1st 2nd inversions Root \/ \/ -----------------------| E | -----------------------| C C | -----------------------| | -----------G-----G---G-| - V (Fifth degree) | -----------E-----E-----| - III (Third degree) -----------C--------- - I (Root Note)
The naming of inversions is actually a complex matter because various companies will assign different names to the same element. What Arrow Dynamics calls a boomerang is called a sidewinder by Vekoma; what Vekoma calls a boomerang is called a cobra roll by Intamin and Bolliger & Mabillard and a batwing by Arrow - B&M also has an element called the batwing, but it is not the same as Arrow's nor is it a cobra roll. Vekoma does have an element called the cobra roll, but it doesn't resemble a B&M cobra roll or an Arrow batwing! And so on.
Inversion is the geometrical process by which points P are transformed to their corresponding inverses P'. Inversion is performed around a circle of inversion, in which all points outside the circle go inside, and all inside go outside. Points on the circle stay put.
To invert a point P outside the circle centered at O with radius r, (you might want to get out a piece of paper for this) draw the line segment OP. Now draw a tangent line to the circle that includes P (call the point it touches the circle T). Finally, draw the line segment OT. The foot of the altitude of OTP from T is the inverted point P' (That is, where the perpendicular to TP intersects OP).
Similarly, to invert a point P inside the circle, first draw the line containing OP. Draw a perpendicular line to OP at P. The point at which this intersects the circle is T. Finally, draw a tangent from the circle at T, and the point where T intersects the line containing OP is P', the inverted point.
Practice it a few times, just to get a feel for it. I'd include a diagram if I could.
Inversion does a lot of cool stuff. For instance, inverting an (infinitely long) line outside the circle of inversion turns it into a circle inside the inversion circle. Also, when inverting anything, intersecting points still intersect. For instance, inverting two intersecting lines gives two tangential circles.
Try inverting a square.
Analytically, inversion about the origin with radius r can be described with the vector equation:
x' = r2x / |x|2 (Eric Weisstein's World of Mathematics, http://mathworld.wolfram.com/Inversion.html)
Naturally, this can be extended to any circle of inversion through simple translation.
If lines are treated as circles of infinite radius, all circles invert to circles.
Many geometrical proofs can be done by inverting the figure in question about a suitable circle of inversion. It can turn massively complex concepts into relatively simple ones.
Inverting a parabola about its focus gives a cardioid. Inverting a logarithmic spiral about its center gives yet another logarithmic spiral.
What happens when you invert a hyperbola?
In*ver"sion (?), n. [L. inversio: cf. F. inversion. See Invert.]
1.
The act of inverting, or turning over or backward, or the state of being inverted.
2.
A change by inverted order; a reversed position or arrangement of things; transposition.
It is just the inversion of an act of Parliament; your lordship first signed it, and then it was passed among the Lords and Commons. Dryden.
3. Mil.
A movement in tactics by which the order of companies in line is inverted, the right being on the left, the left on the right, and so on.
4. Math.
A change in the order of the terms of a proportion, so that the second takes the place of the first, and the fourth of the third.
5. Geom.
A peculiar method of transformation, in which a figure is replaced by its inverse figure. Propositions that are true for the original figure thus furnish new propositions that are true in the inverse figure. See Inverse figures, under Inverse.
6. Gram.
A change of the usual order of words or phrases; as, "of all vices, impurity is one of the most detestable," instead of, "impurity is one of the most detestable of all vices."
7. Rhet.
A method of reasoning in which the orator shows that arguments advanced by his adversary in opposition to him are really favorable to his cause.
8. Mus. (a)
Said of intervals, when the lower tone is placed an octave higher, so that fifths become fourths, thirds sixths, etc.
Said of a chord, when one of its notes, other than its root, is made the bass.
Said of a subject, or phrase, when the intervals of which it consists are repeated in the contrary direction, rising instead of falling, or vice versa.
Said of double counterpoint, when an upper and a lower part change places.
9. Geol.
The folding back of strata upon themselves, as by upheaval, in such a manner that the order of succession appears to be reversed.
10. Chem.
The act or process by which cane sugar (sucrose), under the action of heat and acids or ferments (as diastase), is broken or split up into grape sugar (dextrose), and fruit sugar (levulose); also, less properly, the process by which starch is converted into grape sugar (dextrose).
⇒ The terms invert and inversion, in this sense, owe their meaning to the fact that the plane of polarization of light, which is rotated to the right by cane sugar, is turned toward the left by levulose.
© Webster 1913.
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