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I am wondering whether there is any general method of approximating the midpoint of a given curve, given the coordinates of the endpoints and the equation of the curve. I know the calculus method of finding it exactly, but I am looking for a purely algebraic method that works for all elementary functions.

Considering it, I have come up with 4 methods:

  1. Finding the intersection of the vertical line down from the midpoint of the straight line connecting the two points, and the curve;

  2. finding the intersection of the horizontal line down from the midpoint of the straight line connecting the two points, and the curve;

  3. finding the intersection of the perpendicular line to the straight line connecting the two points and that passes through that line's midpoint, and the curve;

  4. guessing and checking via numerical integration.

    The first 2 are clearly very poor approximations, the 3rd only somewhat better, and the last, is of course undesirable as a guess and check method.

The definition I am using is based on arc length, and I just want the method to work for as many functions as possible, I don’t really care if they are elementary or not.

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  • $\begingroup$ What exactly is the midpoint, how do you define it? Is it by arc length? What types of curves are you considering, as you try to use algebraic methods, are the curves also some kind of algebraic? Is the curve $(x,\exp(x))$ elementary in your definition of it? $\endgroup$
    – Lutz Lehmann
    Mar 2, 2019 at 14:50

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If you are trying to find the midpoint of the arc from the (upper, lower, eastern or western) most point of a circle to any adjacent most point of a circle given the radius and center of a circle, then you just divide the radius of the circle by the square root of 2 to find the distance the x and y travelled along the slope from the center. Mostly its logic. If u draw an x, y coordinate plane making the same center as the circle, then u can easily add or substract to find the midpoint.

For example, if we are trying to find the midpoint of the arc between the uppermost point and the easternmost point of the circle, then we first divide the radius by the square root of 2. Then we will try to figure out if x and y increase or decrease along the slope. This can easily be done by seeing the plane, in this case both x and y increase because the line continues from the center to the NE direction. So finally we add the answer we got from dividing the radius to the square of 2, to both x and y. Then we will have the most approximate answer.

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