learn-zig-series-49-build-a-shell-built-in-commands) - [Learn Zig Series (#50) - Build a Shell: Job Control and Signals](https://hive.blog/hive-196387/@scipio/learn-zig-series-50-build-a-shell-job-control-and-signals) - [Learn Zig Series (#51) - HTTP Server: Accept Loop and Parsing](https://hive.blog/hive-196387/@scipio/learn-zig-series-51-http-server-accept-loop-and-parsing) - [Learn Zig Series (#52) - HTTP Server: Router and Responses](https://hive.blog/hive-196387/@scipio/learn-zig-series-52-http-server-router-and-responses) - [Learn Zig Series (#53) - HTTP Server: Static Files and MIME](https://hive.blog/hive-196387/@scipio/learn-zig-series-53-http-server-static-files-and-mime) - [Learn Zig Series (#54) - HTTP Server: Middleware and Logging](https://hive.blog/hive-196387/@scipio/learn-zig-series-54-http-server-middleware-and-logging) - [Learn Zig Series (#55) - ECS Game Engine: Architecture](https://hive.blog/hive-196387/@scipio/learn-zig-series-55-ecs-game-engine-architecture) - [Learn Zig Series (#56) - ECS Game Engine: Component Storage](https://hive.blog/hive-196387/@scipio/learn-zig-series-56-ecs-game-engine-component-storage) - [Learn Zig Series (#57) - ECS Game Engine: Systems and Queries](https://hive.blog/hive-196387/@scipio/learn-zig-series-57-ecs-game-engine-systems-and-queries) - [Learn Zig Series (#58) - ECS Game Engine: Terminal Rendering](https://hive.blog/hive-196387/@scipio/learn-zig-series-58-ecs-game-engine-terminal-rendering) - [Learn Zig Series (#59) - Assembler: Instruction Encoding](https://hive.blog/hive-196387/@scipio/learn-zig-series-59-assembler-instruction-encoding) - [Learn Zig Series (#60) - Assembler: Two-Pass Assembly](https://hive.blog/hive-196387/@scipio/learn-zig-series-60-assembler-two-pass-assembly) - [Learn Zig Series (#61) - Assembler: Disassembler and Binary Inspector](https://hive.blog/hive-196387/@scipio/learn-zig-series-61-assembler-disassembler-and-binary-inspector) - [Learn Zig Series (#62) - File Systems: Reading Directories and Metadata](https://hive.blog/hive-196387/@scipio/learn-zig-series-62-file-systems-reading-directories-and-metadata) - [Learn Zig Series (#63) - File Watching: Detecting Changes](https://hive.blog/hive-196387/@scipio/learn-zig-series-63-file-watching-detecting-changes) - [Learn Zig Series (#64) - Process Management: Fork, Exec, Wait](https://hive.blog/hive-196387/@scipio/learn-zig-series-64-process-management-fork-exec-wait) - [Learn Zig Series (#65) - Pipes and Inter-Process Communication](https://hive.blog/hive-196387/@scipio/learn-zig-series-65-pipes-and-inter-process-communication) - [Learn Zig Series (#66) - Shared Memory and Semaphores](https://hive.blog/hive-196387/@scipio/learn-zig-series-66-shared-memory-and-semaphores) - [Learn Zig Series (#67) - Signal Handling Deep Dive](https://hive.blog/hive-196387/@scipio/learn-zig-series-67-signal-handling-deep-dive) - [Learn Zig Series (#68) - Unix Domain Sockets](https://hive.blog/hive-196387/@scipio/learn-zig-series-68-unix-domain-sockets) - [Learn Zig Series (#69) - Daemonization: Background Services](https://hive.blog/hive-196387/@scipio/learn-zig-series-69-daemonization-background-services) - [Learn Zig Series (#70) - Timers and Scheduling](https://hive.blog/hive-196387/@scipio/learn-zig-series-70-timers-and-scheduling) - [Learn Zig Series (#71) - Resource Limits and Capabilities](https://hive.blog/hive-196387/@scipio/learn-zig-series-71-resource-limits-and-capabilities) - [Learn Zig Series (#72) - System Call Wrappers](https://hive.blog/hive-196387/@scipio/learn-zig-series-72-system-call-wrappers) - [Learn Zig Series (#73) - seccomp and Sandboxing](https://hive.blog/hive-196387/@scipio/learn-zig-series-73-seccomp-and-sandboxing) - [Learn Zig Series (#74) - ptrace: Process Tracing](https://hive.blog/hive-196387/@scipio/learn-zig-series-74-ptrace-process-tracing) - [Learn Zig Series (#75) - Reading Kernel State from /proc and /sys](https://hive.blog/hive-196387/@scipio/learn-zig-series-75-reading-kernel-state-from-proc-and-sys) - [Learn Zig Series (#76) - Mini Project: Process Monitor](https://hive.blog/hive-196387/@scipio/learn-zig-series-76-mini-project-process-monitor) - [Learn Zig Series (#77) - Mini Project: File Sync Tool - Part 1](https://hive.blog/hive-196387/@scipio/learn-zig-series-77-mini-project-file-sync-tool-part-1) - [Learn Zig Series (#78) - Mini Project: File Sync Tool - Part 2: Delta Transfer](https://hive.blog/hive-196387/@scipio/learn-zig-series-78-mini-project-file-sync-tool-part-2-delta-transfer) - [Learn Zig Series (#79) - Mini Project: File Sync Tool - Part 3: Network Protocol](https://hive.blog/hive-196387/@scipio/learn-zig-series-79-mini-project-file-sync-tool-part-3-network-protocol) - [Learn Zig Series (#80) - Mini Project: File Sync Tool - Part 4: Polish](https://hive.blog/hive-196387/@scipio/learn-zig-series-80-mini-project-file-sync-tool-part-4-polish) - [Learn Zig Series (#81) - UDP Sockets and Datagrams](https://hive.blog/hive-196387/@scipio/learn-zig-series-81-udp-sockets-and-datagrams) - [Learn Zig Series (#82) - DNS Resolver from Scratch](https://hive.blog/hive-196387/@scipio/learn-zig-series-82-dns-resolver-from-scratch) - [Learn Zig Series (#83) - DNS Server Implementation](https://hive.blog/hive-196387/@scipio/learn-zig-series-83-dns-server-implementation) - [Learn Zig Series (#84) - HTTP/1.1 Deep Dive](https://hive.blog/hive-196387/@scipio/learn-zig-series-84-http11-deep-dive) - [Learn Zig Series (#85) - HTTP/2 Frames and Streams](https://hive.blog/hive-196387/@scipio/learn-zig-series-85-http2-frames-and-streams) - [Learn Zig Series (#86) - TLS via C Interop](https://hive.blog/hive-196387/@scipio/learn-zig-series-86-tls-via-c-interop) - [Learn Zig Series (#87) - WebSocket Protocol](https://hive.blog/hive-196387/@scipio/learn-zig-series-87-websocket-protocol) - [Learn Zig Series (#88) - WebSocket Server](https://hive.blog/hive-196387/@scipio/learn-zig-series-88-websocket-server) - [Learn Zig Series (#89) - MQTT Messaging Protocol](https://hive.blog/hive-196387/@scipio/learn-zig-series-89-mqtt-messaging-protocol) - [Learn Zig Series (#90) - Protocol Buffers Serialization](https://hive.blog/hive-196387/@scipio/learn-zig-series-90-protocol-buffers-serialization) - [Learn Zig Series (#91) - MessagePack Format](https://hive.blog/hive-196387/@scipio/learn-zig-series-91-messagepack-format) - [Learn Zig Series (#92) - gRPC Service in Zig](https://hive.blog/hive-196387/@scipio/learn-zig-series-92-grpc-service-in-zig) - [Learn Zig Series (#93) - SOCKS5 Proxy](https://hive.blog/hive-196387/@scipio/learn-zig-series-93-socks5-proxy) - [Learn Zig Series (#94) - NAT Traversal and Hole Punching](https://hive.blog/hive-196387/@scipio/learn-zig-series-94-nat-traversal-and-hole-punching) - [Learn Zig Series (#95) - Mini Project: Chat Server - Protocol Design](https://hive.blog/hive-196387/@scipio/learn-zig-series-95-mini-project-chat-server-protocol-design) - [Learn Zig Series (#96) - Mini Project: Chat Server - Server Core](https://hive.blog/hive-196387/@scipio/learn-zig-series-96-mini-project-chat-server-server-core) - [Learn Zig Series (#97) - Mini Project: Chat Server - Client TUI](https://hive.blog/hive-196387/@scipio/learn-zig-series-97-mini-project-chat-server-client-tui) - [Learn Zig Series (#98) - Mini Project: Chat Server - Rooms and History](https://hive.blog/hive-196387/@scipio/learn-zig-series-98-mini-project-chat-server-rooms-and-history) - [Learn Zig Series (#99) - Mini Project: DNS-over-HTTPS Proxy](https://hive.blog/hive-196387/@scipio/learn-zig-series-99-mini-project-dns-over-https-proxy) - [Learn Zig Series (#100) - Mini Project: Port Scanner](https://hive.blog/hive-196387/@scipio/learn-zig-series-100-mini-project-port-scanner) - [Learn Zig Series (#101) - Mini Project: HTTP Load Tester - Part 1](https://hive.blog/hive-196387/@scipio/learn-zig-series-101-mini-project-http-load-tester-part-1) - [Learn Zig Series (#102) - Mini Project: HTTP Load Tester - Part 2](https://hive.blog/hive-196387/@scipio/learn-zig-series-102-mini-project-http-load-tester-part-2) - [Learn Zig Series (#103) - Mini Project: Reverse Proxy - Routing](https://hive.blog/hive-196387/@scipio/learn-zig-series-103-mini-project-reverse-proxy-routing) - [Learn Zig Series (#104) - Mini Project: Reverse Proxy - Load Balancing](https://hive.blog/hive-196387/@scipio/learn-zig-series-104-mini-project-reverse-proxy-load-balancing) - [Learn Zig Series (#105) - Mini Project: Reverse Proxy - Health Checks](https://hive.blog/hive-196387/@scipio/learn-zig-series-105-mini-project-reverse-proxy-health-checks) - [Learn Zig Series (#106) - Linked Lists: Singly and Doubly](https://hive.blog/hive-196387/@scipio/learn-zig-series-106-linked-lists-singly-and-doubly) - [Learn Zig Series (#107) - Skip Lists](https://hive.blog/hive-196387/@scipio/learn-zig-series-107-skip-lists) - [Learn Zig Series (#108) - B-Trees](https://hive.blog/hive-196387/@scipio/learn-zig-series-108-b-trees) - [Learn Zig Series (#109) - Red-Black Trees](https://hive.blog/hive-196387/@scipio/learn-zig-series-109-red-black-trees) - [Learn Zig Series (#110) - Tries: Prefix Trees](https://hive.blog/hive-196387/@scipio/learn-zig-series-110-tries-prefix-trees) - [Learn Zig Series (#111) - Bloom Filters](https://hive.blog/hive-196387/@scipio/learn-zig-series-111-bloom-filters) - [Learn Zig Series (#112) - Cuckoo Filters](https://hive.blog/hive-196387/@scipio/learn-zig-series-112-cuckoo-filters) - [Learn Zig Series (#113) - Ring Buffers: Lock-Free](https://hive.blog/hive-196387/@scipio/learn-zig-series-113-ring-buffers-lock-free) - [Learn Zig Series (#114) - Memory Pools](https://hive.blog/hive-196387/@scipio/learn-zig-series-114-memory-pools) - [Learn Zig Series (#115) - Slab Allocators](https://hive.blog/hive-196387/@scipio/learn-zig-series-115-slab-allocators) - [Learn Zig Series (#116) - Sorting Algorithms in Zig](https://hive.blog/hive-196387/@scipio/learn-zig-series-116-sorting-algorithms-in-zig) - [Learn Zig Series (#117) - Binary Search Variations](https://hive.blog/hive-196387/@scipio/learn-zig-series-117-binary-search-variations) - [Learn Zig Series (#118) - Graph Representation](https://hive.blog/hive-196387/@scipio/learn-zig-series-118-graph-representation) - [Learn Zig Series (#119) - BFS and DFS](https://hive.blog/hive-196387/@scipio/learn-zig-series-119-bfs-and-dfs) - [Learn Zig Series (#120) - Dijkstra and A](https://hive.blog/hive-196387/@scipio/learn-zig-series-120-dijkstra-and-a) - [Learn Zig Series (#121) - Topological Sort](https://hive.blog/hive-196387/@scipio/learn-zig-series-121-topological-sort) - [Learn Zig Series (#122) - Union-Find](https://hive.blog/hive-196387/@scipio/learn-zig-series-122-union-find) - [Learn Zig Series (#123) - LRU Cache](https://hive.blog/hive-196387/@scipio/learn-zig-series-123-lru-cache) - [Learn Zig Series (#124) - Consistent Hashing](https://hive.blog/hive-196387/@scipio/learn-zig-series-124-consistent-hashing) - [Learn Zig Series (#125) - Mini Project: Search Engine - Inverted Index](https://hive.blog/hive-196387/@scipio/learn-zig-series-125-mini-project-search-engine-inverted-index) - [Learn Zig Series (#126) - Mini Project: Search Engine - TF-IDF](https://hive.blog/hive-196387/@scipio/learn-zig-series-126-mini-project-search-engine-tf-idf) - [Learn Zig Series (#127) - Mini Project: Search Engine - Query Parser](https://hive.blog/hive-196387/@scipio/learn-zig-series-127-mini-project-search-engine-query-parser) - [Learn Zig Series (#128) - Mini Project: Database Engine - Page Storage](https://hive.blog/hive-196387/@scipio/learn-zig-series-128-mini-project-database-engine-page-storage) - [Learn Zig Series (#129) - Mini Project: Database Engine - B-Tree Index](https://hive.blog/hive-196387/@scipio/learn-zig-series-129-mini-project-database-engine-b-tree-index) - [Learn Zig Series (#130) - Mini Project: Database Engine - SQL Parser](https://hive.blog/hive-196387/@scipio/learn-zig-series-130-mini-project-database-engine-sql-parser) - [Learn Zig Series (#131) - Lexing a Simple Language](https://hive.blog/hive-196387/@scipio/learn-zig-series-131-lexing-a-simple-language) - [Learn Zig Series (#132) - Recursive Descent Parsing](https://hive.blog/hive-196387/@scipio/learn-zig-series-132-recursive-descent-parsing) - [Learn Zig Series (#133) - AST Design and Traversal](https://hive.blog/hive-196387/@scipio/learn-zig-series-133-ast-design-and-traversal) - [Learn Zig Series (#134) - Type Checking](https://hive.blog/hive-196387/@scipio/learn-zig-series-134-type-checking) - [Learn Zig Series (#135) - Bytecode Design](https://hive.blog/hive-196387/@scipio/learn-zig-series-135-bytecode-design) - [Learn Zig Series (#136) - Stack-Based Virtual Machine](https://hive.blog/hive-196387/@scipio/learn-zig-series-136-stack-based-virtual-machine) - [Learn Zig Series (#137) - Closures and Upvalues](https://hive.blog/hive-196387/@scipio/learn-zig-series-137-closures-and-upvalues) - [Learn Zig Series (#138) - Garbage Collection: Mark and Sweep](https://hive.blog/hive-196387/@scipio/learn-zig-series-138-garbage-collection-mark-and-sweep) - [Learn Zig Series (#139) - Garbage Collection: Generational](https://hive.blog/hive-196387/@scipio/learn-zig-series-139-garbage-collection-generational) - [Learn Zig Series (#140) - JIT Compilation Basics](https://hive.blog/hive-196387/@scipio/learn-zig-series-140-jit-compilation-basics) - [Learn Zig Series (#141) - Regex: Thompson NFA](https://hive.blog/hive-196387/@scipio/learn-zig-series-141-regex-thompson-nfa) - [Learn Zig Series (#142) - Regex: NFA to DFA](https://hive.blog/hive-196387/@scipio/learn-zig-series-142-regex-nfa-to-dfa) - [Learn Zig Series (#143) - Regex: Matching Engine](https://hive.blog/hive-196387/@scipio/learn-zig-series-143-regex-matching-engine) - [Learn Zig Series (#144) - Code Generation: AST to Machine Code](https://hive.blog/hive-196387/@scipio/learn-zig-series-144-code-generation-ast-to-machine-code) - [Learn Zig Series (#145) - Register Allocation](https://hive.blog/hive-196387/@scipio/learn-zig-series-145-register-allocation) - [Learn Zig Series (#146) - Mini Project: Calculator - Lexer/Parser](https://hive.blog/hive-196387/@scipio/learn-zig-series-146-mini-project-calculator-lexerparser) - [Learn Zig Series (#147) - Mini Project: Calculator - Interpreter](https://hive.blog/hive-196387/@scipio/learn-zig-series-147-mini-project-calculator-interpreter) - [Learn Zig Series (#148) - Mini Project: Calculator - Bytecode Compiler](https://hive.blog/hive-196387/@scipio/learn-zig-series-148-mini-project-calculator-bytecode-compiler) - [Learn Zig Series (#149) - Mini Project: Calculator - VM with Debugger](https://hive.blog/hive-196387/@scipio/learn-zig-series-149-mini-project-calculator-vm-with-debugger) - [Learn Zig Series (#150) - Mini Project: Lisp - Reader](https://hive.blog/hive-196387/@scipio/learn-zig-series-150-mini-project-lisp-reader) - [Learn Zig Series (#151) - Mini Project: Lisp - Evaluator](https://hive.blog/hive-196387/@scipio/learn-zig-series-151-mini-project-lisp-evaluator) - [Learn Zig Series (#152) - Mini Project: Lisp - Special Forms and Macros](https://hive.blog/hive-196387/@scipio/learn-zig-series-152-mini-project-lisp-special-forms-and-macros) - [Learn Zig Series (#153) - Mini Project: Lisp - Standard Library](https://hive.blog/hive-196387/@scipio/learn-zig-series-153-mini-project-lisp-standard-library) - [Learn Zig Series (#154) - Mini Project: Regex Engine - NFA](https://hive.blog/hive-196387/@scipio/learn-zig-series-154-mini-project-regex-engine-nfa) - [Learn Zig Series (#155) - Mini Project: Regex Engine - Matching](https://hive.blog/hive-196387/@scipio/learn-zig-series-155-mini-project-regex-engine-matching) - [Learn Zig Series (#156) - Framebuffer Basics](https://hive.blog/hive-196387/@scipio/learn-zig-series-156-framebuffer-basics) - [Learn Zig Series (#157) - Line Drawing: Bresenham](https://hive.blog/hive-196387/@scipio/learn-zig-series-157-line-drawing-bresenham) - [Learn Zig Series (#158) - Circle and Ellipse Rasterization](https://hive.blog/hive-196387/@scipio/learn-zig-series-158-circle-and-ellipse-rasterization) - [Learn Zig Series (#159) - Polygon Filling: Scanline](https://hive.blog/hive-196387/@scipio/learn-zig-series-159-polygon-filling-scanline) - [Learn Zig Series (#160) - 2D Transform Matrices](https://hive.blog/hive-196387/@scipio/learn-zig-series-160-2d-transform-matrices) (this post)
Learn Zig Series (#160) - 2D Transform Matrices
Last episode ended with a confession: every shape we can now draw -- lines, circles, ellipses, arbitrary filled polygons -- is pinned to hand-typed pixel coordinates. We wrote the triangle's corners as (2,2), (16,4), (8,17) by hand, and the star's ten vertices with a calculator. That is fine for a demo and hopeless for anything that has to move. The moment you want to slide a sprite across the screen, spin it around its centre, or draw the same arrow at ten sizes, you do not want to recompute vertices by hand -- you want one small, composable operation that transforms a whole bag of points at once. That operation is a matrix multiply, and by the end of today you will have written the whole thing from scratch. Here we go!
Solutions to Episode 159 Exercises
Three exercises last time, all pushing the scanline fill a little further. Here are my solutions.
Exercise 1 -- fill, then stroke. The task was to combine last episode's fillPolygonAET with episode 157's drawLine, so a filled polygon gets a crisp border in a second colour. Fill first, then walk the vertex loop drawing an edge between each consecutive pair. The test proves the two colours end up where they should -- a vertex pixel takes the stroke colour, a deep-interior pixel keeps the fill:
zig const std = @import("std");
const Rgba = packed struct { r: u8, g: u8, b: u8, a: u8 = 255 }; const Point = struct { x: i64, y: i64 };
const Framebuffer = struct { pixels: []Rgba, width: usize, height: usize,
fn setPixel(self: Framebuffer, x: i64, y: i64, color: Rgba) void { if (x = self.width or uy >= self.height) return; self.pixels[uy self.width + ux] = color; }
fn getPixel(self: Framebuffer, x: i64, y: i64) ?Rgba { if (x = self.width or uy >= self.height) return null; return self.pixels[uy self.width + ux]; } };
fn drawLine(fb: Framebuffer, x0: i64, y0: i64, x1: i64, y1: i64, color: Rgba) void { var x = x0; var y = y0; const dx: i64 = if (x1 > x0) x1 - x0 else x0 - x1; const dy: i64 = if (y1 > y0) -(y1 - y0) else -(y0 - y1); const sx: i64 = if (x0 = dy) { err += dy; x += sx; } if (e2 = lo and y = self.width or uy >= self.height) return; self.pixels[uy self.width + ux] = color; }
fn getPixel(self: Framebuffer, x: i64, y: i64) ?Rgba { if (x = self.width or uy >= self.height) return null; return self.pixels[uy self.width + ux]; } };
fn fillPolygon(fb: Framebuffer, points: []const Point, color: Rgba) void { if (points.len = lo and y = self.width or uy >= self.height) return; self.pixels[uy self.width + ux] = color; }
fn getPixel(self: Framebuffer, x: i64, y: i64) ?Rgba { if (x = self.width or uy >= self.height) return null; return self.pixels[uy self.width + ux]; } };
const Crossing = struct { x: i64, dir: i32 };
fn lessCrossing(: void, a: Crossing, b: Crossing) bool { return a.x = lo and y hole filled
var pb: [20 20]Rgba = undefined; var b = Framebuffer{ .pixels = &pb, .width = 20, .height = 20 }; @memset(b.pixels, .{ .r = 0, .g = 0, .b = 0 }); fillContoursWinding(&b, &.{ &outer, &inneropp }, white); try std.testing.expectEqual(@as(u8, 0), b.getPixel(10, 10).?.r); // opposite winding => hole empty }
That is the whole reason SVG exposes both fill-rule="nonzero" (the default) and fill-rule="evenodd": they disagree exactly when contours nest, and a font's counters -- the hole in an o, the two holes in a B -- rely on getting that disagreement right.
The core idea: one operation for move, spin and scale
Look at the three things you actually want to do to a shape. Translate it: add (tx, ty) to every point. Scale it: multiply every point by (sx, sy). Rotate it by angle t: the classic x' = xcos(t) - ysin(t), y' = xsin(t) + ycos(t). Two of those three are matrix multiplications already -- scale and rotate are linear, they can be written as a 2x2 matrix times the point. But translation is stubborn: adding a constant is not a linear operation, so it will not fit in the same 2x2 box. And if translate lives in a different world from rotate and scale, you cannot compose them uniformly, which defeats the whole purpose.
The fix is a genuinely beautiful trick called homogeneous coordinates: pretend every 2D point (x, y) is really the 3D point (x, y, 1). That extra 1 gives translation somewhere to hide. A 3x3 matrix whose bottom row is {0, 0, 1} can now express translate, scale, rotate and any combination of them, all as the same matrix point multiply. Here is the type, with identity and translation to start -- and apply, which treats the point's third coordinate as an implicit 1:
zig const std = @import("std");
pub const Vec2 = struct { x: f32, y: f32 };
pub const Mat3 = struct { // row-major 3x3; for an affine 2D transform the bottom row is always {0, 0, 1} m: [3][3]f32,
pub fn identity() Mat3 { return .{ .m = .{ .{ 1, 0, 0 }, .{ 0, 1, 0 }, .{ 0, 0, 1 }, } }; }
pub fn translation(tx: f32, ty: f32) Mat3 { return .{ .m = .{ .{ 1, 0, tx }, .{ 0, 1, ty }, .{ 0, 0, 1 }, } }; }
// transform a point, treating its third homogeneous coordinate as an implicit 1 pub fn apply(self: Mat3, p: Vec2) Vec2 { return .{ .x = self.m[0][0] p.x + self.m[0][1] p.y + self.m[0][2], .y = self.m[1][0] p.x + self.m[1][1] p.y + self.m[1][2], }; } };
test "identity leaves a point alone; translation slides it" { const p = Vec2{ .x = 3, .y = 4 }; const same = Mat3.identity().apply(p); try std.testing.expectEqual(@as(f32, 3), same.x); try std.testing.expectEqual(@as(f32, 4), same.y);
const moved = Mat3.translation(10, -2).apply(p); try std.testing.expectEqual(@as(f32, 13), moved.x); try std.testing.expectEqual(@as(f32, 2), moved.y); }
Why f32 and not the i64 we have used for pixels all series? Because rotation and scale produce fractional coordinates, and forcing them to integers mid-calculation throws away accuracy that compounds badly once you compose transforms. We keep the whole transform pipeline in floating point and round to integers only at the very end, right before setPixel -- the same discipline as computing then rounding, never rounding then computing.
Building the rest from scratch
Now the other three constructors and the operation that ties them together: the matrix product. Scaling puts the factors on the diagonal, rotation is the 2x2 rotation block lifted into the top-left corner, and mul is the textbook triple loop -- row i of the left matrix dotted with column j of the right:
zig const std = @import("std");
pub const Vec2 = struct { x: f32, y: f32 };
pub const Mat3 = struct { m: [3][3]f32,
pub fn identity() Mat3 { return .{ .m = .{ .{ 1, 0, 0 }, .{ 0, 1, 0 }, .{ 0, 0, 1 } } }; }
pub fn translation(tx: f32, ty: f32) Mat3 { return .{ .m = .{ .{ 1, 0, tx }, .{ 0, 1, ty }, .{ 0, 0, 1 } } }; }
pub fn scaling(sx: f32, sy: f32) Mat3 { return .{ .m = .{ .{ sx, 0, 0 }, .{ 0, sy, 0 }, .{ 0, 0, 1 } } }; }
pub fn rotation(radians: f32) Mat3 { const c = @cos(radians); const s = @sin(radians); return .{ .m = .{ .{ c, -s, 0 }, .{ s, c, 0 }, .{ 0, 0, 1 } } }; }
// matrix product: self other. Applying (self.mul(other)) means "do other, then self". pub fn mul(self: Mat3, other: Mat3) Mat3 { var r: Mat3 = .{ .m = undefined }; for (0..3) |i| { for (0..3) |j| { var sum: f32 = 0; for (0..3) |k| sum += self.m[i][k] other.m[k][j]; r.m[i][j] = sum; } } return r; }
pub fn apply(self: Mat3, p: Vec2) Vec2 { return .{ .x = self.m[0][0] p.x + self.m[0][1] p.y + self.m[0][2], .y = self.m[1][0] p.x + self.m[1][1] p.y + self.m[1][2], }; } };
test "a quarter turn sends the x-axis unit vector onto the y-axis" { const rot = Mat3.rotation(std.math.pi / 2.0); const p = rot.apply(.{ .x = 1, .y = 0 }); try std.testing.expectApproxEqAbs(@as(f32, 0), p.x, 1e-6); try std.testing.expectApproxEqAbs(@as(f32, 1), p.y, 1e-6); }
test "scale composes with translate as a single matrix" { // scale by 2 about the origin, THEN translate by (5, 5) const t = Mat3.translation(5, 5).mul(Mat3.scaling(2, 2)); const p = t.apply(.{ .x = 3, .y = 4 }); try std.testing.expectApproxEqAbs(@as(f32, 11), p.x, 1e-6); // 32 + 5 try std.testing.expectApproxEqAbs(@as(f32, 13), p.y, 1e-6); // 42 + 5 }
Read that mul order carefully, because it is the thing people get wrong for years. A.mul(B) produces the matrix that, applied to a point, does B first and A second -- because (AB)p = A(Bp). So "scale then translate" is translation(...).mul(scaling(...)), translate on the left. Matrix multiplication is not commutative, and that is not a mathematical annoyance -- it is the whole reason matrices are useful. Rotating a shape and then sliding it right lands somewhere completely different from sliding it right and then rotating (the second spins it around the origin, dragging it in a big arc). The matrix product lets you bake whichever sequence you mean into one object and reuse it.
Letting the type system carry the weight
Tags: #stem#stemsocial#steemstem#zig#programming