Abstract

A classification of projective transformations of the plane with respect to their fixed elements is introduced into the whole universe of projective transformations of the CIE-chromaticity diagram. On the basis of the given classification and by using some theorems of projective geometry a graphical method is developed for constructing chromaticity coordinates x′, y′ which correspond to given chromaticity coordinates x, y by a given transformation matrix. The principle of the graphical method holds for all types of projective transformations. In connection with some examples which demonstrate the theory, the intrinsic significance of the fixed points of a projective transformation is discussed with respect to colorimetric problems.

© 1956 Optical Society of America

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References

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  1. G. Wyszecki, J. Opt. Soc. Am. 44, 727 (1954).
  2. D. B. Judd, J. Opt. Soc. Am. 25, 24 (1935).
    [Crossref]
  3. D. L. MacAdam, J. Opt. Soc. Am. 27, 294 (1937).
    [Crossref]
  4. R. S. Hunter, Natl. Bur. Standards Circ. C429 (1942).
  5. Helson, Judd, and Warren, Illum. Eng. 47, 221 (1952).
  6. G. Wyszecki, Die Farbe 3, 93 (1954).
  7. W. L. Brewer, J. Opt. Soc. Am. 44, 207 (1954).
    [Crossref] [PubMed]

1954 (3)

G. Wyszecki, J. Opt. Soc. Am. 44, 727 (1954).

G. Wyszecki, Die Farbe 3, 93 (1954).

W. L. Brewer, J. Opt. Soc. Am. 44, 207 (1954).
[Crossref] [PubMed]

1952 (1)

Helson, Judd, and Warren, Illum. Eng. 47, 221 (1952).

1942 (1)

R. S. Hunter, Natl. Bur. Standards Circ. C429 (1942).

1937 (1)

1935 (1)

Brewer, W. L.

Helson,

Helson, Judd, and Warren, Illum. Eng. 47, 221 (1952).

Hunter, R. S.

R. S. Hunter, Natl. Bur. Standards Circ. C429 (1942).

Judd,

Helson, Judd, and Warren, Illum. Eng. 47, 221 (1952).

Judd, D. B.

MacAdam, D. L.

Warren,

Helson, Judd, and Warren, Illum. Eng. 47, 221 (1952).

Wyszecki, G.

G. Wyszecki, Die Farbe 3, 93 (1954).

G. Wyszecki, J. Opt. Soc. Am. 44, 727 (1954).

Die Farbe (1)

G. Wyszecki, Die Farbe 3, 93 (1954).

Illum. Eng. (1)

Helson, Judd, and Warren, Illum. Eng. 47, 221 (1952).

J. Opt. Soc. Am. (4)

Natl. Bur. Standards Circ. (1)

R. S. Hunter, Natl. Bur. Standards Circ. C429 (1942).

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Figures (5)

Fig. 1
Fig. 1

Graphical method for a projective transformation of Type (1).

Fig. 2
Fig. 2

Graphical method for a projective transformation of Type (2).

Fig. 3
Fig. 3

Graphical method for a projective transformation of Type (3).

Fig. 4
Fig. 4

Graphical method for a projective transformation of Type (4).

Fig. 5
Fig. 5

Graphical method for a projective transformation of Type (5).

Equations (15)

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x = a 11 x + a 12 y + a 13 a 31 x + a 32 y + a 33 , y = a 21 x + a 22 y + a 23 a 31 x + a 32 y + a 33 .
x x 1 x 3 ,             y x 2 x 3 ,
ρ x 1 = a 11 x 1 + a 12 x 2 + a 13 x 3 ρ x 2 = a 21 x 1 + a 22 x 2 + a 23 x 3 ρ x 3 = a 31 x 1 + a 32 x 2 + a 33 x 3 .
A = ( a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 )
A 1 ( α 0 0 0 β 0 0 0 γ ) ,             A 2 ( α 0 0 0 α 0 0 0 β ) ,             A 3 ( α 1 0 0 α 0 0 0 β ) , A 4 ( α 1 0 0 α 0 0 0 α ) ,             A 5 ( α 1 0 0 α 1 0 0 α ) .
0 = ( a 11 - ρ ) x 1 + a 12 x 2 + a 13 x 3 , 0 = a 21 x 1 + ( a 22 - ρ ) x 2 + a 23 x 3 , 0 = a 31 x 1 + a 32 x 2 + ( a 33 - ρ ) x 3 ,
r = 2.7760 x + 2.1543 y + 0.1192 - 1.0000 x + 6.3553 y + 1.5405 , g = - 2.9446 x + 5.0323 y + 0.8283 - 1.0000 x + 6.3553 y + 1.5405
D ( ρ ) ρ 3 - 9.3488 ρ 2 + 26.9586 ρ - 16.5258 = 0.
x 1 = 0.157 x 2 = 0.494 - 0.179 i x 3 = 0.494 + 0.179 i y 1 = - 0.087 y 2 = 0.506 + 0.179 i y 3 = 0.506 - 0.179 i .
u = 2 x - x + 6 y + 1.5 , v = 3 y - x + 6 y + 1.5 .
x 1 = - 0.5 x 2 = 0.0 x 3 = 0 y 1 = 0.0 y 2 = 0.25 y 3 = 0.
x = x + 4 y 12 y + 1 ,             y = 5 y 12 y + 1 .
x = x + y - x + 2 ,             y = y - x + 2 .
x = x + y y + 1 ,             y = y y + 1 .
x = 2 x + y x + 2 ,             y = 2 y x + 2 .