PROJECTION AND

This article appears in Volume V11, Page 691 of the Encyclopedia Britannica.

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PROJECTION AND

CROSS

CROSS

-RATIOS
12. If we join a point A to a point S, then the point where the line SA cuts a fixed plane ,r is called the

projection

PROJECTION

of A on the plane a from S as centre of

projection

PROJECTION

. If we have two planes it and Zr and a point S, we may project every point A in 7 to the other plane, If A' is the projection of A, then A is also the projection of A', so that the relations are reciprocal. To every figure in ,r we get as its projection a corresponding figure in .
We shall determine such properties of figures as remain true for the projection, and which are called projective properties. For this purpose it will be sufficient to consider at first only constructions in one plane.
Let us suppose we have given in a plane two lines p and p' and a centre S (fig. 4); we may then project the points in p from S to p',
Let A', B' .. be the projections of A, B . . ., the point at infinity in p which we shall denote by I will be projected into a finite point
single point in the line AB.
[Relations between segments of lines are interesting as showing an application of algebra to geometry. The

genesis

GENESIS
GENESIS (Gr. 'ybeo-es, becoming; the
term
- TERM
being used in
English
- ENGLISH
as a synonym for origin or process of coming into being)

of such relations

C
I' in p', viz. into the point where the parallel to p through S cuts
Similarly one point J in p will be projected into the point JJ' at infinity in p'. This point J is of course the point where the parallel to p' through S cuts p. We thus see that every point in p is projected into a single point in p'.
Fig. 5 shows that a segment AB will be projected into a segment A'B' which is not equal to it, at least not as a

rule

RULE

; and also that the ratio AC: CB is not equal to the ratio A'C': C'B' formed by the projections. These ratios will become equal only if p and p' are parallel, for in this

case

CASE
CASE, JOHN (d. 1600)

the triangle SAB is similar to the triangle SA'B'. Between three points in a line and their pro-
tctions there exists therefore in general no relation. But between four points a relation does exist. i. Let A, B, C, D be four points in p, A', B', C', D' their projections in p', then the ratio of the two ratios AC:CB and AD:DB into which C and D

divide

DIVIDE

the segment AB is equal to the corresponding expression between A', B', C', D'. In symbols we have
AC AD A'C' A'D'
CB DB = C'B' D'B' '
This is easily proved by aid of similar triangles.
Through the points A and B on p draw parallels to p', which cut the projecting rays in
s C2, D2, B2 and Al, Cl,
DI, as indicated in fig. 6. The two triangles
s ACC2 and BCC' will be similar, as will also be the triangles ADD2 and BDDI.
The proof is left to A the reader.
This result of fundamental importance.
The expression A' c d _s' AC/CB:AD/DB has been
called by Chasles the
" anharmonic ratio of the
four points A, B, C, D."

Professor

PROFESSOR (the Latin noun formed from the verb profiteri, to declare publicly, to acknowledge, profess)

Clifford pro-
posed the shorter name of "

cross

CROSS

-ratio." We shall adopt the latter. We have then the

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