TinyTv is a framework for generating ntsc video using two programmed i/o pins of an Atmel AVR AtTiny12 microcontroller to drive a resistor network serving as an digital to analog converter.
We will use a second order polynomial in x and y to describe a circle. The following spreadsheet explores the incremental evaluation of the polynomial as a series of addition operations on its coefficients.
http://c2.com/doc/Codosome2/VideoCircles.xls
On Sheet2 of this workbook we work out a sample calculation by both direct evaluation (with a polynomial in each cell) and incrementally (with an sum of neighbors in each cell). The two methods are shown to produce the same results. Here are the coefficients for the sample:
TinyTv.Coefficients 

a 
1 
b 
1 
c 
10 
d 
20 
e 
50 
And here are the values that we compute from the polynomial. The formulas before each table are the formulas in the spreadsheet cells.
The y coefficients are only computed once per horizontal scan. References (y2, y1 and y0) are to values from the previous horizontal scan.
startVert = b, startHorz = y2
TinyTv.Values 
y2 
2 

2 

2 

2 
startVert = b + d, startHorz = y2+y1
TinyTv.Values 
y1 
19 

17 

15 

13 
startVert = e, startHorz = y1+y0
TinyTv.Values 
y0 
50 

31 

14 

1 
The x coefficients are computed throughout the horizontal scan. Note x0 depends on y0 which is assumed to be already computed for the current row. Other references (x2, x1 and x0) are to values from the previous horizontal increment.
startHorz = 2a, incHorz = x2
TinyTv.Values 
x2 

2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
startHorz = a+c, incHorz = x2+x1
TinyTv.Values 
x1 

9 
7 9 actual 
5 9 actual 
3 9 actual 
9 5 actual 
7 3 actual 
5 1 actual 
3 1 actual 
9 5 actual 
7 7 actual 
5 9 actual 
3 9 actual 
9 13 actual 
7 15 actual 
5 17 actual 
3 17 actual 
startHorz = y0, incHorz = x1+x0
TinyTv.Values 
x0 

50 
41 50 actual 
34 50 actual 
29 50 actual 
31 34 actual 
22 29 actual 
15 26 actual 
10 25 actual 
14 29 actual 
5 34 actual 
2 41 actual 
7 50 actual 
1 70 actual 
10 83 actual 
17 98 actual 
22 115 actual 
Note that the origin is in the upperleft hand corner of the results. This value, for x=0 and y=0, is simply the constant term, e=50.
Note also that the spreadsheet computes in floating point while TinyTv operates with 8bit integer arithmetic. For the purposes of this test we have chosen integer coefficients that do not drive the calculation out of range over the small section of interest.
Finally Note that the TinyTv implementation increments e on each frame in order to effect an animation that is beyond the capability of the spreadsheet model.