curves

If you paddle along a curved path, you still have some (all right, an infinity of) options about how your canoe pivots. If your canoe pivots in the same direction and at the same rate as the curve, then it’s an ordinary circle (fig.9); that’s assuming that your canoe started out facing in the direction you’ll be paddling. These figures show where the canoe goes in a variety of circumstances.

Click any figure to see an animation of that manoeuvre.

Figure 1. The canoe pivots at the same rate as the curve, but in the opposite direction.

Figure 2. The canoe pivots at three times the rate of the curve.

Figure 3. The canoe pivots at half the rate of the curve, but in the opposite direction. (takes two circles to complete)

Figure 4. The canoe pivots at 2.5 times the rate of the curve. (takes two circles to complete)

Figure 5. The canoe doesn’t pivot. (a displaced curve)

Figure 6. The canoe pivots at twice the rate of the curve.

Figure 7. The canoe pivots at half the rate of the curve. (takes two circles to complete)

Figure 8. The canoe pivots at 1.5 times the rate of the curve. (takes two circles to complete)

Figure 9. The canoe pivots at the same rate as the curve, with the canoe starting at a tangent to the circle. Your ordinary, garden variety forward circle.

Figure 10. The canoe pivots at the same rate as the curve, with the canoe starting pointing 45° toward the centre of the circle.

Figure 11. The canoe pivots at the same rate as the curve, with the canoe starting pointing 45° away from the centre of the circle.

Figure 12. The canoe pivots at the same rate as the curve, with the canoe starting pointing toward the centre of the circle.

Figure 13. The canoe pivots at the same rate as the curve, with the canoe starting pointing away from the centre of the circle.

Figure 14. The canoe pivots at the same rate as the curve, with the canoe starting pointing 45° toward the centre of the circle and away from the direction of travel.

Figure 15. The canoe pivots at the same rate as the curve, with the canoe starting pointing 45° away from the centre of the circle and away from the direction of travel.

Figure 16. The canoe pivots at the same rate as the curve, with the canoe starting at a tangent to the circle and facing away from the direction of travel. Your ordinary, garden variety backward circle.

Of course, all these examples assume that the pivot point is fixed in the centre of the canoe and the rate of curve and pivot is fixed. One could, with a bit of hard thinking, imagine: pivot points elsewhere in the boat (eg at the stems); pivot points outside the boat; pivot points that move; variable curves (eg spirals); and variable pivot rates.