Linear Biasing

In linear biasing, the biasing intensity corresponds to the positive slope of a straight line over the interval [0,1] of the Real Line. This interval is uniformly divided into as many subintervals as specified by the element density and they are mapped along the edge so that the length of the image interval is proportional to the height of the line over the midpoint of the source interval. Each image interval corresponds to the side of an element.

Specifically, let n be the element density and let .

We want a node placement function x(s) taking values in [0,1] with x(0) = 0 and x(1) = 1. If m is the slope of the line, and b is its y-intercept, then:

Using x(0) = 0, and x(1) = 1, we find: so,

For this, m is the absolute value of the biasing intensity. If the biasing intensity is negative, the nodes are placed according to 1 - x(s). Thus, a positive biasing intensity puts small elements at the start of the interval.

We can use b to scale the behavior of the function so that convenient values are in the range [0,20]. The value used is b = 1.5.