SDFs

Signed distance functions (SDFs) are a mathematical representation of shapes used in computer graphics and related fields. For any point p in space, an SDF returns the oriented distance to the surface of the shape: positive if p lies outside, negative if p lies inside, and zero when p is on the surface. Thus the surface is the zero level set of the function. SDFs can be defined in 2D or 3D and are valued for their smooth, continuous representation of geometry.

Shapes are built from primitive SDFs for simple forms such as spheres, boxes, planes, cylinders, and toruses. Complex shapes are formed by combining primitives with operations that analogize union, intersection, and subtraction, typically implemented as min, max, and negation of the component distances. This makes SDFs convenient for constructive solid geometry and procedural modeling.

Rendering and interaction with SDFs often use ray marching, a technique that advances a ray by the distance returned by the SDF until it approaches zero or exceeds a threshold. Normals for lighting are approximated from the SDF gradient. SDFs are resolution-independent and can be smoothly deformed, blended, and animated, enabling smooth transitions between shapes.

Applications include real-time rendering, collision detection in physics engines, procedural content generation, and tools for sculpting and 3D texturing. They provide a compact, adaptable representation for implicit surfaces, volume effects, and level-set methods, and are used in both entertainment and scientific visualization.

Limitations include the difficulty of exactly representing very detailed surfaces without numerous primitives or high-resolution samples, potential performance costs for large scenes, and challenges near sharp concavities. Nonetheless, SDFs remain a widely used and active area in computational geometry and computer graphics.