The goal of this problem is to slice a unit circle into two pieces which can be fit into a rectangle of minimal area.
Consider these three methods:

One: Slice the circle across its diameter and slide the pieces along each other a little.
What distance between the radii of the two semicircles gives the smallest rectangle? This rectangle has a smaller area than square that circumscribes the original circle. What is the ratio of rectangle to square?

Two: Slice the circle along a diameter and place these into a rectangle so the straight edges of each semicircle are along opposite edges of the rectangle.
What is the ratio of rectangle to square this time?

Three: Slice a segment off of the circle and place this segment to the side. The large piece's straight edge is along one side of the rectangle. The straight edge of the segment is tangent to the large piece but not necessarily parallel to a side of the rectangle.
What chord length minimizes the rectangle? What is the ratio?

Note: Part Three is may be particularly difficult to find an exact solution for. If you have a method of finding a good approximation feel free to share your results.

There are quite a few complexities in calculating the resulting rectangular area, and it is quite prone to error. So the derivation below should be checked, as well as the coding to see it matches the derivation. I've done my checking, but other eyes might see more.

Let the segment be sliced off the left side, for sake of argument. Let x be the width of the thickest part (middle) of the segment and theta be half the angle at the circle's center subtended by the segment.

theta = arccos(1-x)

Let alpha be the angle clockwise from the top of the circle, where the flat portion of the segment will be placed tangent to the circle.  We'll assume this will be no larger than 90, as we whatever could be placed on bottom could be placed on top instead. In the negative direction, it can't go any further than where it would go outside the left side of the rectangle as determined by where the circle was cut off.

The center of the arc of the segment so placed will be at (x*sin(alpha),x*cos(alpha)); it still retains its radius of curvature of 1.

If |alpha| exceeds theta, the height of the rectangle will be determined by the height of an endpoint of the segment, and is 1 + cos(alpha) + |sin(theta)*sin(alpha)|, using the absolute value as the opposite end of the segment will be determinative for positive or negative alpha. The 1 at the beginning is the bottom radius of the circle.

If, on the other hand, theta exceeds |alpha|, the highest point will be along the arc of the segment. The high point is then at 1 + 1 + yc, where yc is the x*cos(alpha) y-coordinate of the center of curvature of the arc portion of the segment.

In no instance, however, can the height be less than 2, which is the height if |alpha| is so large that no part of the segment extends above the top of the circle.

The width is done similarly, generally with sin(alpha) and cos(alpha) switched. But a check has to be made to see that the portion of the circle to the left of the slice is not exceeded.

The following program produces a table, and a summary of the best value (least area) found. The lowAlpha, highAlpha, lowX and highX are in their final states after homing in on a minimum.

DECLARE FUNCTION acos# (x#)
DEFDBL A-Z
DIM SHARED pi
pi = ATN(1) * 4

CLS

min = 999
lowAlpha = 12.03875#: highAlpha = 12.03876#: stSize = (highAlpha - lowAlpha) / 36#: highAlpha = highAlpha + stSize / 4
lowX = .4124#: highX = .4126#: xStep = (highX - lowX) / 8#: highX = highX + xStep / 4
FOR alpha = lowAlpha TO highAlpha STEP stSize
  LOCATE 2 + (alpha - lowAlpha) / stSize, 1
  PRINT USING "###.#### "; alpha;
  FOR x = lowX TO highX STEP xStep
    LOCATE 1, INT((x - lowX) / xStep + 1.5) * 7 + 2
    PRINT USING " #.####"; x;
    theta = acos(1 - x)
    sc = SIN(theta)
    cx = x * SIN(alpha * pi / 180)
    cy = x * COS(alpha * pi / 180)
    IF ABS(alpha * pi / 180) > theta THEN
      h = 1 + COS(alpha * pi / 180) + ABS(sc * SIN(alpha * pi / 180))
    ELSE
      h = 1 + cy + 1
    END IF
    IF h < 2 THEN h = 2
   
    IF (90 - ABS(alpha)) * pi / 180 > theta THEN
      exW = ABS(SIN(alpha * pi / 180)) + sc * COS(alpha * pi / 180)
    ELSE
      exW = ABS(cx) + 1
    END IF
    exW2 = SIN(alpha * pi / 180) - sc * COS(alpha * pi / 180)
    IF exW > (1 - x) AND alpha < 0 OR -exW2 > (1 - x) AND alpha >= 0 THEN
     ' doesn't work; extends beyond left slice
    ELSE
      IF alpha < 0 THEN
        w = 2 - x
      ELSE
        IF exW < 1 THEN exW = 1
        w = 1 - x + exW
      END IF
      LOCATE 2 + (alpha - lowAlpha) / stSize, INT((x - lowX) / xStep + 1.5) * 7 + 2
      PRINT USING "##.####"; w * h
      IF w * h < min THEN
        min = w * h: minAlpha = alpha: minX = x
        mincx = cx: mincy = cy
        minw = w: minh = h: mintheta = theta * 180 / pi
      END IF
    END IF
   
  NEXT
NEXT
LOCATE 45, 1: PRINT min, minAlpha, minX
PRINT mincx, mincy
PRINT minw, minh, mintheta

FUNCTION acos (x)
 IF ABS(x) = 1 THEN
   a = pi / 2 - pi / 2 * (SGN(x))
 ELSEIF x = 0 THEN
   a = pi / 2
 ELSE
   a = ATN(SQR(1 - x * x) / x)
 END IF
 acos = a
END FUNCTION

 

producing

      
      x: 0.0500 0.1000 0.1500 0.2000 0.2500 0.3000 0.3500 0.4000 0.4500
alpha   
-90.0000
-85.0000
-80.0000
-75.0000
-70.0000
-65.0000
-60.0000
-55.0000
-50.0000
-45.0000 3.9000
-40.0000 3.9000
-35.0000 3.9000
-30.0000 3.9432 3.9595
-25.0000 3.9746 3.9722
-20.0000 3.9907 3.9785 3.9608
-15.0000 3.9942 3.9835 3.9680
-10.0000 3.9960 3.9871 3.9733 3.9545
 -5.0000 3.9971 3.9893 3.9764 3.9586 3.9358
  0.0000 3.9975 3.9900 3.9775 3.9600 3.9375
  5.0000 3.9971 3.9893 3.9764 3.9586 3.9358 3.9081
 10.0000 3.9960 3.9871 3.9733 3.9545 3.9309 3.9023 3.8687
 15.0000 3.9942 3.9835 3.9680 3.9477 3.9226 3.8926 3.8578 3.8935 3.9333
 20.0000 3.9907 3.9785 3.9608 3.9383 3.9111 3.9091 3.9734 4.0242 4.0627
 25.0000 3.9746 3.9722 3.9515 3.9263 3.9457 4.0209 4.0814 4.1289 4.1644
 30.0000 3.9432 3.9595 3.9403 3.9544 4.0403 4.1094 4.1643 4.2067 4.2376
 35.0000 3.9000 3.9314 3.9352 4.0357 4.1128 4.1739 4.2214 4.2571 4.2827
 40.0000 3.9000 3.8878 3.9911 4.0935 4.1627 4.2140 4.2526 4.2833 4.3125
 45.0000 3.9000 3.8600 4.0128 4.1165 4.1861 4.2299 4.2646 4.2982 4.3309
 50.0000 3.9000 3.8925 3.9999 4.1033 4.1732 4.2260 4.2677 4.3030 4.3375
 55.0000 3.9000 3.9383 3.9526 4.0554 4.1351 4.2000 4.2532 4.2966 4.3325
 60.0000 3.9443 3.9679 3.9598 3.9851 4.0762 4.1518 4.2150 4.2681 4.3125
 65.0000 3.9765 3.9813 3.9719 3.9625 3.9968 4.0815 4.1535 4.2148 4.2672
 70.0000 3.9930 3.9879 3.9819 3.9759 3.9698 3.9898 4.0688 4.1370 4.1959
 75.0000 3.9966 3.9932 3.9898 3.9864 3.9830 3.9796 3.9761 4.0354 4.0994
 80.0000 3.9985 3.9970 3.9954 3.9939 3.9924 3.9909 3.9894 3.9878 3.9863
 85.0000 3.9996 3.9992 3.9989 3.9985 3.9981 3.9977 3.9973 3.9970 3.9966
 90.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000 4.0000

 3.857822164681937           15            .35
indicating the best for this particular table is an area of 3.857822164681937  at alpha=15 degrees and x = 0.35.

The idea is then to concentrate on numbers around those values. The program was modified to be more generic with regard to the upper and lower bounds of x and alpha.

        0.3000 0.3250 0.3500 0.3750 0.4000 0.4250 0.4500 0.4750 0.5000
10.0000 3.9023 3.8861 3.8687 3.8501
10.2778 3.9018 3.8856 3.8682 3.8496
10.5556 3.9014 3.8852 3.8677 3.8491
10.8333 3.9009 3.8847 3.8672 3.8485 3.8286
11.1111 3.9004 3.8842 3.8667 3.8480 3.8280
11.3889 3.9000 3.8837 3.8661 3.8474 3.8274
11.6667 3.8995 3.8831 3.8656 3.8468 3.8268
11.9444 3.8990 3.8826 3.8650 3.8462 3.8261
12.2222 3.8984 3.8820 3.8644 3.8456 3.8255
12.5000 3.8979 3.8815 3.8638 3.8449 3.8248
12.7778 3.8974 3.8809 3.8632 3.8443 3.8274
13.0556 3.8968 3.8803 3.8626 3.8436 3.8359 3.8574
13.3333 3.8963 3.8797 3.8619 3.8429 3.8444 3.8658
13.6111 3.8957 3.8791 3.8613 3.8423 3.8528 3.8742
13.8889 3.8951 3.8785 3.8606 3.8416 3.8611 3.8825
14.1667 3.8945 3.8778 3.8599 3.8447 3.8693 3.8907
14.4444 3.8939 3.8772 3.8592 3.8529 3.8774 3.8989
14.7222 3.8933 3.8765 3.8585 3.8609 3.8855 3.9069
15.0000 3.8926 3.8758 3.8578 3.8690 3.8935 3.9149 3.9333
15.2778 3.8920 3.8751 3.8571 3.8769 3.9014 3.9228 3.9412
15.5556 3.8913 3.8744 3.8570 3.8848 3.9093 3.9306 3.9490
15.8333 3.8907 3.8737 3.8648 3.8926 3.9170 3.9384 3.9567
16.1111 3.8900 3.8730 3.8725 3.9003 3.9247 3.9461 3.9644
16.3889 3.8893 3.8723 3.8802 3.9079 3.9323 3.9537 3.9719
16.6667 3.8886 3.8715 3.8878 3.9155 3.9399 3.9612 3.9794
16.9444 3.8879 3.8707 3.8954 3.9230 3.9473 3.9686 3.9868 4.0021
17.2222 3.8871 3.8719 3.9028 3.9304 3.9547 3.9759 3.9941 4.0094
17.5000 3.8864 3.8793 3.9102 3.9377 3.9620 3.9832 4.0014 4.0166
17.7778 3.8856 3.8867 3.9175 3.9450 3.9693 3.9904 4.0085 4.0237
18.0556 3.8849 3.8939 3.9248 3.9522 3.9764 3.9975 4.0156 4.0308
18.3333 3.8841 3.9012 3.9319 3.9593 3.9835 4.0045 4.0226 4.0377
18.6111 3.8833 3.9083 3.9390 3.9664 3.9905 4.0115 4.0295 4.0446
18.8889 3.8825 3.9154 3.9460 3.9733 3.9974 4.0183 4.0363 4.0513 4.0636
19.1667 3.8883 3.9224 3.9530 3.9802 4.0042 4.0251 4.0430 4.0580 4.0702
19.4444 3.8953 3.9294 3.9599 3.9870 4.0109 4.0318 4.0497 4.0646 4.0768
19.7222 3.9022 3.9362 3.9667 3.9937 4.0176 4.0384 4.0562 4.0711 4.0832
20.0000 3.9091 3.9430 3.9734 4.0004 4.0242 4.0449 4.0627 4.0776 4.0896

3.824829444556757           12.50000000000001           .4000000000000001

bringing us down to area = 3.825.

This process was continued, leading to

area = 3.815441597154761   alpha = 12.03875805555555   x = .4125
 (.0860364931384987,.4034277777352832) as the coordinates of the center of curvature of the arc of the segment in its new place.
width of rectangle = 1.5875    height of rectangle = 2.403427777735283  theta = 54.02019940252578

The chord length = 2 * sqrt(1 - (1-.4125)^2) ~= 1.618448330963951

The area relative to the original square is about .95386.

Edited on August 23, 2007, 10:57 am

Edited on August 23, 2007, 12:18 pm

Posted by Charlie on 2007-08-23 10:55:42