Add comprehensive documentation on coordinate system conversions in note.md. Explain the necessity and order of transformations for world and camera coordinates, clarifying the distinction between Z-up to Y-up and OpenCV to OpenGL conversions.

note.md

@@ -107,6 +107,56 @@ The actual sequence is:
 
 Thank you for catching this! It's crucial to understand the correct order of operations, especially when dealing with coordinate system transformations.
 
+Let me break this down more clearly:
+
+## Why do we need these conversions?
+
+1. **World Coordinate Conversion (Z-up to Y-up)**:
+   - Your original data uses a Z-up coordinate system (Z is "up" in the real world)
+   - Three.js uses a Y-up coordinate system (Y is "up" in Three.js)
+   - We need to convert between these two world coordinate systems
+
+2. **Camera Coordinate Conversion (OpenCV to OpenGL)**:
+   - Camera coordinate systems define how the camera "sees" the world
+   - OpenCV uses a right-handed system where Z points forward from the camera
+   - OpenGL/Three.js uses a system where Z points backward from the camera
+   - We need to convert between these camera conventions
+
+## What's the difference?
+
+- **World conversion** (Z-up to Y-up) affects how we represent points in the world
+- **Camera conversion** (OpenCV to OpenGL) affects how the camera interprets what it sees
+
+These are completely separate transforms that operate on different coordinate spaces:
+- One operates on the world before the camera sees it
+- The other operates on the camera's view of the world
+
+## Why this specific order?
+
+The order matters because we're dealing with a transformation pipeline:
+
+1. First, we want to convert the world from Z-up to Y-up (`worldCvt`)
+   - This makes the world match Three.js's expectations
+
+2. Then, apply the camera's extrinsic matrix (`Rt`)
+   - This transforms world points into the camera's coordinate system
+   - At this point, we're in OpenCV camera space
+
+3. Finally, convert from OpenCV to OpenGL camera conventions (`cameraCvt`)
+   - This converts the camera space to match Three.js's expectations
+
+If we changed the order:
+- Applying camera conversion before the camera transformation would incorrectly transform world points
+- Applying world conversion after the camera transformation would transform points that are already in camera space
+
+Think of it as a pipeline:
+1. Fix the world coordinates (world conversion)
+2. View the world through the camera (camera extrinsic)
+3. Adjust how the camera interprets what it sees (camera conversion)
+
+The fact that matrix multiplication is associative means we can compute this entire pipeline as a single matrix operation, but the conceptual order still matters for getting the correct result.
+
+
 ---
 
 [Matrix4.multiply](https://threejs.org/docs/#api/en/math/Matrix4.multiply)

src/App.tsx

@@ -119,12 +119,14 @@ const Scene = () => {
     let Rt: THREE.Matrix4
     if (extrinsic instanceof THREE.Matrix4) {
       Rt = extrinsic
-    } else {
+    } else if (Array.isArray(extrinsic)) {
+      console.assert(extrinsic.length === 16, "extrinsic must be a 4x4 matrix")
       Rt = new THREE.Matrix4()
       // @ts-expect-error 16 elements
       Rt.set(...extrinsic)
+    } else {
+      throw new Error("extrinsic must be a 4x4 matrix or an array of 16 elements")
     }
-    console.assert(Rt.elements.length === 16, "extrinsic must be a 4x4 matrix")
     const font = new FontLoader().parse(HelvetikerRegular)
     const camera = new THREE.PerspectiveCamera(fov ?? 60, aspect ?? 4 / 3, near ?? DEFAULT_NEAR, far ?? DEFAULT_FAR)
     const helper = <cameraHelper args={[camera]} />
@@ -161,17 +163,25 @@ const Scene = () => {
     )
   }
 
-  const preProcessExtrinsic = (extrinsic: number[]) => {
-    // Create the initial world-to-camera transform
-    const worldToCamera = new THREE.Matrix4()
+  const preProcessExtrinsic = (extrinsic: number[] | THREE.Matrix4) => {
+    let Rt: THREE.Matrix4
+    if (extrinsic instanceof THREE.Matrix4) {
+      Rt = extrinsic
+    } else if (Array.isArray(extrinsic)) {
+      console.assert(extrinsic.length === 16, "extrinsic must be a 4x4 matrix")
+      Rt = new THREE.Matrix4()
       // @ts-expect-error 16 elements
-    worldToCamera.set(...extrinsic)
+      Rt.set(...extrinsic)
+    } else {
+      throw new Error("extrinsic must be a 4x4 matrix or an array of 16 elements")
+    }
 
-    // Convert from Z-up to Y-up first (this affects world coordinates)
-    const worldZupToYup = Z_UP_TO_Y_UP.clone()
 
     // Then handle OpenCV to OpenGL camera convention
-    const cameraConversion = CV_TO_GL_MAT.clone()
+    const cameraCvt = CV_TO_GL_MAT.clone()
+
+    // Convert from Z-up to Y-up first (this affects world coordinates)
+    const worldCvt = Z_UP_TO_Y_UP.clone()
 
     // Final transformation:
     // 1. Convert world from Z-up to Y-up
@@ -179,9 +189,9 @@ const Scene = () => {
     // 3. Convert camera coordinates from OpenCV to OpenGL
     const final = new THREE.Matrix4()
     final
-      .multiply(cameraConversion)
-      .multiply(worldToCamera)
-      .multiply(worldZupToYup)
+      .multiply(cameraCvt)
+      .multiply(Rt)
+      .multiply(worldCvt)
 
     // Invert to get the camera-to-world transform
     final.invert()