Finite Differences

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Functions defining primitives often need a lot of computation to determine the next pixel. However we can use the forward difference of the function $f$ at $x_i$ to save operations.

Examples

Implicit line function

Given $l(x,y)\triangleq ax+by+c=0$. Instead of finding the next $x$ value by calculating $l(x+1,y)$ (which requires two multiplications and two additions), we can use the functions.

$$\begin{array}{r}{\delta }_{x}l(x,y)=l(x+1,y)-l(x,y)=a\\ {\delta }_{y}l(x,y)=l(x,y+1)-l(x,y)=b\end{array}$$

We can then increment the line function by computing

$$l(x,y)+{\delta }_{i}l(x,y)=l(x,y)+\{a,b\}$$

Reducing it to only one addition for $x$, and one addition for $y$.