Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

The top right quadrant of a circle
consists of an
upper 45o arc and lower
45o arc. Choose a circle with radius
√2.

The lower arc is shown below at left with four uniform steps in the x direction. A chord is constructed for each step.

Note the increasing separation between the chord and the arc as the chord lengthens.

Shown lower, for ten equal steps in x, is the sequence of chord step values, in red, and their sum.

At left is shown the effect of using variable size steps in x to create more uniform steps in the chord sizes, again using just ten steps.

Ideally, the increasing sum would show as a straight line.

The final step is smaller because the sum of the preceding chords came close to the end of the 45o arc.

The method employed for adjusting step size used a "set" value of √2 for chord length. If, in a previous step, the set chord length was exceeded, then the step in x was halved.

If x was about to exceed √2, then the final step in x was adjusted to just reach √2 at the end of the 45o arc.

This simple procedure was chosen because it was presumed that, for this problem, step size in x would not need to be increased.

The estimated ratios of the circumference of a circle to its diameter reached by the two methods are distinctly different. With 10 uniform steps in x, the estimate was 2.82842.. . With ten non-uniform steps it was 3.14064.. .

From our earlier work, using 10,000 uniform steps, we know that the ratio must be greater than 3.1415926527.

There are many possible algorithms to adjust step size but extra calculation is required to decide to change step size.

Highly accurate adjustment might require an amount of calculation greater than that required to achieve the same accuracy by simply reducing step size.

The spreadsheet cells for adaptive step size follow.

Relevant expressions can be viewed by clicking the (column, row) addresses following:

More will be said in subsequent topics about choice of step size.

### Next

In this topic we developed a lower bound to the ratio of circumference to diameter.
The
next topic deals both lower and upper bounds and with the ratio of a circle's
area to the square of its radius.

The lower arc is shown below at left with four uniform steps in the x direction. A chord is constructed for each step.

Note the increasing separation between the chord and the arc as the chord lengthens.

Shown lower, for ten equal steps in x, is the sequence of chord step values, in red, and their sum.

At left is shown the effect of using variable size steps in x to create more uniform steps in the chord sizes, again using just ten steps.

Ideally, the increasing sum would show as a straight line.

The final step is smaller because the sum of the preceding chords came close to the end of the 45o arc.

The method employed for adjusting step size used a "set" value of √2 for chord length. If, in a previous step, the set chord length was exceeded, then the step in x was halved.

If x was about to exceed √2, then the final step in x was adjusted to just reach √2 at the end of the 45o arc.

This simple procedure was chosen because it was presumed that, for this problem, step size in x would not need to be increased.

The estimated ratios of the circumference of a circle to its diameter reached by the two methods are distinctly different. With 10 uniform steps in x, the estimate was 2.82842.. . With ten non-uniform steps it was 3.14064.. .

From our earlier work, using 10,000 uniform steps, we know that the ratio must be greater than 3.1415926527.

There are many possible algorithms to adjust step size but extra calculation is required to decide to change step size.

Highly accurate adjustment might require an amount of calculation greater than that required to achieve the same accuracy by simply reducing step size.

The spreadsheet cells for adaptive step size follow.

Relevant expressions can be viewed by clicking the (column, row) addresses following:

. | . |

I32 | J32 | K32 | L32 | M32 | N32 | O32 | I33 |

More will be said in subsequent topics about choice of step size.