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Surface Explorer calculates geometry in R4 (X, Y, Z, P) and projects it into R3 for visualization.
In addition to classic 3D rotations, three hyperspatial rotations involving the P axis are available:
- Omega (X-P): Rotates the X plane onto the P axis.
- Phi (Y-P): Rotates the Y plane onto the P axis.
- Psi (Z-P): Rotates the Z plane onto the P axis.
The three of them act on parametric surfaces, which are generated in 4D and then projected, and on the implicit surfaces of the Cross Section sub-tab, where they tilt the plane that slices a 4D surface — see Implicit Surfaces (Ray Marching). Implicit surfaces written in the 3D sub-tab have no fourth coordinate, so the rotations do not apply to them.
4D rotations in Stereographic mode: stereographic projection sends the pole P = +1 to infinity, so a surface passing near that pole is drawn at a huge distance. The engine caps that distance to keep the geometry inside the far plane, which flattens the affected vertices onto a single sphere and shows up as a rippled or serrated rim.
If your surface comes close to the pole, prefer Psi for continuous rotation and avoid Omega and Phi: those two mix a spatial coordinate with P and can push part of the surface onto the pole, while Psi often leaves the maximum P untouched. Raising Steps makes the ripple finer but never removes it, because the affected region is an area of the domain rather than a few isolated vertices. Switching to Perspective or Orthogonal avoids the effect entirely, since neither projection has a pole at infinity.
Lighting Modes (3D vs 4D) |
To seamlessly switch between classic 3D lighting and advanced 4D lighting, simply change the active motion dock:
- 3D Lighting (3D Motion Dock): When the 3D Motion dock is active, the engine uses standard 3D lighting. It ignores 4D depth, lighting the object as if it were a standard 3D shape and preserving classic shadows and volumes.
- 4D Lighting (4D Motion Dock): When the 4D Motion dock is opened, hyperspatial lighting takes over. You can cycle through the different 4D lighting models by clicking the dedicated Lighting Button inside the dock:
- Directional Lighting: Simulates a global "4D Sun" with parallel rays from infinity, providing even, studio-like illumination across the hyper-surface.
- Observer: Simulates a "light" positioned at the 4D observation point, dynamically brightening or darkening the surface based on its proximity in the 4th dimension while preserving 3D volume.
- Slice: Highlights the intersection with the current 3D observation hyperplane (useful for cross-sections) with soft volumetric transitions.
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