Watching someone enter a perfect pivot and then glide out again is just like poetry in motion. It is beautiful to watch and gives the feeling of interconnectedness through paddle, body, boat and water. Beyond the fun of watching and executing pivots the skill serves many useful purposes.
A few other clips
Think of a pivot as a small circle (really small that is) or alternatively a circle is just a pivot that has gotten away. Just like the moon circling the earth paddling a circle includes one pivot for every circuit. If you can do a pivot well then you can paddle in circles, turn, land, and even go in a nice straight line.
To execute a nice pivot sit near the middle (front to back) and heel the canoe over so the gunwhale is close to the water line. By sitting in the middle you will have good control over both the bow and the stern. Heeling the canoe 'plants' or fixes the centre of the canoe in one place and pulls the ends up and out of the water. It is easier to spin when you are not dragging your tail. It is possible to pivot sitting towards the bow or stern but it means alittle more concentration and an adjustment of the force of each stroke (keep it simple to start with and sit near the middle).
Rotate your upper body to face the side of the canoe, I tend to turn my knees to the side alittle as well - similar to how I sit for side displacement. By facing the side of the canoe you have equal reach and better control over the draw/push towards the front or back of the canoe.
Choke up on the paddle (way up). This gives you good reach forward and backward. Remember the further the blade [tip] is away from your pivot point the more leaverage you have to turn the canoe. Make sure the paddle grip is pretty much outside of the canoe so the blade can be close, or even extend under, the canoe.
A basic starting point for pivoting the canoe involves a draw and a pushaway done a good distance away from the pivot point. Between each the paddle is sliced through the water without any pitch on the blade. This is called a Box stroke. The important thing to watch is that the draw/or push is perpendicular to the line of the canoe. The most common problem that you will run into when trying to pivot is doing a draw towards you, instead of the canoe, which will cause the canoe to move forward and cause to to go in a circle. With the push away the problem is usually almost opposite with the push going away from you, or to much of a push that transitions into a short sweep.
With really a small modification of box stroke you can paddle in a circle (sometimes called stirring the pot). Doing a circle stroke can be difficult since it involves getting the blade under the canoe.
It is possible to do a pivot with just a sweep stroke - choke up lots and use short strokes as far from your pivot point as you can reach. Many people think that doing on outside pivot is easiest using just a forward sweep. This can actually be quite tricky and will usually send you in a [small] circle instead.
You can also use cross-strokes (draw/push) to complete a pivot as well. These can be difficult to do in wider boats but they are very helpful in smaller boats.
Start slowly with small strokes and speedup as you go moving from a box, to circle. Next time you are out paddling try paddling into a pivot from an inside spiral. The smallest point of the spiral is a pivot. Go around a couple of times an then exit the opposite diretion following an outside spiral.
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