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Homology II-L '

Homology II

The geometric relationship between the two images of the point A and A ', following two distinct projections from two different centers of projection C and C 'outside of the plan is said homology . Before
perspectivist: ω1 = C, u, AA °
Second perspectivist: ω2 = C ', u, A' A °
Homology = product perspectivist = ω1 = ω = X ω2 U, U, AA '
The overlay planes π and π' are in actually a ruse to clarify the concept of product of two perspectivist, but the concepts remain the same if, instead of two superimposed levels, if one takes into account only. In this case we can consider A and A 'as the corresponding points on the same level as calculated by the projection centers C and C'. You get that a correspondence between points in the same plane, by projecting them on a another level of two distinct centers of projection.

• The intersection of the straight line joining the two centers of projection, with the plane π, find the point U said c by homology .
• The straight line joining A and A 'U passes. Corresponding points are in fact aligned with the center of homology. The line u , intersection of the two plans is the 'axis of homology .

Property homology

• Lines corresponding homology meet u axis (line of intersection between the planes π and π 0 - united line - locus of points combined);
• corresponding points are aligned with the center of homology U (the intersection of the straight line joining the two centers of projection with the plane π). A

perspectivist that has the properties described above is called homology and because these properties are found only in the plane figures is also called homology level.

A homology is found when, in a plan, are assigned the center U homology, its axis and a pair of corresponding points (or a pair of corresponding lines).

Homology III

One of the cases we encounter frequently is two floors above the other (π = π '), the plan π 0 and two centers of projection C and C ∞' ∞ (improper willing indefinitely, at a right angle to the plane π = π 'and floor plans bisettore of the dihedral formed by the π and π 0 ). Projecting a point A plan π 0 0 center improper C ∞ plane π, we obtain the point A, then projecting the same point A ⁰ from the center C '∞ we get point A' yea then said that the points A and A 'are legati da una corrispondenza biunivoca di centro C∞ e C’∞.
L'asse dell'omologia è rappresentato dalla retta U, (intersezione fra i piani π e π ₀ ) e il centro dell'omologia è rappresentato dall'intersezione della retta congiungente i due centri di proiezione con il piano π. Poiché i due centri di proiezione sono entrambi impropri, l’unione di essi definisce sul piano un punto improprio che si rappresenta nel punto U∞ la cui direzione è definita dall'intersezione del piano formato dai due centri C∞ e C’∞ con il piano π. Anche in questo caso la coppia di punti corrispondenti A e A' è allineata il centro dell'omologia U∞. Questa prospettività, caratterizzala dal centro improprio, prende il nome di affinità omologica .