A sinusoid is an interesting shape (understatement of the day)..
In the image below you see a sinusoid with period 40 and amplitude 1 (the image scale is incorrect). It consists of 4 identical segments: if you reflect the first, blue segment about the vertical line x=10 , you get the green segment, which you then reflect about the X-axis, and about the vertical line x=20: it returns the red segment, which after a reflection about the vertical line x=30 returns the yellow segment:
Point A: is the green dot on the blue curve: the point where the horizontal line of y=0.5 (half the Amplitude of the sinusoid) intersects with the sine curve. It can be seen that the X-axis coordinate of this intersection point is at 1/3rd the length of the segment. Remember that 1 segment is 1/4th the total wave's period. 1/3rd of 1/4th is 1/12th the period. In plain English: half the height of the segment is covered by one third its length, and the next half of the height requires two thirds of the segment. So there is structure in the deceleration of the ascent of the curve.
Point B: is the orange point on the blue curve: the point where the vertical line of x=5 (half the length of the segment) intersects with the sine curve. It can be calculated that the Y-axis coordinate of the intersection point with the sinusoid and the vertical line at half the segment length, is at sqrt(((A)^2)/2) of the amplitude. In plain English: after you pass half the length of the segment, you will increase in height by a factor of the square root of half the squared amplitude.
These 2 points: half the height for 1/3rd the length, and half the length for sqrt(((A)^2)/2) the height (with A = the amplitude).
This is true for any period / amplitude, and I have not seen this documented anywhere.
Do you know of any other interesting geometric properties of sinusoids that are interesting or less commonly known? Maybe at other pi/.. points of the segment? Or other multiples of the amplitude (i.e. 0.25 or 0.75)?