Coloring And Assigning Attributes to Objects Defined by Algebraic Fields
Original Publication Date: 1991-Apr-01
Included in the Prior Art Database: 2005-Apr-02
An algorithm for work in graphics systems that define objects as a scaler field is described. It permits the mixing of colors according to a rule similar to those of differentiation.
Coloring And Assigning Attributes to Objects Defined by
for work in graphics systems that define
objects as a scaler field is described. It permits the mixing of
colors according to a rule similar to those of differentiation.
WINSOM2 system defines an object as a scalar field;
that is, a scalar value associated with every point in
three-dimensional space. To draw an object, it creates a solid of
all points in space for which the value is not positive. This
definition of objects permits many useful operations, such as
blending and averaging objects, which are performed by applying
conventional arithmetic operators such as addition pointwise at every
position in space. However, it is not clear how to color the blend or
average of a red and a green object.
defines an algorithm to assign colors to objects
resulting from arithmetic operations. These colors behave in a
sensible way both mathematically and intuitively. An example of
mathematical sensibleness is that obj1 color(1) - obj2 color(2) is
colored the same as obj1 color(1) + (-1) * obj2 color(2)). An
example of intuitive sensibleness is that the color of a blended pair
of objects takes the color of the underlying objects as the blend
falls close to them, and smoothly changes to intermediate colors over
the significant surface of the blend. In cases where only the set
operators are used, the algorithm returns the same color as early
versions of WINSOM did, except possibly for a point precisely on the
intersection of the two objects. The algorithm is appropriate to any
attribute, solid or surface, that is to be assigned to an object. It
would work on either density, or on hue, saturation.
object obj is made by the combination of several
colored objects, obj.i of color.i for i = 1 to n. We assign weights
to each of the colors col.i according to how much the value obj.i
effects the value of obj. We call this w.i. The color assigned to
(sigma w.i*colorj) / (sigma w.i).
w.i. is the
partial derivative of obj with respect to obj.i,
and it is this that gives the method mathematical sensibleness. The
formula above does not specify a point p in space. Like the formula
that defined obj itself, it is assumed that the formula to derive its
color is defined pointwise over space. Not only can the values obj.i
vary over space, but so may the values color.i. Thus,
cylinder(1,1,0,0) color x is colored according to the x value, and
fred color sphere(1) is colored by distance from the origin.
(cylinder(1,1,0,0) color x) + (fred color sphere(1)) thus acquires a
Even if obj.i
is a constant, we can still measure what the
effect of changing it would have on obj. Thus, we can talk about
(sphere(1) color 1) + (2 color 3). The differentials used measure
the effect of a change in obj.i on a change on obj. They are not
to be confuse...