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Kevin's Corner

<spacer> <spacer> Taxicab Geometry > Point to Line Distance

In Euclidean geometry, the distance between a point A and a line l is found by 1) constructing the line perpendicular to the original line that passes through the point, 2) identifying the point B of intersection of the two lines, and 3) measuring the distance between the original point A and the point of intersection B of the two lines (Figure 1).


FIGURE 1: Distance from a point to a line in Euclidean geometry
is measured along a line perpendicular to the line.

Another more visual approach is to inflate an Euclidean circle about the point until the circle just touches (is tangent to) the line (Figure 2). In this case, the distance to the line is the radius of the tangential circle. It is this approach we will use to conceptually explore the distance between a point and a line in taxicab geometry.


FIGURE 2: Visualizing Euclidean distance from a point to a line
in terms of expanding circles about the point.

Consider the situation in Figure 3a. A taxicab circle is inflated about the point until the circle just touches the line. Notice that the distance from the point to the line is the direct vertical distance from the point to the line. But, perhaps this is due to the line being a shallow, or gradual line (a line whose slope has absolute value less than 1). What if the line was relatively steep? This situation is shown in Figure 3b where the slope of the line has absolute value greater than 1. The distance from the point to the line is now the direct horizontal distance from the point to the line. So, the position of the line has an impact on how the distance to a point is computed.


FIGURE 3: Visualizing taxicab distance from a point to a line in terms of expanding taxicab circles
about the point. Three cases appear for a) shallow / gradual lines, b) steep lines, and c) diagonal lines.

To complete our analysis, consider a line of slope 1 (or -1) in Figure 3c. In this case, there are an infinite number of paths that can be followed to compute the distance from the point to the line because the inflated circle actually overlaps the line and does not just touch it at one point.

Just as the position of an Euclidean line segment affects its taxicab length, the position of a line affects the way in which distance is measured from a point to the line. The position of objects affecting their taxicab properties is a common theme in taxicab geometry.

To summarize the cases for computing the distance between a point and a line,
  • If the line is shallow (has slope with absolute value less than 1), measure the distance from the point to the line along a vertical line.
  • If the line is steep (has slope with absolute value greater than 1), measure the distance from the point to the line along a horizontal line.
  • If the line is diagonal (has slope with absolute value equal to 1), measure the distance from the point to the line along either a horizontal or a vertical line.

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References
[1] Krause, Eugene F. Taxicab Geometry: An Adventure in Non-Euclidean Geometry, 1975.
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