9 Spatial Coordinates🔗ℹ

SCENE-3D-A introduces the mathematical coordinate system used by spatial Visuals, SCENE-3D-B makes its wireframe part visible, SCENE-3D-C adds opaque triangle meshes, SCENE-3D-D adds deterministic spatial and camera motion, SCENE-3D-E adds semantic spatial relations plus crisp ordinary 2D labels anchored to projected spatial points, SCENE-3D-F adds finite points, tubes, sampled curves, arrows, axes, grids, and vector diagrams, SCENE-3D-G adds fixed-grid parametric/function surfaces with direct-time calculus helpers, SCENE-3D-I adds clipping, sections, and depth-aware transparency, and SCENE-3D-J adds direct-time linear, affine, pointwise, and homotopy maps. SCENE-3D-K adds immutable prepared 3D ODE trajectories plus deterministic static vector-field, streamline, and particle geometry. SCENE-3D-L adds immutable spatial inspection records, deterministic local BVH traversal, and exact camera-ray triangle picking for a sampled view3d. SCENE-3D-M adds an effectful retained-renderer protocol behind that immutable model. SCENE-3D-N adds explicit indexed-mesh topology diagnostics, orientation repair, and camera-independent compiled geometry resources; renderer caches and metrics remain outside the spatial tree and the deterministic software path remains the conformance reference. SCENE-3D-O adds renderer-independent mathematical strokes, screen-sized points and arrowheads, and camera-dependent feature/silhouette outlines. These marks prepare from immutable centrelines at the requested frame; they do not become camera-dependent author values. SCENE-3D-P adds an optional Racket/OpenGL implementation behind the same renderer protocol. It owns retained GPU resources and an offscreen framebuffer, then reads copied ARGB pixels back into the ordinary 2D composition; the software renderer remains the default reference. It still does not add a second scene timeline: view3d is an ordinary two-dimensional Visual inside the existing immutable scene, with a separate immutable spatial tree.

Spatial coordinates are right-handed:

The coordinate constants are origin3, x-axis3, y-axis3, and z-axis3. In particular, (vec3-cross x-axis3 y-axis3) is z-axis3. A two-dimensional world point embeds in the future spatial coordinate system as (vec3 x y 0).

The algebra, spatial-tree, mesh, and camera values in animate/3d are immutable, finite semantic data. They can be calculated and tested in a headless process just like the existing immutable scene model. Rendering a view3d through animate uses the ordinary Pict renderer; that adapter is the deliberate effectful boundary, not a second rendering system exposed to authors.

The author-oriented transform3 applies its components in this order:

(define transformed

(make-transform3

#:translation (vec3 4 0 0)

#:rotation (axis-angle z-axis3 (/ pi 2))

#:scale (vec3 2 2 2)))

(transform3-apply-point transformed (vec3 1 0 0))

That example first scales the local point, then rotates it counter-clockwise around positive z-axis3, then translates it. A general composition of nonuniform decomposed transforms can create shear, so transform3-compose returns an exact affine3 map rather than silently discarding it.

Current limitation: a view3d can draw depth-tested filled triangles plus mathematical screen/world strokes. Opaque triangles are clipped against all six camera-frustum planes, back-face culled by default, unlit/flat/smooth shaded, and resolved by a deterministic z-buffer. A render-only clip3d adds local half-space clipping, while slice-mesh3d and plane sections produce actual geometry. Transparent triangles are sorted far-to-near against the opaque depth target, which is deterministic but not order-independent transparency. There is no texture mapping, specular response, shadows, or arbitrary slice cap generation. Stage L picking operates on indexed mesh triangles (including generated curve and surface meshes), not analytic implicit shapes, UVs, or a GPU selection pass; its preview overlays are diagnostic-only and never enter a rendered frame. The default retained backend caches reference camera-space preparation but is not GPU accelerated. The optional OpenGL backend requires an explicit animate/3d/opengl choice in a GUI-capable Racket process and uses one serialized context, FBO readback, and tolerance-based rather than bit-exact software conformance. It has no direct GL presentation, GPU picking, textures, shadows, specular/roughness shading, or order-independent transparency. Stage F diagrams have deterministic physical-radius tubes and direct-time curve animation; Stage O adds screen-space widths, caps, joins, dashes, visible and hidden depth modes, and screen-sized marker primitives. The software coverage is deterministic rather than analytically antialiased, and hidden-line classification deliberately ignores transparent surfaces. Stage G surfaces use fixed rectangular topology, deterministic normal fallbacks, and direct-time reveal/morph. Adaptive/implicit surfaces and general solids remain later work. Projected labels are 2D overlays with fixed pixel offsets—not occlusion-aware 3D billboards—but may opt into opaque-depth 'hide or 'fade behaviour and can overlap. ‘move3d-*‘, ‘rotate3d-*‘, ‘scale3d-*‘, and ‘camera3d-*‘ requests are finite immutable clips: a frame is calculated from its requested time rather than the previous frame. Camera aspect comes from the viewport being rendered, so a camera is reusable in viewports of different sizes. Linear and affine map requests retain exact indexed topology and can apply to a whole named spatial subtree. Pointwise and homotopy maps currently operate on unwrapped mesh3d values only; they sample the authored vertices and do not adaptively remesh. A non-injective map can make degenerate or self-intersecting triangles. The default invalid-point policy is an error; 'drop-triangle deliberately leaves holes and does not cap or repair them. Prepared 3D ODE fields must return finite vec3 values. Their vector-field grids and streamline seeds are explicit finite samples, not adaptive field-line topology; there is no event detection or 3D ODE source inspector.