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<spacer> <spacer> Length / Area / Volume > Area

Before studying this section, please make sure you have read the section Concept of Area which explains the basic assumptions and concept of area in taxicab geometry in two and three dimensions.

Area in Two Dimensions

The standard approach to the computation of two-dimensional area in undergraduate calculus courses is integration. Estimates of area under a functional curve involve increasing numbers of rectangles that each approximate a portion of the area under the curve. Since Euclidean and taxicab geometry agree on length in the horizontal and vertical directions, these area approximations in the two geometries will agree. Therefore, we should expect the calculation of area under a curve by integration to transfer seamlessly to taxicab geometry. This is a further argument to those in Concept of Area to view area in two dimensions consistently between the two geometries.

Surface Area in Three Dimensions

As stated in Concept of Area, we should expect the taxicab area of a "flat" surface in three dimensions to differ from the area of the "same" figure (in a Euclidean sense) in one of the coordinate planes. This follows from the pattern seen with line segments in one and two dimensions. To discover how the area will differ, consider a plane rotated upward from the xy-plane along the y-axis (Figure 1).


FIGURE 1: A plane rotated up from the xy-plane along the y-axis will have different
Euclidean and taxicab scales of measurement in the x-direction.

In a manner similar to Figure 2 on the Linear Length page, the taxicab scale in the x-direction will be compressed in the rotated plane compared to the Euclidean scale. The angle θ formed in a plane parallel to the xz-plane will determine the scaling effect seen in the one dimension of the rotated plane with the maximal difference occurring at 45 degrees. The result is the "same" figure (in a Euclidean sense) in the xy-plane and in the rotated plane will have different taxicab areas. (Interestingly, the taxicab coordinate grid in the rotated plane is not even composed of Euclidean squares but of Euclidean rectangles since the taxicab scale in the y-direction is unaffected by the specific rotation performed.)

As a specific example, a unit square at the origin in the xy-plane will have a Euclidean and taxicab area of 1. If we rotate the square upward 45 degrees along the y-axis, the Euclidean area will be unchanged but the taxicab area will now be sqrt(2) since one side will now have taxicab length cos(45) + sin(45) = sqrt(2).

If the plane is instead rotated along the x-axis, the taxicab area is again found by scaling the Euclidean area. When rotation occurs in both manners, the scaling factors compound each other. So, the taxicab area of a figure in the rotated plane can be found from its Euclidean area using Equation 1 on the page Linear Length as a guide for each direction of rotation: the scaling factors are sums of the (Euclidean) cosine and sine of the angles α and β formed by lines in cross-sectional planes parallel to the xz-plane and the yz-plane similar to the single rotation illustrated in Figure 1,

(1)

Surfaces of Revolution

As an application of our approach to surface area in three dimensions, we would like to find a formula for the surface area of a solid of revolution. (Please see the Solids of Revolution page for a discussion of how to create a solid of revolution in taxicab geometry.) For a differentiable function f over an interval [a, b] divide the interval into n subintervals. Using a linear approximation of the curve over each subinterval, we revolve each line segment about the x-axis. In the Euclidean derivation, this creates the frustrum of a cone. In taxicab geometry, the Solids of Revolution discussion shows us this will create the frustrum of a taxicab cone, which is equivalent to the frustrum an Euclidean right square pyramid (partially shown in Figure 2). To find the taxicab surface area of the frustrum, we will compute the Euclidean area and then scale the area by examining the rotations of the frustrum sides relative to the xy-plane.


FIGURE 2: A portion of the revolution of a line segment about the x-axis in a taxicab manner
with a close-up of the shaded triangle. (The full revolution creates the frustrum of an Euclidean
right square pyramid.)

The Euclidean surface area of the frustrum of a pyramid is (1/2)(p1 + p2)s where p1 and p2 are the perimeters of the "top" and "bottom" edges of the frustrum and s is the slant height. For the kth subinterval, the Euclidean perimeters are 4*sqrt(2)*f(xk-1) and 4*sqrt(2)*f(xk). To compute the slant height, we project the shaded triangle in Figure 2 onto a plane parallel to the yz-plane. The slant height s is the hypotenuse of the right triangle ABC. The projected triangle is an isoceles right triangle with legs of length |f(xk) - f(xk-1)| (labeled df in the figure), so the altitude of this triangle is (sqrt(2)/2)*|f(xk) - f(xk-1)|. The other leg of triangle ABC has length Δxk (labeled dx in the figure). Therefore, the slant height is


So, we have the Euclidean surface area of kth frustrum of the approximation:


The sides of this frustrum form Euclidean angles of 45 degrees with the xy-plane using a cross-sectional plane parallel to the xz-plane. This gives one scaling factor for the taxicab surface area of cos(45) + sin(45) = sqrt(2). The other scaling factor is dependent on the sum of the cosine and sine of the angle of the linear approximation of the function and the x-axis. This gives another scaling factor of


Therefore, using Equation 1 above the taxicab surface area of the kth frustrum of the approximation is


As the derivation now continues along the same lines as the Euclidean version, the Intermediate Value Theorem and the Mean Value Theorem give us values xk* and xk** such that f(xk*) = (1/2)*(f(xk) + f(xk-1)) and f(xk) - f(xk-1) = f '(xk**)Δxk. Therefore,


Accumulating the frustrums and taking the limit yields the formula for the taxicab surface area of a solid of revolution (where πt = 4 is the value for π in taxicab geometry):


This is a very interesting formula. The portion outside the radical is very reminiscent of the Euclidean formula with the expected changes in the circle circumference factor (2πt*f(x)) and the arc length factor (1 + |f '(x)|) due to the taxicab metric. The extra radical represents a scaling factor based on the ratio of the slant height to the linear approximation of the curve. The greater the difference between these two quantities the more the frustrum sides are rotated with respect to the xy-plane thus requiring more scaling. If these two quantities are close, the frustrum sides are rotated very little and therefore require little scaling.

To complete the analysis inspired by Janssen described on the Solids of Revolution page, half of a taxicab sphere of radius r is obtained by revolving the function f(x) = -x + r over the interval [0, r]. Doubling this result yields the surface area of a whole taxicab sphere:


Such a taxicab sphere is composed of two Euclidean right square pyramids of height (r / sqrt(6))*2 and base side length r*sqrt(2). The Euclidean surface area is therefore 4*r2*sqrt(3). Since the sides are at 45 degree angles to the xy-plane in both directions, the taxicab area scaling factors are both sqrt(2). This gives a taxicab surface area of 8*r2*sqrt(3) in perfect agreement with the formula obtained from the solid of revolution above.

Other Examples

Using the standard definition of a taxicab parabola (described in [3]), half of a (horizontally) parallel case of the parabola with focus (0, a) and directrix y = -a is given by


Restricting the total "height" of the parabola to h and revolving the curve about the y-axis yields an open-top taxicab paraboloid (Figure 2) with surface area


As we would expect based on Figure 3, this is the sum of the surface area of a cylinder with radius a and height h - a and half a sphere of radius a.


FIGURE 3: A taxicab paraboloid.

In the defining paper concerning conics in taxicab geometry (see [4]), nondegenerate (or "true") two-foci taxicab ellipses are described as taxicab circles, hexagons, and octagons. If we revolve half of one of these figures about the x-axis we obtain a taxicab ellipsoid (Figure 4).


FIGURE 4: Nondegenerate, two-foci taxicab ellipses. Shown are the upper halves of a) the octogonal ellipse,
b) the hexagonal ellipse, and c) the circular ellipse (a taxicab circle).

If we consider a taxicab ellipse with major axis length 2a, minor axis length 2b, and s the sum of the distances from a point on the ellipse to the foci, the function


describes the upper half of an ellipse centered at the origin. This function will generally cover the sphere (s = 2a and a = b), hexagon (s = 2a and a > b), and octagon (s > 2a) cases for a taxicab ellipse. In Euclidean geometry, there is not a closed form for the surface area an ellipsoid. Since a taxicab ellipse is composed of straight lines, this problem is avoided in taxicab geometry. The surface area of an taxicab ellipsoidal solid of revolution is


For a hexagonal ellipsoid, the first term is the surface area of the ends which combine to equal a taxicab sphere of radius b; the second and last terms are zero; and, the third term is the area of a taxicab cylinder of radius b and length s - 2b composing the middle of the ellipsoid. For the octogonal ellipsoid (Figure 4a), the first term is an overestimate since the top of the sphere is not present on either end. This is corrected by the subtraction of the second term amounting to a taxicab sphere of radius (s/2) - a. (A similar correcion term is seen in the volume formula for the taxicab ellipsoid.) The last term accounts for the area of the ends of the ellipsoid which are taxicab circles of radius (s/2) - a.

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References
[1] 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.
[2] Janssen, Christina. Taxicab Geometry: Not the Shortest Ride Around Town. Dissertation (unpublished), Iowa State University, Jul 2007.
[3] Laatsch, Richard. Pyramidal Sections in Taxicab Geometry, Mathematics Magazine, Vol. 55, No. 4 (Sep 1982), pp. 205-212.
[4] Kaya, Rüstem; Ziya Akça; I. Günalti; and Münevver Özcan. General Equation for Taxicab Conics and Their Classification, Mitt. Math. Ges. Hamburg, Vol. 19 (2000), pp. 135-148.
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