329:
765:
283:
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631:
78:
264:
660:
345:
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743:
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pole of the lens on a cross-section of the eye can approximately scale the retina over more than an entire hemisphere. It is only in the 2000s that the limitations of this approximation have become apparent, with an exploration into why some intraocular lens (IOL) patients see dark shadows in the far periphery (negative dysphotopsia, which is probably due to the IOL being much smaller than the natural lens.)
1007:
1303:
are the points where each optical surface crosses the optical axis. They are important primarily because they are physically measurable parameters for the optical element positions, and so the positions of the cardinal points of the optical system must be known with respect to the surface vertices to
750:
The eye itself has a second special use of the nodal point that tends to be obscured by paraxial discussions. The cornea and retina are highly curved, unlike most imaging systems, and the optical design of the eye has the property that a "direction line" that is parallel to the input rays can be used
382:
to have crossed the front principal plane, as viewed from the front of the lens. This means that the lens can be treated as if all of the refraction happened at the principal planes, and rays travel parallel to the optical axis between the planes. (Linear magnification between the principal planes is
59:
systems, the basic imaging properties such as image size, location, and orientation are completely determined by the locations of the cardinal points; in fact, only four points are necessary: the two focal points and either the principal points or the nodal points. The only ideal system that has been
1461:
elsewhere. For example, object rays are real on the object side of the optical system, while image rays are real on the image side of the system. In stigmatic imaging, an object ray intersecting any specific point in object space must be conjugate to an image ray intersecting the conjugate point in
687:
since the nodal points and principal points coincide in this case. This is a valuable addition in its own right to what has come to be called "Gaussian optics", and if the image was in fluid instead, then that same ray would refract into the new medium, as it does in the diagram to the right. A ray
294:
or "stop" at the rear focal plane of a lens can be used to filter rays by angle, since an aperture centred on the optical axis there will only pass rays that were emitted from the object at a sufficiently small angle from the optical axis. Using a sufficiently small aperture in the rear focal plane
1286:
error. These claims generally arise from confusion about the optics of camera lenses, as well as confusion between the nodal points and the other cardinal points of the system. A better choice of the point about which to pivot a camera for panoramic photography can be shown to be the centre of the
755:
in 1836, but most discussions incorrectly imply that paraxial properties of rays extend to very large angles, rather than recognizing this as a unique property of the eye's design. This scaling property is well-known, very useful, and very simple: angles drawn with a ruler centred on the posterior
648:
The front and rear nodal points of a lens have the property that a ray aimed at one of them will be refracted by the lens such that it appears to have come from the other with the same angle to the optical axis. (Angular magnification between nodal points is +1.) The nodal points therefore do for
1387:
Rotational symmetry greatly simplifies the analysis of optical systems, which otherwise must be analyzed in three dimensions. Rotational symmetry allows the system to be analyzed by considering only rays confined to a single transverse plane containing the optical axis. Such a plane is called a
830:
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to find the magnification or to scale retinal locations. This line passes approximately through the 2nd nodal point, but rather than being an actual paraxial ray, it identifies the image formed by ray bundles that pass through the centre of the pupil. The terminology comes from
576:
656:, was the first to describe the nodal points in 1845 to evaluate the human eye, where the image is in fluid. The cardinal points were all included in a single diagram as early as 1864 (Donders), with the object in air and the image in a different medium.
649:
angles what the principal planes do for transverse distance. If the medium on both sides of an optical system is the same (e.g., air or vacuum), then the front and rear nodal points coincide with the front and rear principal points, respectively.
302:
Similarly, the allowed range of angles on the output side of the lens can be filtered by putting an aperture at the front focal plane of the lens (or a lens group within the overall lens), and a sufficiently small aperture will make the lens
68:
the behavior of real optical systems. Cardinal points provide a way to analytically simplify an optical system with many components, allowing the imaging characteristics of the system to be approximately determined with simple calculations.
726:. For collimated light, a lens could be placed in air at the second nodal point of an optical system to give the same paraxial properties as an original lens system with an image in fluid. The power of the entire eye is about 60
1377:
of the system. Optical systems can be folded using plane mirrors; the system is still considered to be rotationally symmetric if it possesses rotational symmetry when unfolded. Any point on the optical axis (in any space) is an
1002:{\textstyle {\frac {\overline {R_{2}}}{\overline {OC_{2}}}}={\frac {\overline {R_{1}}}{\overline {OC_{1}}}}\rightarrow {\frac {\overline {R_{2}}}{\overline {R_{1}}}}={\frac {\overline {OC_{2}}}{\overline {OC_{1}}}}}
315:
sensors. The pixels in these sensors are more sensitive to rays that hit them straight on than to those that strike at an angle. A lens that does not control the angle of incidence at the detector will produce
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The transformation between object space and image space is completely defined by the cardinal points of the system, and these points can be used to map any point on the object to its conjugate image point.
688:
through the nodal points has parallel input and output portions (blue). A simple method to find the rear nodal point for a lens with air on one side and fluid on the other is to take the rear focal length
1202:
434:
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in image space. Focal systems also have an axial object point F such that any ray through F is conjugate to an image ray parallel to the optical axis. F is the object space focal point of the system.
1465:
Geometrical similarity implies the image is a scale model of the object. There is no restriction on the image's orientation; the image may be inverted or otherwise rotated with respect to the object.
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systems have no focal points, principal points, or nodal points. In such systems an object ray parallel to the optical axis is conjugate to an image ray parallel to the optical axis. A system is
780:
The optical center of a spherical lens is a point such that If a ray passes through it, then its lens-exiting angle with respect to the optical axis is not deviated from the lens-entering angle.
439:
279:
at the rear focal plane. For an object at a finite distance, the image is formed at a different location, but rays that leave the object parallel to one another cross at the rear focal plane.
260:. The rear (or back) focal point of the system has the reverse property: rays that enter the system parallel to the optical axis are focused such that they pass through the rear focal point.
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The nodal points characterize a ray that goes through the centre of a lens without any angular deviation. For a lens in air with the aperture stop at the principal planes, this would be a
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161:
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each other. This term also applies to corresponding pairs of object and image points and planes. The object and image rays, points, and planes are considered to be in two distinct
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if an object ray parallel to the axis is conjugate to an image ray that intersects the optical axis. The intersection of the image ray with the optical axis is the focal point F
230:
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are defined as the planes, perpendicular to the optic axis, which pass through the front and rear focal points. An object infinitely far from the optical system forms an
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of the system is determined by the distance from an object to the front principal plane and the distance from the rear principal plane to the object's image. The
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in air, the principal planes both lie at the location of the lens. The point where they cross the optical axis is sometimes misleadingly called the
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are the radii of curvature of its surfaces. Positive signs indicate distances to the right of the corresponding vertex, and negative to the left.
1291:. On the other hand, swing-lens cameras with fixed film position rotate the lens about the rear nodal point to stabilize the image on the film.
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In some optical systems imaging is stigmatic for one or perhaps a few object points, but to be an ideal system imaging must be stigmatic for
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of the lens. For a real lens the principal planes do not necessarily pass through the centre of the lens and can even be outside the lens.
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of the system. In the more general case, the distance to the foci is the focal length multiplied by the index of refraction of the medium.
234:
Aperture effects are ignored: rays that do not pass through the aperture stop of the system are not considered in the discussion below.
1829:
1335:
entering an optical system, a single and unique image ray exits from the system. In mathematical terms, the optical system performs a
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of an optical system, by definition, has the property that any ray that passes through it will emerge from the system parallel to the
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image space. A consequence is that every point on an object ray is conjugate to some point on the conjugate image ray.
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1902:
571:{\displaystyle {\begin{aligned}H&=-{\frac {f(n-1)d}{r_{2}n}}\\H'&=-{\frac {f(n-1)d}{r_{1}n}},\end{aligned}}}
296:
304:
130:. The paraxial approximation assumes that rays travel at shallow angles with respect to the optical axis, so that
1981:
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168:
134:
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1336:
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1971:
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1976:
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that maps every object ray to an image ray. The object ray and its associated image ray are said to be
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The image of an object confined to a plane normal to the axis is geometrically similar to the object.
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1927:
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653:
383:+1.) The principal planes are crucial in defining the properties of an optical system, since the
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to have crossed the rear principal plane at the same distance from the optical axis that the ray
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Gauss's original 1841 paper only discussed the main rays through the focal points. A colleague,
1274:, where it is commonly asserted that the light rays "intersect" at "the nodal point", that the
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328:
127:
699:(EFL) of the lens. The EFL is the distance from the rear nodal point to the rear focal point.
1428:
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764:
406:), then the distance from each principal plane to the corresponding focal point is just the
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8:
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Rotationally symmetric optical systems; optical axis, axial points, and meridional planes
38:
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1873:
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An optical system is rotationally symmetric if its imaging properties are unchanged by
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The two principal planes of a lens have the property that a ray emerging from the lens
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object point. In an ideal system, every object point maps to a different image point.
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of an optical system. Each point is defined by the effect the optical system has on
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1702:
731:
395:
317:
30:
1944:
734:, has the same definition for power, with an average value of about 21 dioptres.
22:
663:
Cardinal point diagram for an optical system with different media on each side.
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243:
43:
1965:
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when they are in the part of the optical system to which they apply, and are
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meet the lens surfaces. As a result, dashed lines tangent to the surfaces at
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Rays that leave the object with the same angle cross at the back focal plane.
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Various lens shapes, and the location of the principal planes for each. The
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1410:, rotationally symmetric, optical imaging system must meet three criteria:
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rotation about some axis. This (unique) axis of rotational symmetry is the
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of the lens is located there, and that this is the correct pivot point for
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1738:"Scaling the retinal image of the wide-angle eye using the nodal point"
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with respect to the respective lens vertices are given by the formulas
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object point converge to a single and unique image point; imaging is
684:
608:
1929:
Revolving Table Method of
Measuring Focal Lengths of Optical Systems
630:
77:
1657:
Hecht, Eugene (2017). "Chapter 6.1 Thick Lenses and Lens
Systems".
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659:
391:
are the points where the principal planes cross the optical axis.
16:
Six points which determine imaging properties of an optical system
1308:
742:
727:
263:
730:, for example. Similarly, a lens used totally in fluid, like an
344:
1449:, optical rays extend to infinity in both directions. Rays are
403:
1361:; additional intermediate optical spaces may be used as well.
332:
Principal planes of a thick lens. The principal points H and H
1197:{\textstyle {\frac {\overline {OC_{2}}}{\overline {OC_{1}}}}}
413:
For a single lens surrounded by a medium of refractive index
276:
679:
for effective focal length. The chief ray is shown in purple
1903:"Theory of the "No-Parallax" Point in Panorama Photography"
768:
A diagram showing the optical center of a spherical lens.
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and divide it by the image medium index, which gives the
399:
286:
Angle filtering with an aperture at the rear focal plane.
1323:
Modeling optical systems as mathematical transformations
1139:
are also same. As a result, the optical center location
1149:
1118:
1091:
1023:
833:
623:"Nodal point" redirects here. Not to be confused with
206:
171:
137:
1402:
Ideal, rotationally symmetric, optical imaging system
1227:
may contain excessive or inappropriate references to
1057:
437:
1781:. European Conference on Visual Perception. Cyprus.
1427:
Object planes perpendicular to the optical axis are
1685:Simpson, M. J. (2022). "Nodal points and the eye".
1830:"The Proper Pivot Point for Panoramic Photography"
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1770:
1196:
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1104:
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1043:
1001:
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394:If the medium surrounding an optical system has a
224:
189:
155:
1800:Hecht, Eugene (2017). "Focal Points and Planes".
64:, however the cardinal points are widely used to
1963:
1204:on the optical axis, is fixed for a given lens.
644:The front and rear nodal points of a thick lens.
1932:. Optical Convention. London. pp. 168–171.
791:are where parallel lines of radii of curvature
1896:
1894:
1892:
323:
1863:
1861:
1636:(4th ed.). Addison Wesley. p. 155.
1573:
1571:
1270:The nodal points are widely misunderstood in
746:Use of the nodal point in analysis of the eye
1085:are same and the curvature center locations
81:The cardinal points of a thick lens in air.
1900:
1889:
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827:are similar (i.e., their angles are same),
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1858:
1568:
1431:to image planes perpendicular to the axis.
237:
1761:
1680:
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1258:Learn how and when to remove this message
813:are also parallel. Because two triangles
1777:Strasburger, H.; Simpson, M. J. (2023).
763:
741:
658:
629:
420:, the locations of the principal points
343:
327:
281:
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190:{\textstyle \tan \theta \approx \theta }
156:{\textstyle \sin \theta \approx \theta }
76:
1870:"Misconceptions in photographic optics"
1779:Is visual angle equal to retinal angle?
1735:
1729:
1684:
1606:
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336:and front and rear focal points F and F
41:, focal, optical system. These are the
1964:
1925:
1867:
1823:
1821:
1804:(5th ed.). Pearson. p. 169.
1675:
1661:(5th ed.). Pearson. p. 257.
1469:Focal and afocal systems, focal points
1315:are called the anterior and posterior
352:of the lens surfaces are indicated as
1799:
1656:
1631:
1238:by removing references to unreliable
126:that pass through that point, in the
1942:
1936:
1901:Littlefield, Rik (6 February 2006).
1827:
1495:
1311:, the surface vertices of the eye's
1242:where they are used inappropriately.
1211:
1818:
1294:
1078:{\displaystyle {\overline {R_{2}}}}
13:
225:{\textstyle \cos \theta \approx 1}
14:
1993:
1828:Kerr, Douglas A. (4 April 2019).
1580:Field Guide to Geometrical Optics
1491:
759:
582:is the focal length of the lens,
103:front and rear principal points;
1499:
1216:
1044:{\textstyle {\overline {R_{1}}}}
783:In the right figure, the points
114:front and rear surface vertices.
1919:
618:
118:The cardinal points lie on the
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1609:Aberrations of Optical Systems
1600:
1207:
915:
537:
525:
473:
461:
72:
1:
1578:Greivenkamp, John E. (2004).
271:The front and rear (or back)
92:front and rear focal points;
55:; there are two of each. For
1552:Radius of curvature (optics)
1414:All rays "originating" from
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1036:
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7:
1945:"Anatomy of the Human Body"
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324:Principal planes and points
10:
1998:
1017:, the radii of curvatures
776:are the lens nodal points.
622:
241:
60:achieved in practice is a
29:consist of three pairs of
1926:Searle, G. F. C. (1912).
1582:. SPIE Field Guides vol.
1763:10.3390/photonics8070284
1009:. In whatever choice of
738:Nodal points and the eye
307:. This is important for
297:object-space telecentric
1736:Simpson, M. J. (2021).
1607:Welford, W. T. (1986).
1586:. SPIE. pp. 5–20.
1143:, defined by the ratio
305:image-space telecentric
238:Focal points and planes
1982:Science of photography
1632:Hecht, Eugene (2002).
1229:self-published sources
1198:
1133:
1106:
1079:
1045:
1003:
777:
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706:of a lens is equal to
697:effective focal length
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645:
625:Nodal admissions point
586:is its thickness, and
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371:
341:
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226:
191:
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128:paraxial approximation
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39:rotationally symmetric
1787:10.31219/osf.io/tuy68
1304:describe the system.
1280:panoramic photography
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675:for Nodal Point, and
671:for Principal point,
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1943:Gray, Henry (1918).
1563:Notes and references
1547:Pinhole camera model
1398:through the system.
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1754:2021Photo...8..284S
1699:2022ApOpt..61.2797S
1447:rays in mathematics
295:will make the lens
1972:Geometrical optics
1868:van Walree, Paul.
1511:. You can help by
1329:geometrical optics
1194:
1132:{\textstyle C_{2}}
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1105:{\textstyle C_{1}}
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350:radii of curvature
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1977:Geometric centers
1811:978-1-292-09693-3
1707:10.1364/AO.455464
1693:(10): 2797–2804.
1668:978-1-292-09693-3
1557:Vergence (optics)
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1282:, so as to avoid
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667:for Focal point,
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1876:on 19 April 2015
1872:. Archived from
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1839:. Archived from
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23:Gaussian optics
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1947:. p. 1019
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1918:
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1857:
1846:on 2 July 2024
1817:
1810:
1792:
1769:
1728:
1687:Applied Optics
1674:
1667:
1649:
1642:
1624:
1617:
1599:
1592:
1566:
1564:
1561:
1560:
1559:
1554:
1549:
1544:
1537:
1534:
1527:
1526:
1520:September 2013
1506:
1504:
1493:
1492:Transformation
1490:
1480:
1474:
1470:
1467:
1457:
1451:
1441:
1436:
1435:
1432:
1425:
1417:
1409:
1403:
1400:
1390:
1380:
1372:
1366:
1363:
1357:
1351:
1347:optical spaces
1341:
1337:transformation
1324:
1321:
1296:
1293:
1289:entrance pupil
1276:iris diaphragm
1266:
1265:
1224:
1222:
1215:
1209:
1206:
1190:
1184:
1180:
1176:
1170:
1164:
1160:
1156:
1126:
1122:
1099:
1095:
1072:
1067:
1063:
1038:
1033:
1029:
995:
989:
985:
981:
975:
969:
965:
961:
954:
948:
943:
939:
933:
928:
924:
917:
911:
905:
901:
897:
891:
886:
882:
875:
869:
863:
859:
855:
849:
844:
840:
824:
817:
802:
795:
761:
760:Optical center
758:
739:
736:
654:Johann Listing
620:
617:
613:optical centre
600:
591:
563:
557:
552:
548:
542:
539:
536:
533:
530:
527:
524:
518:
515:
512:
510:
507:
504:
500:
499:
493:
488:
484:
478:
475:
472:
469:
466:
463:
460:
454:
451:
448:
446:
444:
441:
440:
381:
377:
366:
357:
325:
322:
244:Focus (optics)
239:
236:
221:
218:
215:
212:
209:
186:
183:
180:
177:
174:
152:
149:
146:
143:
140:
74:
71:
67:
15:
9:
6:
4:
3:
2:
1994:
1983:
1980:
1978:
1975:
1973:
1970:
1969:
1967:
1946:
1939:
1931:
1930:
1922:
1904:
1897:
1895:
1893:
1875:
1871:
1864:
1862:
1842:
1838:
1831:
1824:
1822:
1813:
1807:
1803:
1796:
1788:
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1732:
1724:
1720:
1716:
1712:
1708:
1704:
1700:
1696:
1692:
1688:
1681:
1679:
1670:
1664:
1660:
1653:
1645:
1643:0-321-18878-0
1639:
1635:
1628:
1620:
1618:0-85274-564-8
1614:
1610:
1603:
1595:
1593:0-8194-5294-7
1589:
1585:
1581:
1574:
1572:
1567:
1558:
1555:
1553:
1550:
1548:
1545:
1543:
1540:
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1533:
1523:
1514:
1510:
1507:This section
1505:
1502:
1498:
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1489:
1483:
1477:
1466:
1463:
1460:
1454:
1448:
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1430:
1426:
1423:
1422:
1415:
1413:
1412:
1411:
1407:
1399:
1397:
1396:cross-section
1393:
1385:
1383:
1376:
1370:
1362:
1360:
1354:
1348:
1344:
1338:
1334:
1330:
1320:
1319:of the lens.
1318:
1314:
1310:
1305:
1302:
1292:
1290:
1285:
1281:
1277:
1273:
1262:
1259:
1251:
1241:
1237:
1231:
1230:
1225:This section
1223:
1214:
1213:
1205:
1182:
1178:
1174:
1162:
1158:
1154:
1142:
1124:
1120:
1097:
1093:
1065:
1061:
1031:
1027:
1016:
1012:
987:
983:
979:
967:
963:
959:
952:
941:
937:
926:
922:
903:
899:
895:
884:
880:
873:
861:
857:
853:
842:
838:
823:
816:
812:
808:
801:
794:
790:
786:
781:
775:
771:
766:
757:
754:
744:
735:
733:
729:
724:
717:
705:
700:
698:
686:
678:
674:
670:
666:
661:
657:
655:
650:
643:
636:
632:
626:
616:
614:
610:
605:
599:
590:
561:
555:
550:
546:
540:
534:
531:
528:
522:
516:
513:
511:
505:
502:
491:
486:
482:
476:
470:
467:
464:
458:
452:
449:
447:
442:
417:
411:
409:
405:
401:
397:
392:
390:
386:
385:magnification
379:
375:
365:
356:
351:
346:
330:
321:
319:
314:
310:
306:
300:
298:
293:
284:
280:
278:
274:
265:
261:
259:
255:
249:
245:
235:
219:
216:
213:
210:
207:
184:
181:
178:
175:
172:
150:
147:
144:
141:
138:
129:
125:
121:
113:
106:
102:
95:
91:
84:
79:
70:
65:
63:
58:
54:
50:
46:
45:
40:
36:
32:
28:
24:
19:
1949:. Retrieved
1938:
1928:
1921:
1909:. Retrieved
1878:. Retrieved
1874:the original
1848:. Retrieved
1841:the original
1836:
1801:
1795:
1778:
1772:
1745:
1741:
1731:
1690:
1686:
1658:
1652:
1633:
1627:
1608:
1602:
1583:
1579:
1530:
1517:
1513:adding to it
1508:
1479:
1473:
1472:
1464:
1456:
1450:
1444:
1437:
1419:
1405:
1389:
1386:
1379:
1375:optical axis
1368:
1356:
1352:object space
1350:
1342:conjugate to
1340:
1326:
1316:
1306:
1300:
1298:
1269:
1254:
1245:
1234:Please help
1226:
1140:
1014:
1010:
821:
814:
810:
806:
799:
792:
788:
784:
782:
779:
773:
769:
749:
719:
712:
701:
682:
676:
672:
668:
664:
651:
647:
638:
634:
619:Nodal points
612:
606:
597:
588:
415:
412:
408:focal length
398:of 1 (e.g.,
393:
388:
373:
363:
354:
309:DSLR cameras
301:
289:
273:focal planes
272:
270:
258:optical axis
253:
251:
248:Focal length
120:optical axis
117:
108:
104:
97:
93:
86:
82:
62:plane mirror
56:
53:nodal points
52:
48:
44:focal points
42:
35:optical axis
26:
20:
18:
1951:12 February
1837:The Pumpkin
1381:axial point
1358:image space
1331:, for each
1299:In optics,
1272:photography
1248:August 2024
1208:Photography
340:are marked.
254:focal point
73:Explanation
66:approximate
1966:Categories
1911:14 January
1748:(7): 284.
1542:Film plane
1394:; it is a
1333:object ray
1236:improve it
252:The front
242:See also:
51:, and the
1880:1 January
1850:25 August
1742:Photonics
1723:247300377
1429:conjugate
1421:stigmatic
1287:system's
1189:¯
1169:¯
1071:¯
1037:¯
994:¯
974:¯
947:¯
932:¯
916:→
910:¯
890:¯
868:¯
848:¯
685:chief ray
609:thin lens
532:−
517:−
468:−
453:−
292:diaphragm
217:≈
214:θ
211:
185:θ
182:≈
179:θ
176:
151:θ
148:≈
145:θ
142:
1886:Item #6.
1715:35471355
1536:See also
1284:parallax
753:Volkmann
728:dioptres
506:′
380:appeared
1750:Bibcode
1695:Bibcode
1611:. CRC.
1458:virtual
1445:Unlike
1309:anatomy
1240:sources
376:appears
311:having
1808:
1802:Optics
1721:
1713:
1665:
1659:Optics
1640:
1634:Optics
1615:
1590:
1475:Afocal
607:For a
578:where
404:vacuum
47:, the
31:points
25:, the
1906:(PDF)
1844:(PDF)
1833:(PDF)
1719:S2CID
1481:focal
1440:every
1408:ideal
1317:poles
708:1/EFL
704:power
277:image
57:ideal
37:of a
1953:2009
1913:2007
1882:2007
1852:2024
1806:ISBN
1711:PMID
1663:ISBN
1638:ISBN
1613:ISBN
1588:ISBN
1584:FG01
1452:real
1416:each
1355:and
1313:lens
1112:and
1051:and
1013:and
820:and
809:and
798:and
787:and
772:and
702:The
595:and
424:and
361:and
246:and
199:and
124:rays
1783:doi
1758:doi
1703:doi
1515:.
1406:An
1371:any
1327:In
1307:In
822:OAC
815:OBC
710:or
677:efl
418:= 1
402:or
400:air
313:CCD
208:cos
173:tan
139:sin
21:In
1968::
1891:^
1860:^
1835:.
1820:^
1756:.
1744:.
1740:.
1717:.
1709:.
1701:.
1691:61
1689:.
1677:^
1570:^
1384:.
1349:,
774:N'
673:NP
637:,
299:.
290:A
107:,
96:,
85:,
1955:.
1915:.
1884:.
1854:.
1814:.
1789:.
1785::
1766:.
1760::
1752::
1746:8
1725:.
1705::
1697::
1671:.
1646:.
1621:.
1596:.
1522:)
1518:(
1486:′
1424:.
1261:)
1255:(
1250:)
1246:(
1232:.
1183:1
1179:C
1175:O
1163:2
1159:C
1155:O
1141:O
1125:2
1121:C
1098:1
1094:C
1066:2
1062:R
1032:1
1028:R
1015:B
1011:A
988:1
984:C
980:O
968:2
964:C
960:O
953:=
942:1
938:R
927:2
923:R
904:1
900:C
896:O
885:1
881:R
874:=
862:2
858:C
854:O
843:2
839:R
825:1
818:2
811:B
807:A
803:2
800:R
796:1
793:R
789:B
785:A
770:N
722:′
720:f
718:/
715:′
713:n
692:′
690:f
669:P
665:F
641:′
639:N
635:N
627:.
601:2
598:r
592:1
589:r
584:d
580:f
562:,
556:n
551:1
547:r
541:d
538:)
535:1
529:n
526:(
523:f
514:=
503:H
492:n
487:2
483:r
477:d
474:)
471:1
465:n
462:(
459:f
450:=
443:H
428:′
426:H
422:H
416:n
370:.
367:2
364:r
358:1
355:r
338:′
334:′
232:.
220:1
197:,
163:,
111:′
109:V
105:V
100:′
98:P
94:P
89:′
87:F
83:F
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