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The value returned from the Math.sqrt method can be considered to be either positive or negative. Only the positivevalue is returned by this method. When the returned value is considered to be positive and the results are plotted, one-half of a circle is displayed as shownby the BLUE line in Figure 10 . Similarly, when the returned value is considered to be negative and the results are plotted, the other half of the circle isdisplayed as shown by the GREEN line in Figure 10 .
double function(double rVal,double xVal){
double yVal = Math.sqrt(rVal*rVal - xVal*xVal + 0.0000000001);return yVal;
}//end function If the expression passed to the sqrt method is negative, the value returned by that method will not be valid. Because the computations inthis program are performed as type double , and because all floating point computations are only estimates of the truth, the computed differencebetween the radius squared and the x-value squared can actually be an extremely small negative value when it should be zero. A small positive fudge factor wasadded to prevent that value from going negative due to small floating point computational errors. (This sort of thing is often required when doing a lot of floating point computations. There are various ways to do it and there may bebetter ways than the one used here.)
The code in Listing 11 draws the BLUE half of the circle shown in Figure 10 .
for(int cnt=0; cnt<=100;cnt++,xVal += xInc){
//Get a y-value for the given x-value.yVal = function(rVal,xVal);
//Apply the offsets and scale the resultscol = (int)((xOff+xVal)*xScale);
row = (int)((yOff+yVal)*yScale);//Move to the first point without drawing a line because the// pen is not down. Translate the origin to the center in the
// process.turtle.moveTo(col + world.getWidth()/2,
row + world.getHeight()/2);//Lower the pen in order to draw a line from each point to the
// next point.turtle.penDown();
}//end for loop Having initialized xVal and xInc on the basis of the radius in Listing 9 , the code in Listing 11 is essentially the same as the code in the earlier programs in this lesson.
The code that draws the GREEN half of the circle is shown in Listing 18 . It is essentially the same as the code in Listing 10 except that the sign on yVal is flipped from positive to negative as discussed above .
One other difference that I haven't mentioned yet is trivial but interesting. You may have noticed that the line width in Figure 10 is about twice that in Figure 9 . I purposely did that to illustrate that one of the useful features of graphing functions with a turtle is that you can control the width of the line.The importance of that capability will become apparent in the next section.
That concludes the interesting differences between this program and the previous programs.
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