Metaballs

Each metaball is defined as a functioninn dimensions (e.g., for three dimensions, ${\displaystyle f(x,y,z)}$ ; three-dimensional metaballs tend to be most common, with two-dimensional implementations popular as well). A thresholding value is also chosen, to define a solid volume. Then,

$${\displaystyle \sum _{i}{\mbox{metaball}}_{i}(x,y,z)\leq {\mbox{threshold}}}$$ 

determines whether the volume enclosed by the surface defined by the metaballs is filled at ${\displaystyle (x,y,z)}$  or not.

A more Informal definition could be, That if you take 2 circles in 2D, and at point P, circle 1's influence(1/distance) is X and circle 2's influence is Y.

If X+Y>threshold. point P is part of Metaball.. And then you calculate it for all Points, obviously there are Graphing techniques to do that. Interactive Metaball with Neat Function

A typical function chosen for metaballs is the inverse-square law, that is, the contribution to the thresholding function falls off in a bell-shaped curve as the distance from the centre of the metaball increases.

For the three-dimensional case,

${\displaystyle f(x,y,z)=1/{\sqrt {(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}}}$ 

where ${\displaystyle (x_{0},y_{0},z_{0})}$  is the center of the metaball. The fast inverse square root technique may be used in this calculation.

Various other falloff functions have historically been used for reasons of computational efficiency. Desirable properties of the function include:

• Finite support. A function with finite support goes to zero at a maximum radius. When evaluating the metaball field, any points beyond their maximum radius from the sample point can be ignored. Nearest neighbor search can ensure only adjacent metaballs need to be evaluated regardless of the total number in the field.
• Smoothness. Because the isosurface is the result of adding the fields together, its smoothness is dependent on the smoothness of the falloff curves.

More complicated models use a Gaussian potential constrained to a finite radius or a mixture of polynomials to achieve smoothness. The Soft Object model by the Wyvill brothers provides higher degree of smoothness.^{[citation needed]}

A simple generalization of metaballs is to apply the falloff curve to distance-from-lines or distance-from-surfaces.

There are a number of ways to render the metaballs to the screen. In the case of three dimensional metaballs, the two most common are brute force raycasting and the marching cubes algorithm.

2D metaballs were a very common demo effect in the 1990s. The effect is also available as an XScreensaver module.

• NURBS
• Bézier surface

• Blinn, J. F. (July 1982). "A Generalization of Algebraic Surface Drawing". ACM Transactions on Graphics. 1 (3): 235–256. doi:10.1145/357306.357310. S2CID 24838292.

• Implicit Surfaces article by Paul Bourke
• Meta Objects article from Blender wiki
• Metaballs article from SIGGRAPH website
• "Exploring Metaballs and Isosurfaces in 2D", 3 September 2008, Stephen Whitmore, gamedev.net