Signed Distance Function (SDF)

A Signed Distance Function (SDF) is a mathematical construct used to model shapes by representing the distance from any point in space to the nearest point on a shape’s surface. The function is characterized by its sign: it is negative inside the shape, positive outside, and zero on the boundary. This property makes SDFs particularly useful in various applications, including computer graphics, computational geometry, and 3D printing.

Overview

SDFs are implicit functions that provide a way to describe complex geometries and surfaces. For any given point in space, an SDF returns the shortest distance to the surface of a shape. The sign of the distance indicates whether the point is inside or outside the shape. This feature allows SDFs to efficiently model and manipulate shapes in a computational environment.

Mathematical Representation

For a simple example, consider a circle in a 2D plane. The SDF for a circle centered at the origin with radius 1 can be defined as:

def circle(x, y):
    return (x**2 + y**2)**0.5 - 1

This function calculates the Euclidean distance from any point ((x, y)) to the nearest point on the circle’s boundary. The result is negative if the point is inside the circle, positive if outside, and zero if on the boundary.

Visual Representation

The concept of SDFs can be visualized through figures that illustrate how the function behaves in different regions of space:

• Figure 10.2 shows a disk and its SDF, highlighting the negative values inside the disk and positive values outside.

  Figure 10.2 A disk and its SDF: negative inside, positive outside

• Figure 10.3 compares an SDF of a circle with an equivalent model that isn’t an SDF. Both functions model the same circle, but the SDF provides additional information about the sign of the distance.

  Figure 10.3 An SDF of a circle (left) and an equivalent model that isn’t an SDF (right). Both functions model the same circle.

Applications

The versatility of SDFs extends to various fields, with one of the most prominent being 3D printing. SDFs allow for the creation of complex forms that can be programmed to exhibit specific properties, such as stress distribution, heat dissipation, or integration with biological tissues.

• Figure 10.1 illustrates a femur model generated from an SDF, designed to be porous. This porosity is crucial for implants to integrate properly with living tissue.

  Figure 10.1 This femur model, generated from an SDF, is specifically made to be porous. Porosity is important so that implants can grow properly into living tissue.

In summary, Signed Distance Functions provide a powerful tool for modeling and manipulating shapes in a computational setting, with applications ranging from graphics to advanced manufacturing techniques like 3D printing.