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The equation x = a ø , y = a(1 - cos ø) can be transformed very easily using a unit circle to say x = ø , y = (1 - cos (ø + pi/2) the equation is now in the form of the sin. So we measure and angle from the graph of the cycloid (point P) pi/2 (90 degrees). We then draw a line that intersects with the circle. Then draw a line perpendicular to the line that touches the circle. The line that is perpendicular to the horizontal center line intersects the line just drawn is where the sine path lies. For a simpler picture refer to the drawing. Note the position of this line is not stationery it changes by x distance OB = arc AB. So as the circle moves so does the position of the cycloids curve. If the point were stationary the curve formed would just be a variation of the ordinary cycloid. The cosine is given by 1+ or - the vertical length to point “P”.
It is added if P is in the lower quadrant and subtracted if P is in
the upper quadrant. |
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