A cycloid is a trajectory formed when a fixed point on the boundary of a circle moves along a straight line. It is an ordinary trochoid.

The initial position of the fixed point on the circle is the coordinate origin, and the straight line is the X axis. When the circle rolls by J angle, the fixed point on the circle reaches the position of P from the position of O point. When the circle rolls once, that is, J changes from O to 2π, the first arch of the cycloid is drawn on a fixed point on the moving circle. Scroll forward for another week, draw a second arch on the moving circle, and continue to scroll to get the third arch and the fourth arch ... All these arches are exactly the same in shape, with an arch height of 2a (the diameter of the circle) and an arch width of 2πa (the circumference of the circle).

Extended data properties:

1, its length is equal to 4 times the diameter of the circle of revolution. What is particularly interesting is that its length is a rational number that does not depend on π.

2. The area under the arc is three times that of the circle of revolution.

3. The points on the circle depicted by the cycloid have different speeds-in fact, it is even at rest somewhere.

When the marbles are released from different points of the cycloidal container, they will reach the bottom at the same time.

Formula:

If k is an integer, then the curve is closed, and the curve has k peaks (that is, sharp corners, and the curve is non-differentiable). Especially, for k = 2, the curve is a straight line and the circle is called cardano circle. Cardano circle first described hypocycloid and its application in high-speed printing.

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