Implicit Surfaces

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Flubber

The image to the right is of the character “Flubber” from the 1997 Disney movie of the same title.

Basic Description

The Flubber image rendered is an implicit surface. Implicit equations are useful in computer graphics for representing these smooth shapes which we call implicit surfaces. The character is defined by these implicit equations which a rendering method uses to produce the output of this image.

A More Mathematical Explanation

An implicit function is a function in which the dependent variable has not been given explicitly in t [...]

An implicit function is a function in which the dependent variable has not been given explicitly in terms of the independent variable. An implicit equation is different from an explicit equation in that one variable may depend on the value of another, but one is not given explicitly (e.g. ).
The equation $x^2 + y^2 + z^2 - 1 = 0$ produces a sphere, all of the points are an equal distance from the center of the object. Similarly, the equation $x^2 + (y-1.5)^2 + z^2 - .5 = 0$ produces a smaller spherical object. The equation $(x^2 + y^2 + z^2 - 1)\times( x^2 + (y-1.5)^2 + z^2 - .5) - .015 = 0$ produces a model of the two objects blended together. $2 - (cos(x + T \times y) + cos(x - T \times y) + cos(y + T \times z) + cos(y - T \times z) + cos(z - T \times x) + cos(z + T \times x))$, where $T$ is the golden ratio, produces an even more complicated form. In this way, complex implicit functions can be made to describe a variety of complex objects, which has proved useful in computer-aided design.

Applications for Implicit Surfaces

Implicit surfaces are useful in the modeling of medical data, fluid flow (for engineering), interactive characters, and numerous other applications. These surfaces which have been developing since the early 1970’s can be rendered using ray tracing or various other algorithms and can range in complexity from simple geometric objects, to complex objects which create full scenes. Implicit methods simplify some modeling operations like blending, warping, collision detection, et al. By using implicit methods to define 3-D objects, it has become easier to represent smooth objects and curves. Originally, trying to model smooth 3-D objects required using geometric primitives (lines, points, pyramids, etc.), which often were unable to create the smoothness which was desired. Implicit methods have eased the process of modeling smooth 3-D objects.

Related Tools

At the top half of the applet the metaballs are visible. The graph at the bottom corresponds to the a cross section of the metaball field at the red line in the top half.

If you can see this message, you do not have the Java software required to view the applet.

• Do It Yourself- See this applet if you would like to try graphing your own implicit surfaces.

References

http://en.wikipedia.org/wiki/Implicit_function
http://local.wasp.uwa.edu.au/~pbourke/miscellaneous/implicitsurf/
http://iat.ubalt.edu/summers/math/platsol.htm
Shirley, Peter, et al. Fundametals of Computer Graphics. 3rd ed. Natick, Massachusetts: A K Peters, 2009. Print.