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<spacer> <spacer> Angles and Trigonometry > Parallax

Parallax, the apparent shift of an object due to the motion of the observer, is a commonly used method for estimating the distance to a nearby object. The method of stellar parallax was used extensively to find the distances to nearby stars in the 19th and early 20th centuries. We now wish to explore the method and results of parallax in taxicab geometry and examine how these differ from the Euclidean method and results. We will discover that the taxicab method yields the same formula commonly used in the Euclidean case with the exception that the taxicab formula is exact.

Suppose that as a citizen of Taxicab Land you wish to find the distance to a nearby object Q in the first quadrant, and that there is also a distant reference object P "at infinity" essentially in the same direction as Q with reference angle θ (Figure 1). The distant reference object should be far enough away so that it appears stationary when you move small distances. We may assume without loss of generality that the object Q does not lie on either axis, for if it did, we could move a small distance to get the object in the interior of the first quadrant.


FIGURE 1: A parallax diagram in taxicab geometry.

Initially standing at A, measure the angle α between Q and P using the taxicab equivalent of a cross staff or sextant. Now, for reasons to be apparent later, you should move a small distance (relative to the distance to the object) in such a way that the distance to the object does not change. This can be accomplished by moving in either of two directions, and, provided you move only a small distance, the object remains in the interior of the first quadrant. Furthermore, exactly one of these directions results in the angle between Q and Q being increased, so that the situation depicted in Figure 1 is generic.

You have therefore moved from A to B in one of the following directions: NW, NE, SW, or SE. Now measure the new angle β between the two objects. With this information we can now find the taxicab distance to the object Q. Construct the point Q' such that QQ' is parallel to AB and len(QQ' ) = len(AB) = s. The angle PAQ' has measure β - α since it is merely a translation of angle QBP. Thus, angle QAQ' has measure β - α. Now, the lengths of AQ and BQ are equal since the direction of movement from A to B was shrewdly chosen so that the distance to the object remained constant. Since translations do not affect taxicab lengths, this implies AQ and AQ' have equal lengths. Hence, the points Q and Q' lie on a taxicab circle of radius d centered at A. Using the formula for the length of a taxicab arc,

(1)

where s and d are taxicab distances and β - α is a taxicab angle. This formula is identical to the Euclidean distance estimation formula with d and s Euclidean distances and β - α a Euclidean angle. However, as we shall now see, the commonly used Euclidean version is truly an approximation and not an exact result. This realization is necessary to logically link the commonly used Euclidean formula and the taxicab formula.

Using Figure 1 but with all distances and angles now Euclidean, we isolate triangle QAB. Using the law of sines and the fact that


we have

(2)

This formula can be simplified by moving from A to B in a direction perpendicular to the line of sight to Q from A rather than in one of the four prescribed directions above. In this case, we see from Figure 2 that


Thus, the law of sines gives the Euclidean parallax formula


If we now apply the approximation that the tangent function is equal to the identity function for small angles we obtain the commonly used Euclidean parallax formula


which is not exact.


FIGURE 2: A parallax diagram in Euclidean geometry.

It is interesting to note the quite different movement requirements in the two geometries to obtain the best possible approximations of the distance to the object. This difference lies in the methods of keeping the distance to the object as constant as possible. In the Euclidean case, moving small distances on the line tangent to the circle of radius d centered at the object (i.e. perpendicular to the radius of this circle) essentially leaves the distance to the object unchanged. In taxicab geometry, moving in one direction along either y = x or y = -x keeps the distance to the object exactly unchanged.

We are now in a position to justify the link between results (1) and (2). Since the line segment QQ' of taxicab length s lies on a taxicab circle,


The distance d to the object is given by


since the Euclidean angle between the line of sight AQ and the x-axis is (alpha + theta). Using the Euclidean-Taxicab Angle Measure Corollary with

   

the taxicab measure of β - α is given by


Using these substitutions, formula (1) becomes formula (2).

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References
[1] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 11, No. 2 (Spring 2000), pp. 87-96.
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