Metaballs

Each metaball is defined as a function in n dimensions (e.g., for three dimensions, $f(x,y,z)$; 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,

$$\backslash sum\_\{i\}\; \backslash mbox\{metaball\}\_i(x,y,z)\; \backslash geq\; \backslash mbox\{threshold\}$$

that is, all points larger than the threshold are inside the metaball.

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Image:Metaball contact sheet.png.
Note that the two smaller metaballs combine to create one larger object.]]

A typical function chosen for metaballs is simply inverse distance, that is, the contribution to the thresholding function falls off asymptotically toward zero as the distance from the centre of the metaball increases:

:$f(r)\; =\; \backslash frac\{a\}\{r\}$

where $a$ scales the magnitude of the contribution (changing the radius of a metaball) and $r$ is the distance from the ball's center, calculated from cartesian coordinates as:

:$r\; =\; \backslash sqrt\{(x-x\_0)^2\; +\; (y-y\_0)^2\; +\; (z-z\_0)^2\}$

where $(x\_0,\; y\_0,\; z\_0)$ is the center of the metaball.

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 an inverse square law, or a Gaussian potential constrained to a finite radius or a mixture of polynomials to achieve smoothness.{{Cite journal |last=Blinn |first=James Frederick |date=1982 |title=A generalization of algebraic surface drawing |url=https://dl.acm.org/doi/10.1145/357306.357310 |journal=ACM Transactions on Graphics |volume=1 |issue=3 |pages=235–256}} The Soft Object model by the Wyvill brothers provides higher degree of smoothness.{{Cite journal |last=Wyvill |first=Geoff |last2=McPheeters |first2=Craig |last3=Wyvill |first3=Brian |date=1986-08-01 |title=Data structure forsoft objects |url=https://link.springer.com/article/10.1007/BF01900346 |journal=The Visual Computer |language=en |volume=2 |issue=4 |pages=227–234 |doi=10.1007/BF01900346 |issn=1432-2315}}

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

{{Reflist}}

• {{cite journal |last1=Blinn |first1=J. F. |title=A Generalization of Algebraic Surface Drawing |doi=10.1145/357306.357310 |journal=ACM Transactions on Graphics |volume=1 |issue=3 |pages=235–256 |date=July 1982 |s2cid=24838292 |url=https://dl.acm.org/doi/pdf/10.1145/357306.357310}}

• [https://www.desmos.com/calculator/hkvv0eydjf Interactive 2D metaballs] using the online Desmos graphing calculator
• [http://paulbourke.net/geometry/implicitsurf/index.html Implicit Surfaces article] by Paul Bourke
• [https://docs.blender.org/manual/en/latest/modeling/metas/introduction.html Meta Objects article] from Blender wiki
• [https://web.archive.org/web/20141103023010/http://www.siggraph.org/education/materials/HyperGraph/modeling/metaballs/metaballs.htm Metaballs article] from SIGGRAPH website
• "[https://www.gamedev.net/tutorials/_/technical/graphics-programming-and-theory/exploring-metaballs-and-isosurfaces-in-2d-r2556 Exploring Metaballs and Isosurfaces in 2D]", 3 September 2008, Stephen Whitmore, gamedev.net

Category:3D computer graphics

Category:Demo effects

Category:Implicit surface modeling