← Geodesic Flow & Non-Euclidean Geometries  ·  Contents  ·  Rendering & Textures →

In addition to parametric surfaces, Surface Explorer 4D supports Implicit Surfaces defined by mathematical equations in the form of f(x, y, z) = 0 (e.g., x^2 + y^2 + z^2 = 1). These are rendered using an advanced graphics technique called Ray Marching.

The Implicit tab holds two sub-tabs, each with its own equation editor:

• 3D — the ordinary case, an equation in x, y, z. This is what the rest of this page describes.
• Cross Section — an equation in four variables, x, y, z, p. The surface it defines lives in 4D space, and what you see on screen is the three-dimensional slice the camera cuts through it.

Clicking the sub-tab you are already on reloads its default surface: the quickest way back to a known state.

Write an equation in x, y, z, p and the app renders the slice of it that lies in the observer's three-dimensional space. Two groups of controls decide which slice:

• Omega, Phi, Psi (4D dock, and the Omega/Phi/Psi keys in the 3D dock) — rotate the surface in 4D, which changes the inclination of the cutting plane. The slice changes shape as you turn, often dramatically: a single 4D surface has a whole family of three-dimensional sections. The Speed 4D slider animates the rotation.
• P+ / P− (4D dock) — move the plane along the fourth coordinate, changing its height rather than its tilt. This is what lets you scan the hypersurface from one end to the other and watch the section appear, grow, split and vanish. Without it you would only ever see the central slice.

The 4D camera path works here too, so the slicing plane can be flown along a trajectory instead of moved by hand.

Instead of generating a grid of polygons (vertices and faces) on the CPU, the Ray Marching engine shoots rays from the virtual camera into the 3D space to find the exact point where the mathematical formula equals zero. This allows for mathematically perfect curves, infinite repeating structures, and complex boolean operations (additions, subtractions, and intersections of shapes) that would be impossible with standard geometry.

• Equation: The main formula. Use the variables x, y, z. Example: x^2 + y^2 + z^2 - 1.0 = 0.0 (the = 0.0 part is handled automatically if you omit it).
• Displacement / Variations: You can inject GLSL code here to distort the space or add bumps to the surface before it is evaluated.
• Ray Steps: Defines the maximum number of steps a ray can take to find the surface. Higher values (e.g., 200-500) improve rendering accuracy but reduce performance (framerate). The slider stops at 1000: for more, type the value in the field (up to 5000 on desktop, 2000 on mobile devices).
• Step Relax (S): A safety multiplier for the ray step size. If you see visual artifacts, "holes," or ghosting on highly distorted or fractal shapes, lower this value (e.g., to 0.4 or 0.2) to force the rays to advance more cautiously.
• Spatial Limits (Bounding Box): X, Y and Z Min/Max cut the shape inside an invisible box. Like the U/V limits they accept the constants A–F and S, and like the equation they apply with Run.
• Constant sliders (A–F): name a constant anywhere in the equation, in 3D Variation or in Texture Code and its slider unlocks. Textures written in the Shadertoy form (mainImage()) cannot reach the constants. The sliders are one set shared by the surface, its texture and the background — see the Scripting Guide.
• Solid vs Shell: Solid treats the object as filled, Shell renders it as a hollow shell with a wall of its own.
• Shell Thickness: the wall thickness in Shell mode, in scene units — the same real length for every equation.

Where a cut surface gets its lid. When the spatial limits cut through the object, the flat disc that closes the cut is not part of your equation: it is the face of the cutting box, drawn where the box passes through the surface. What the marcher renders is the boundary of the intersection between the two, and that boundary has two pieces — the curved part of your surface inside the box, and the part of the box inside your surface. This is why a cut sphere looks capped rather than open, and why, if you move the camera inside it, you find the cap from behind and the shell hollow: there was never any material in between, only the skin and the lid.

Troubleshooting Visual Artifacts

When exploring arbitrary implicit equations, you might encounter visual artifacts such as infinite "blades", floating planes, or holes in the geometry. This happens because some mathematical functions grow too fast, causing the ray to "overshoot" and completely jump past the surface.

Shell vs. Solid Stability: These artifacts are especially prominent in Shell mode. Because the shell is a thin wall, a fast-traveling ray can jump right through it without detecting a hit, which shows up as holes that open and close as the camera moves. The first thing to try is the Precise marcher (see below): it finds the wall by where the equation changes sign, so it misses it far less often. Raising Shell Thickness works too, by giving the ray a wider wall to land on. If you don't specifically need the hollow shell effect, Solid mode is more robust still. The 0.1 multiplier and a lower Step Relax (S) remain useful too.

These two radio buttons choose how the rays look for the surface. They matter most on equations of high degree — quartics, products of several factors, the 4D sections of Cross Section.

• Fast — sphere tracing, the method every surface has always used. It estimates how far the surface is from the value of the equation and jumps ahead by that much. Quick, and right almost everywhere.
• Precise — finds the surface by where the equation changes sign, then narrows it down by bisection. It cannot report a hit in empty space, because the equation has to actually cross zero.

The symptom that calls for Precise is a false weld: a blob of surface joining parts that should be separate, or a thick spur growing out of the object. It appears because the distance estimate Fast relies on becomes unreliable where the equation's gradient nearly vanishes, and the ray stops short of the surface — in mid-air. A good way to confirm it is to lower the transparency slider a little: the weld disappears at once, because transparency uses a different method. If it does, Precise will remove it on the opaque surface too.

Precise costs roughly 70% more work on the surfaces that need it. Fast stays the default because on a few 3D surfaces Precise can nibble the odd pixel along grazing edges; Cross Section surfaces open on Precise.

The Ray Steps and Step Relax sliders still matter with Precise. Before the sign-change search begins, the ray approaches the surface with the same steps Fast uses, and that approach decides two things: where the search starts, and which rays are treated as background. If a long step jumps clean over a thin part of the surface, the search starts beyond it, or the ray is discarded as background, and that part shows up as a hole. If Precise leaves holes or missing pieces, try a lower Step Relax first.

← Geodesic Flow & Non-Euclidean Geometries  ·  Contents  ·  Rendering & Textures →