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Length / Area / Volume > Arc Length
The cases and types of length measurement discussed on the Linear Length page are well known and are well understood to those familiar with taxicab geometry. But, even in two dimensions, there is at least one other type of length measurement in Euclidean geometry: the length of a curve. How is the length of a (functional) curve in two (or even three) dimensions measured in taxicab geometry?
Arc Length in Two Dimensions
In Euclidean geometry, the arc length of a curve described by a function f with a continuous derivative over an interval [a, b] is given by
To find a corresponding formula in taxicab geometry, we may follow a traditional method for deriving arc length. First, split the interval [a, b] into n subintervals with widths Δx1, Δx2, ... , Δxn by defining points x1, x2 , ... , xn-1 between a and b (Figure 1). This allows for definition of points P0, P1, P2, ... , Pn on the curve whose x-coordinates are a, x1, x2, ... , xn-1, b. By connecting these points, we obtain a polygonal path approximating the curve. At this crucial point of the derivation we apply the taxicab metric instead of the Euclidean metric to obtain the taxicab length Lk of the kth line segment,
FIGURE 1: Approximating arc length in taxicab geometry.
By the Mean Value Theorem, there is a point xk* between xk-1 and xk such that f(xk) - f(xk-1) = f'(xk*) Δxk. Thus, the taxicab length of the kth segment becomes
and the taxicab length of the entire polygonal path is
By increasing the number of subintervals while forcing max Δxk to zero, the length of the polygonal path will approach the arc length of the curve. So,
The right side of this equation is a definite integral, so the taxicab arc length of the curve described by the function f over the interval [a, b] is given by
As an example, the northeast quadrant of a taxicab circle of radius r centered at the origin is described by f(x) = -x + r. The arc length of this curve over the interval [0, r] is given by
which is precisely one-fourth of the circumference of a taxicab circle of radius r. A more interesting application involves the northeast quadrant of the Euclidean circle of radius r at the origin:
FIGURE 2: (a) Multiple straight-line paths between two points can have the same length in taxicab geometry.
b) A curve can be approximated by small horizontal and vertical paths.
The distance is the same as if we had traveled along the taxicab circle between the two endpoints! While this is a shocking result to the Euclidean observer who is accustomed to distinct paths between two points generally having different lengths, the taxicab observer merely shrugs his shoulders. A curve such as a Euclidean circle can be approximated with increasingly small horizontal and vertical steps near its path (Figure 2b). As any good introduction to taxicab geometry teaches us, such as Krause, multiple straight-line paths between two points have the same length in taxicab geometry (see Figure 2a). So, each of these approximations of the Euclidean circle will have the same taxicab length. Therefore, we should expect the limiting case to also have the same length. To further make the point, we can follow the Euclidean parabola f(x) = -(1/r)x2 + r in the first quadrant and arrive at the same result (Figure 3).
FIGURE 3: Multiple curves between two points can have the same length in taxicab geometry.
Shown are a) part of the taxicab circle of radius r, b) part of the Euclidean circle of radius r,
and c) part of the curve -(1/r)x2 + r.
To formalize these observations, we have the following theorem.
THEOREM 1: If a function f is monotone increasing or decreasing and differentiable with a continuous derivative over an interval [a, b], then the arc length of f over [a, b] is
(i.e. the path from (a, v=f(a)) to (b, w=f(b)) is independent of the function f under the stated conditions).
Proof: The taxicab arc length of f over [a, b] is
If f is monotone increasing, f'(x) ≥ 0. And, since f has a continuous derivative, we may apply the First Fundamental Theorem of Calculus to get
If f is monotone decreasing, we get the similar result
In either case, the latter difference must be positive so we have
For functions that are not monotone increasing or decreasing, the arc length can be found using this theorem by taking the sum of the arc lengths over the subintervals where the function is monotone increasing or decreasing.
Arc Length in Three Dimensions
The simplest extension of the arc length of a curve to three dimensions is for parametric curves. For a curve in two dimensions, we can parameterize the function f = (x(t), y(t)) and modify Equation 1 above to get
Generalizing this to three dimensions, we have
Intuitively, Theorem 1 generalizes to three-dimensions saying that the length of a three-dimensional "monotonic" curve depends only on its endpoints.
 Thompson, Kevin P. The Nature of Length, Area, and Volume in Taxicab Geometry, International Electronic Journal of Geometry, Vol. 4, No. 2 (2011), pp. 193-207.
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