Element One of the three fundamental, irreducible, unique and distinct qualities of projective, or synthetic, geometry — point, line and plane.
Incidence One of the immediate relationships, or interactions, of the geometric elements. For example, the incidence of two planes is a line.
Infinity This is a constant, cardinal number, or count, which must be an integer, and is commonly supposed to be the distance to an extraordinary place. But places need not be numbered, and numbers need not denote places. So we will have a spatial, geometric infinity only if place and number can be properly conflated.
Elements "at" infinity If two, distinct places are both at infinity, then the element incident with both (a line) is also, in its entirety, "at" infinity. If two lines are "at" infinity, then the single place at which they are incident, and the single plane with which they are incident, are also "at" infinity. However, because lines and planes do not have location (places - a.k.a. 'points' - alone have that as their sole quality), the preposition, "at", in the phrase, "at infinity", when used in reference to lines and planes, should be taken to imply that places incident with such elements will be at infinity.
In fact, there is no projective-geometric infinity, because infinity is a Euclidean notion, one based on assumptions that fail in strict, elementary geometry. There are no extraordinary places or elements in projective geometry. They are all ordinary. The discussion above concerning elements at infinity arises only in connection with a certain hybrid of Euclidean and Projective geometries, attributed to Arthur Cayley and Felix Klein.
bodies as closely in a line as they can be, in any order on that line. Both
Opposition and Conjunction are Alignments. "Closeness" is usually, though not
always, reckoned in respect of differences in Geocentric longitude.
Alignment(2) The degree to which two bodies are aligned, wrt a third considered as vertex. The square of the cosine of the angle subtended by the two other bodies on that vertex.
Epoch An actual, calendar, moment; a moment in Real Time; an actual Date and Time-of-Day. Usually for human convenience given and asked for in "Civil" date form in the Bud Worskshop, sometimes as a Julian Date. The BW always works with Julian epochs internally - easier, mostly because it doesn't need to take account of leap years! A Julian Year has 365.25 days.
Leap Year "If a calendar year, such as 2004, is exactly divisible by 4, but not by 100, then it is a leap year, with 366 days."
Grand Period (GP) The whole period of a bud series, or a whole period within which astronomical events are considered.
Inter-Alignment Period. (IAP) The time interval between two successive alignments.
mean Inter-Alignment Period (mIAP) The arithmetic mean of IAPs over a given Grand Period.
Pole In this context, one of the two real vertices of the Semi-Imaginary Invariant Tetrahedron judged to pertain to a real bud instance.
(IPD) Inter-Pole Distance The Euclidean distance between the top-most, highest or tip Pole, and bottom-most, lowest or base Pole, of a Bud.
Image Unit (iu) The unit of length used by the Bud Workshop for an uncalibrated bud image. The width of the image's frame, whatever it is in pixels, is taken to be of unit length. The BW always uses this unit internally, but converts lengths to the units of the current calibration , if one has been done, for display, and for the records. It is obviously neither a standard nor a constant unit.
Mean Radius Deviation (MRD) Properly, the arithmetic average of the differences between predicted and measured bud radii. In practice, the term is used loosely and is often applied without modification to diameters as well. And sometimes 'MRD' is used to refer to just one radius or diameter deviation, not to the actual Mean, as in, "The MRDs on this bud varied greatly.", meaning that the individual deviations did. It is usually clear from context what is intended.
Standard Error The MRD rendered in terms of Standard Deviation, the usual statistical measure.
Semi-Imaginary Invariant Tetrahedron When space is transformed into itself, some elements remain invariant - they do not move. Together, they form a tetrahedron, which is therefore invariant. If two of the four vertices of this tetrahedron are Real, and two Imaginary, we have a "Semi Imaginary Invariant Tetrahedron." All other points move (transform) in path curves, which, however, do not themselves move. Families of invariant path curves within this particular type of tetrahedron envelope a bud, and, highly probably ( p > 99.9% ), exactly describe its form. This invariant tetrahedron is accordingly the "frame" of the bud
Cross-Ratio Given that some measure is taken as standard, by that measure intervals preserve cross-ratio under projection: this means that for three adjacent intervals, a, b and c, projected from a point into intervals, a', b' and c,' on another line, we have
[(a+b)/c] / [a/(b+c)] = [(a'+b')/c'] / [a'/(b'+c')].
[Proof] Other permutations of these intervals also form cross-ratios: they differ one
from another, but all are constant.
There is a peculiar problem with regard to measurement in pure Projective Geometry: it does not define or preserve absolute sizeof anything. The cross-ratio is the only numeric constant, but, as indicated already, we cannot define it or use it until one of the measures is selected as standard.
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