Dot Product finds Arc Midpoint

Everyone knows the good old midpoint of a line segment formula: if two points in xy-plane are given by A(x₁, y₁)  and B(x₂, y₂), then midpoint C(x₀, y₀) of the line segment AB has coordinates x₀ = (x₁ + x₂)/2  and y₀ = (y₁ + y₂)/2. This is simple enough. Now take these two points to be on a circle, centered at (0,0) and find the coordinates of a midpoint M(x_m, y_m) of the arc AB. Suddenly there is no formula for such a simple problem.

Of course you could use your geometry skills to find the midpoint C of the chord connecting the two given points, then find the equation of the line through C and the origin, and finally find where this line intersects the circle. On any given day this should not take you more than 5 minutes. So, let?s consider a three dimensional case. Using the above method might take as much as 15 minutes. Want to derive a quick formula? This will most likely lead you to discover different cases, and memorizing which one works in which octant is not the most optimal solution (even for a computer algorithm). But if that doesn?t scare you, just think of the n-sphere! Consider an alternative.

Let A(x₁, y₁, ..., z₁) and B(x₂, y₂, ..., z₂) be fixed points on an n-sphere of radius R centered at the origin O(0, 0, ... 0), such that AB is not a diameter. Let M(x_m, y_m, ..., z_m) be midpoint of arc AB of radius R (see diagram for n=2). Consider vectors x = OA and y = OB, and define λ = 1 + (x?y)/R², where ? is the standard dot product. Then vector m = OM is given by m = (x+y)/(2λ)^1/2 

The proof is very elegant. Consider the vector sum, u = x+y = (x₁+x₂, y₁+y₂, ..., z₁+x₂). Since, by construction, m = (R/|u|)u we get:

img src="http://ArcMidFormula.jpg" height="140" width="300"

It is interesting to note that midpoint of line segment is a special case of arc midpoint as R→∞. You can see this since limit as R→∞ of λ is 2. This is expected, considering that arc AB has a finite length (otherwise it would not make sense to talk about its midpoint), and therefore arc loses its curvature as radius is becoming infinitely large.

You may be wondering, aren?t there two arcs joining A and B? Indeed there are. The above result gives the midpoint of the shorter arc. To find the midpoint of the longer arc simply negate all of the coordinates of M.

img src="http://ArcMid.jpg" height="240" width="340"

Alex Akulov

Oleksandr G. Akulov