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The Algebra of Measure (1)

Real End-Points.

standard ruler

Remarks

Against the Standard Reference Measure(SRM), the plane at infinity is supposed to lie literally at an infinite distance from any non-infinite location, since in this case a point at numeric infinity corresponds with and lies on a point (actually, a degenerate double-point) at geometric infinity   [*]. However, there is no "geometric infinity", so this statement is hard to interpret meaningfully.

As a possible indication of the true nature of infinity, it would be expected that infinite physical energy must be expended to reach it
from any location not currently there, and this is probably the true measure of it, as geometry cannot provide one. We may need to look to physics, not geometry, for a proper definition and understanding of infinity.

(1) Cross Ratio * 

Cross Ratio is so important, it merits a page of its own.

(2) Perspective Measure.

The case for which both end-points are real, together (degenerate) and either actually local (i.e., "nearby", as above, not at geometric infinity), or local by virtue of being projected from infinity into a local region of a plane (for example, "in view").

The algebraic expression for the distance, according to a "standard ruler", or SRM, of a point in the perspective measure from the end-point(s) is derived in the panel on the right.

n represents the distance from origin in the native units of the perspective measure, so the nth u is the conversion of that distance, n, into the units of the SRM.

parthenon

Those Ancient Greeks certainly knew how to build!
The columns are very equally spaced, and the collonades are beautifully parallel.
Assuming the structure is level, we can see where the horizon runs, even though land obscures it. Strictly, though, this is the line at infinity for the horizontal planes of the Parthenon.

Measurement with a ruler (SRM!) of the distances (on the photograph) from the columns on the left to the vanishing point gave differences-of-adjacent-reciprocals identical to within 0.5%.

Given the use of the SRM, it may be shown (by, for example, using the theorems of Ceva and Menelaus), that for a quadrilateral ABCD as shown above, the interval formed on line l by the meets of the two pairs of opposite sides is the harmonic mean of the intervals formed on line l by the diagonals, so that—

(3) Geometric (or Exponential) Measure

The case for which the end-points are real and distinct, with one of the two placed locally, and the other at infinity. By the SRM, the measure formed is a geometric series.

The algebraic expression for the distance, according to a "standard ruler", or SRM, of a point in the geometric or exponential measure from the local end-point is derived in the panel on the right.

n represents the distance from origin in the native units of the exponential measure, so the nth u is the conversion of that distance, n, into the units of the SRM.

Again by using Ceva and Menelaus, it may be shown that the distance (by the SRM) of the nth point in this measure from end-point X is the geometric mean of distance n-1 and distance n+1, so that—

(4) Growth Measure

The case for which the end-points are real, distinct and local. It is the most general case.

The name, "Growth", arises from the fact that the heights at which branches, or leaves, emerge from the stems of many plants as they grow conform, at least approximately, to a measure of this kind.

p represents the distance from origin in the native units of the growth measure, so the pth u is the conversion of that distance, p, into the units of the SRM.

X and Y divide the line into two sections. The expression derived on the right applies to the section not including the point at infinity. The expression for a growth measure in the section including the point at infinity is obtained by changing the plus sign in the denominator to a minus sign.

It produces a "hyperbolic" measure, in the sense that, by the SRM, the intervals follow a version of the hyperbolic tangent function [tanh(x)], as indicated below.

tanh conversion

A Growth Measure, as seen by the SRM, is an exponential measure in perspective.

That is to say, if n is the distance from origin in the native units of a perspective measure, we can make n the distance to (say) the pth point in an exponential measure—in other words, a point in the the exponential measure as it appears to the perspective measure. We then convert the result to the SRM.

Imaginary Circling Measure (1) → (2) → (3) →

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