CA1153228A - Progressive power ophthalmic lenses - Google Patents

Abstract

BWP:cm PROGRESSIVE POWER OPHTHALMIC LENSES

ABSTRACT

An ophthalmic lens for the correction of presbyopia having a progressive power surface generated by the line of intersection of an ordered sequence of intersecting spheres and cylinder surfaces, the cylinder surfaces being so chosen as to produce a uniform distribution of aberration and optical power for gently curving smooth optical effect.

Description

11532~8 PROGRESSIVE POWER OPHTHALMIC LENSES

BACKGROUND OE' THE INVENTION
.
Field of the Invention:
This invention relates to ophthalmic lenses in general and is more particularly concerned with improvements in progressive power lenses for thé ~ -correction of presbyopia.
~-- 10 Descri~tion of the Prior Art~
-- The use of pr~gressive power lenses for the correction of presbyopia has become increasingly popular in recent years. In addition to their o~vious ~-cosmetic appeal, progressive lenses provide significant functional benefits to the patient, namely a continuous range o~ focal powers and an unobstructed visual -- field. Such advantages are, however, partially offset by peripheral astigmatism and distortion aberrations that are unavoidably present in all progressive lenses.
The design of progressive lenses thus naturally centers on reducing the unwanted aberrations to minimum effect~
It is generally recognized that the aberrations d can be minimized permitting them to extend over broad areas of the lens including, for example, the peripheral 25 ~ por~tions of the near vision level.; This, of course, implies a sacrifice of acuity in those peripheral areas. ~owever, virtually all modern commercial progres~ive lenses make use of the principle of extended-area aberration control. U.S. Patents ~os.
3,687,528 and 4,056,311 are exemplary It is not enough merely to state that the aberrations shall occupy extended areas of the lens~
lhe manner of their distribution within those areas ~ ~ is critically important. Badly distributed aberratiQns ; :

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~15~2~8 can undo the potential advantage gained by sacrificing acuity within the peripheral areas. For example, if a high value is placed on the re~uirement of orthoscopy (i.e. the maintence of horizontals and verticals in visual field), the designer shapes the peripheral aberrated zones in such a way that the component --; of vertical prism along horizontal lines remains~
constant. The corrected.peripheral areas, however, must be joined to the central portion of the intermediate - ~ -area, and the latter cannot be corrected to preserve orthoscopy. Therefore, a blend zone must be interposed between the inner and outer areas. The blend must ~ `
not be made too abruptly or the visually annoying condensation of aberration within the blend zone will overpower and may effectively negate the ad~antage of orthoscopy gained at the lens periphery. -Progressive lenses heretofore designed for preservation of orthoscopy do not directly address the requirement of uniform distribution of aberrations and it is a principal ob~ect of this invention to fully exploit a technique of extended-area aberration control to achieve smooth and natural optical effect.
More particularly, there is the objective of providing a progressive power ophthalmic lens with progressive surface designed to insure a uniform -- ;
distribution of aberrations and a smooth optical -effect with orthoscopy at least approximately preserved in lateral peripheral areas of the lens and without accrual of strong aberrations elsewhere in the lens.
Still another object is to provide a natural flow of optical lens power which will be readily accepted by emerging and advanced presbyopes alike~

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~SUMMARY OF THE INyENTIoN
The only known method for reduci~n~ the strength of progressive power lens aberrations is to allow a spread over a larger than us.ual area which entails re-definition of boundaries of the spherical di~stance portIon (DP~ and reading portion (RP),zones~.
With many variatiQns poss,ible, including cLr~
cular and parabolic RP's heneath a straight or u~wardl~v co,ncave arc defining the DP boundary, a progres~iye in~
termediate portion CI`Pl i`s; ~enerated by the line o~ l~nter-section of an ordered sequence of i~ntersecting sph.eres and cylinder surfaces with cylinder chosen to produce a gently curving surface insurin~ smooth optIcal e~fect.
According to an aspect of the inventiQn there i`s provided an ophthalmic lens, for the correction of pres~
b.yopia having opposi~te concave and conyex si~des~ one ~f which is characterized by a progressive power fi~rs,~ yiewi'ng surface having a principal meri~dian with curVatu,re i~n~
creasing continuously along the meridian from mini~um ~;~ 20. value uppermost of th.e firs.t surface tQ maxi`mum value i,`n a lower portion of the lens;
:thè lower port~on of the lens haYi~n~ a secQnd vie~ing surface of su~s.tantially spherical con$~ura~t~on ~i`th. defined boundar~ and havin~ approximate.~x the ~axi-mum value of curvature;
the progress~i~ve ~o~er fi;rst surfaçe surrQundi~ng a least a major portion of the boundary o~ t~he $econd vie~ing surface and ~ei~ng generated ~y the ~ine Qf, i`nte,r~
section of an ordered sequence of i`ntersectin~ sphqres 3 ~

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and cylinders for uniform di~stribution of aberrati~ns around the' second viewi~n~ su~fa,ce with at least approxi~-mate preservation o$ orthoscopy.
Details of the invention will become appa~ent from the following descri~pt~n when taken in conjuncti~on with accompanying dra,wi~ngs.
IN~THE DRAWIN~S
Figs. lA and lB illustrate~ in yertica~ e~eya,tion and cross-section respecti~vely, a progressi~,~e power ophtha~c lens of a type dealt with according to the present invention;
Fig. 2 illustrates the eyolute of the meridional line of the lens of Figs. lA, lB;
Fig. 3 is, a schemati;c illustratIon Qf construc~
t'on of a progressi~ve surface of the lens of F~gs~. lA, lB;
Fig. 4 is a vertical elevational view of a prior art progressive power ophthalmic lens showing various viewing zones thereof and the associated power la~;

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~153;2~8 Figs. 5~, 5B, 5C and 5D diagrammatically illustrate some of various definition~ of DP and RP boundaries pos-sible to achieve a reduction of strength of aberrations acc:ording to the invention;
Figs. 6A and 6B demonstrate a geometrical transfor-mation from a prior art IP of lens progressive power to one representative of the present invention;
Fig. 7 schematically illustrates a development of cylindrical surfaces chosen to satisfy aims of the present invention;
Fig. 8 depicts viewing zones of a lens constructed according to principles of the invention;
Fig. 9 is an electronic computer evaluation of one half of a symmetrical lens of the general design depicted in Fig. 8; and Fig. 10 illustrates a grid pattern produced by a lens of the Figs. 7-9 design.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
Lenses under consideration by the present invention are assumed to be made of glass or a pIastic material having a uniform refractive index. The changing curvatures required for progressive power are confined to the convex side of the lens with the concave side being reserved for prescription grinding in the usual way. The convex side of the lens will hereafter be referred to as a "progressive surface". However, there is no intention to limit the invention to lenses having convex progressive surfaces since the present ~rinciples apply equally well to convex or concave progressive surfaces.
The lens design which compri~ses the present i~n-vention is considered an improvement over earlier mab/,., ;,~t --~ ( 115^3;~8 design and exposition of the present design begins with reference to the prior art where Canadian Patent No. 583,087 is exemplary.
~ Prior art lens 10 tFigs. lA and lB) can be described as follows:
With progressive surf~ce 12 tangent~to a vertical : plane 14~at the geometrical center b, a second vertical -- pl~ne-~6 passes through-O at right angles to the first vertical plane and divides the lens in~o two - 10 symmetrical halves. The second plane 16 is called - ~ the principal vertical meridian and its curve of intersection MM' with the progressive surface is called the meridian line 18, Fig. 2.
Functional requirements of a progressive lens -lS dictate that the surface along the meridian line and its partial derivatives, at least through second order and preferably through third order, be continuous.
To provide for progressive power variation, the curvature of the meridian line increases continuously in~a predetermined manner from a minimum value in the -upper half of the lens to a maximum~value in the lower half.
The locus of the centers of curvature of the meridian line 18 comprises a continuous plane curve mm' ~Fig. 2) called the evolute of the meridian line~ -For each point Q of the meridian line there exists a corresponding point q on the evolute. The radius vector qQ connecting two corresponding points (Q,q) is perpendicular to the meridian line 18 at Q and tangent to ~he evolute mm' at ~. -~ig. 3 illustrates the construction o~ the pertinént embodiment of the design. The progressive surface is generated by a circular arc C of horizontal orientation and variable radius which passes successively ' ~
. . , ._ _ _ ... __ . .. .... _ , .. . . .

1153;ZZ8 through all points Q of the meridian line 18. SpecifiCally, the generator C through a given point Q is defined as the line of intersection between a sphere of radius Qq centered at q and a horizontal plane through Q.
~ 5 Thus, the complete progressive surface may be considered to be generated by the line of intersection of an -ordered sequence of intersecting spheres and horizontal ` : planes. As a consequence of this construction, the - principal curvatures at each point of the meridian line are equal, i.e. the surface is free of astigmatism -at the meridian line.
- ~- The progressive surface 12 of this prior art - -~
lens is readily describea in algebraic terms. A
~- rectangular coordinate system (Fig. 1) is defined - 15 whose origin coincides with O, and whose x-y plane - coincides with the tangent plane at O. The x-axis points downward in the direction of increasing optical power.
Letting u denote the x-coordinate of a point Q on the meridian line, the coordinates (~, n. ~) of the corresponding point q on the evolute~ as well as the radius of curvature r = qQ, may be expressed as a function of the parameter u:
: -- ~ 25 ~ = ~(u) :-~ n= o (U) ~1) r - r~u) ~2) The equation of the sphere of radius r~u) centered at q, expressed as an elevation with respect to the x-y plane, may be written .
z = ~(u)-{r (u)-lx-~(u)3 _y }Y2 ~3~
- .
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,, .~ , . .... . .. .. .

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The equation of a horizontal plane through Q is x = u .
~~ , ~GuaLion (3) r-~-esents a family of spheres, and Equation (4) a family of parallel planes. The members of each family are generated by the single parameter u. For each value of u there exists a ~ -- uni~ue sphere and a plane that intersects it. By ~ 10 eliminating u between Equation (3) and Equation (4), -- a generated arc C ~Fig. 3) is created through each ~ -;-point Q of the meridian line, thus producing the -- - required e~uation of t~e progressive surface ~ = ~(x,y), ~ -where -~

f(x,y) = ~(x)-{r (X)-[x-~(x~32-y2~Y~
..
If the meridional power law of lens 10 has the conventional form illustrated in Fig. 4, then

2~ the DP and RP areas of the design are spherical and extend over the full width of the lens. Such a design provides full distance and reading utility, but, -- as is well known, aberrations within the IP area are unacceptably strong. ~ -23 - According to the present inven~ion, and as mentioned heretofore, the only known met~od for actually reducing the strength of the aberrations is to allow them to spread over a larger area of the lens. This entails a redefinition of the boundaries of the spherical DP and RP zones with many variations possible, some of whi~h are illustrated in Figs. 5A, 5B~ 5C, and ~D. In the lens of Fig. 5A, the spherical DP occupies the upper half of the lens (e.g. as in Canadian Patent No. 583rO87~ but the spherical RP is bounded , .
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11532~8 by a circle. The example of Fig. 5B is similar to Fig. 5A, except that the RP boundary is parabolic. In the asymmet-rical example of Fig. 5C, the RP boundary is parabolic and the DP boundary is incli~ed 9 from the horizontal. This boundary becomes horizontal after rotating the lens 9 to pro~ide the traditional inset of the RP. The example of Fig.
5D differs from that of Fig. 5A in that the DP boundary is an upwardly concave circular arc which permits an additional spreading out of the aberrations. The radius of the DP arc must be long enough so that, after rotation of the lens 9, the aberrations on the temporal side do not interfere with lateral eye movement in the distance gaze. In pràctice, this means ~hat the DP arc should not be much less than about 65 milimeters in radius.
With DP and RP boundaries defined, it remains to determine the form of the IP that exists between them.
This is accomplished by applying a geometrical transforma-tion from the prior art, the nature of which is illustrated in Figs. 6A and 6B. ~n Fig. 6A a prior art lens has been illustrated showing the intersectlons of members of the family of planes x = u with the x-y plane. These intersec-tions form a family of parallel straight lines, which are in turn parallel to the DP and RP boundaries. As Fig. 6B
indicates, in passing to an embodiment of the present inven-tion, the family of parallel straight lines transforms i~n-to a family of more or less equally-spaced curYed l~nes.
The curved lines of lens 20 (Fig. 6B) represent the inter-sections of a one-parameter family of cylinders- wi~th the x-y plane. h~or each member of the original family of planes, there exists a corresponding mem~er o~ the mab/7~
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family of cylinders. Corresponding members of the two fami-lies are identified by the same parameter u, where u is the x-coordinate of a point Q on either meridian line. The con-struction of the new progressive surface is generated by the line of intersection of an ordered sequence of intersecting sp'heres and cylindrical surfaces. In particular, the equa-tion of any member of the family of cylindrical surfaces may be written in the form 10x = g(y,u). (6) This equation may be solved for the parameter u, giving an equation of the form u = h(x,y), - (7) which reduces to Equation (4) in the case of the prior art lens. The equation of the progressive surface of the present lens is obtained by eliminating the parameter u between Equations (7) and ~3). Explicitly, 20f(x,y) = ~[h(x,y)]-({r[h(x,y)]}2-{x-~[h(x,y)]}2_y2)~ (8) The detailed form of the resulting progressive sur-face will naturally depend on the form and spacing of the cylindrical surfaces, Equation (6). To satisfy aims of the invention, the cylindrical surfaces must be chosen so as to produce a gently curving surface ensuring a smooth optical effect.
The form of the cylindrical surfaces is determined as follows:
Considering a certain auxiliary function ~(x,y), 30defined on the x-y plane in the space exterior to _ g _ mab/~ ~`
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the curves representing the DP and RP boundaries, which have been mathematically continued to form closed curves as indi-cated in Fig. 7, ~ takes on the constant boundary values c and c2 respectively at the DP and RP boundaries. The smoothest function ~(x,y) consistent with the given geometry and boundary values is determined as follows:
If the problem were one-dimensional, rather than two-dimensional, it would be obvious that if ~(x) has the boundary values ~(0) = cl, ~(1) = c2, then the smoothest function ~(x) between x = 0 and x = 1 is the linear function ~(x) - cl + (c2 - cl)x. This function satisfies the dif-ferential equation d2~
= O
dX 2 Thus, the required function ~(x,y) in the two-dimen-sional case satisfies the two-dimensional Laplace equation:
a2~ a2~:
+ = o aX2 ay2 (10) Functions satisfying Equation (10) are called harmonic functions. This result may be deduced in another way. A criterion for the requirement of smoothness is to require that the average values of the moduli of the deri-vatives a~ax and a~/ay be a minimum. Alternatively, if the average of the sum of the squares of these quantities is considered, i.e., the integral a~ a~
~;[ (_~ 2 ~ (_) 2~ dxdy ax ay (11) then, on application of the Euler-Lagrange principle, ~, -- 10 --mab/~
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Equation 11 is minimized when ~(x,y) satisfies Laplace's equation (Equation 10). Thus Laplace's equation defines the smoothest function between the DP and RP boundaries.
To make use of the auxiliary function ~, we form the level curves ~(x,y) = c (12) which are defined as curves along which ~ has a constant value. These curves may be expressed in the form given by Equation (6) or Equation (7), and may therefore be taken to represent the required family of cylinders.
To summarize, the progressive surfaces of the in-vention is generated by a generating curve C~ which is the line of intersection between an ordered ~equence of spheres of radii qQ centered on the evolute of the meridian line, and a corresponding sequence of cylinders whose generating line runs parallel with the z-axis, and whose intersections with the x-y plane coincide ~ith the level surfaces of the harmonic function ~ which attains constant values at the DP and RP boundarles.
Because the level curyes are derived from harmonic functions, the incorporation of leYel curves into the defi-nition of the progressive surface ensures a uniform distri-bution of aberration and optlcal power.
The theory of har~onic ~unctions provides two well known methods for determining the level curves~ The first requires finding an ortho~onal system of curYilinear coordinates with coordinate curves that coincide wi~th the ~ DP and RP boundaries. The coordinate curves between the :

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DP and the RP boundaries may then be identified with thelevel curves of the system. The second method, conformal mapping, executes a transformation of the level curves of the simpler prior art system into the level curves of the more complex lens comprising the invention. Use of these methods allows construction of a progressive surface with DP and RP boundaries of arbitrary shape.
NUMERICAL EXAMPLE
An example of a lens constructed according to the above principles is as follows:
As depicted in Fig. 8, the spherical DP of lens 22 is bounded by a circular arc 24 and the spherical RP is bounded by a circle 26. The progressive corridor begins at the origin 0. The DP and RP boundaries may be regarded as coordinate lines in a bipolar system of coordinates.
The level curves between the DP and RP boundaries can there-fore be identified with the coordinate lines of the bipolar system.
For generality, define a = radius of RP boundary b = radius of DP boundary h = length of progressive corridor The level curve through an arbitrary point x,y intersects the x-axis at the point u(x,y). After calcula-tion, it is found that /(x-~S) 2+w2+y2 (x-~) 2+W2+y2 u(x,y)=~ + sgn(x-~) _______----- - {[ ]2-w2}~
2Ix-~l 2(x-~) (13) mab/~

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where w2 = (h-~) 2 + 2a(h-~) (14) h2 t 2ah 2(a+b+h) (15) Equation (13) represents a special case of Equation (7).
Defining, rD = radius of curvature of DP sphere -10 rR = radius of curvature of RP sphere The equation of the progressive surface may be written:
Distance Portion:
f(x,y) = rD-(rD2-x2-y2)~ - ~16) Progressive Zone (from Equation (3~):
f(x,y) = ~(u)-~r2(u)-[x-u+r(u)sin9(u)] 2_y2 }~ (17) where 2D ~ sin~(U) - u-~(u) r(u) (18) : u du r(u) (19) ~(u) = r(u)cos~(u) ~ IOtan~(u)du (20) ' 1 - = - + (- - -)(c2u2+c3u3+c4u4+c5uS) (21) ~r(u) rD rR rD

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C2 = 10/3h2 C3 = O
C4 = -5/h4 C5 = 8/3hS
u(x,y) is given by Equation (13);
Reading Portion:
f(x,y) = ~(h)-~rR2-[x-h+rRsin~(h)] 2_y2}~ (22) For simplicity, the above equations have been presented for the case in which the beginning of the pro-gressive corridor coincides with the center, 0 of the lens blank. It may be desirable, however, to decenter the entire progressive surface up or do`wn, right or left, relative to the geometrical center 0. The equation of the - decentered surface relative to the original system of co-ordinates is obtained by replacing x and y in the above equations by x-d~ and y-d2, respectively, where dl and d2 are the x and y values of decentration.
The progressiye surface generally defined by Equations (13) - ~22) wlll now be evaluated for a lens having a reading addition of 3.00 diopters. The lens is assumed to have an index of refraction of 1.523~ and the following values of the parameters are assumed a = 10.00 mm = 91.0 mm h = 16.0 mm rD = 84.319 mm ; rR = 57.285 mm : : ' , ' ., ' mabj,~J

l~S,~2~3 d~ = -2.00 mm d2 = 0.00 mm Fig. 9 shows the results of an electronic com-puter evaluation of the equations, using the given values of the parameters. Because the lens is symmetrical about the vertical meridian, only the right half is shown.
This figure gives the elevation of the surface above the x-y plane, computed at 4 mm intervals. Because the x-y plane is tangent to the lens surface at the point x = -2, y = 0, the elevation at x=y=0 is non-zero.
When a square ~rid is viewed through a progres-sive lens of the invention the distorted pattern of the grid provides information about the distribution and strength of the lens aberrations. The grid pattern pro-duced by the lens described above is depicted in Fig. 10.
In this diagram, the lens was rotated 9, as it would be when mounted in a spectacles fra~e. It will be seen that the grid lines are continuous, smoothly flowi~ng, and uniformly distributed. Note also that the grid lines in the periphery of the temporal side are oriented hori-zontally and vertically; this means that orthoscopy is preserved in that area. While orthoscopy may not be as well preserved in the nasal periphery of the progres-sive zone, this is not objectionable because much of the nasal side is removed by ed~i~ng for spectacles frame glazing.
It is to be understood that the term "lens"
as used herein is intended tQ include the ophthalmic . , :

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product in any and all forms common to the art, i.e.
including lens blanks requiring second side (concave or convex) finishing as well as lenses finished on both sides and "uncut" or "cut" (edged) to a size and shape re~uired for spectacles frame glazing. The present lenses may be formed of glass or any one of the various known and used ophthalmic plastics. If second side finished, i.e.
on the side opposite that having the progressive power surface, the second side may have prescription surface curvatures applied with the lens RP decentered in usual fashion.
Those skilled in the art will readily appreciate that there are various forms and adaptations of the inven- -tion not discussed herein which may be made to suit par-ticular requirements. Accordingly, the foregoing illus-trations are not to be interpreted as restrictive beyond that necessitated by the following claims.

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Claims (15)

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. An ophthalmic lens for the correction of presbyopia having opposite concave and convex sides, one of which is characterized by a progressive power first viewing surface having a principal meridian with curvature increasing continuously along said meridian from minimum value uppermost of said first surface to maximum value in a lower portion of said lens;
said lower portion of said lens having a second viewing surface of substantially spherical configuration with defined boundary and having approximately said maxi-mum value of curvature;
said progressive power first surface surrounding at least a major portion of said boundary of said second viewing surface and being generated by the line of inter-section of an ordered sequence of intersecting spheres and cylinders for uniform distribution of aberrations around said second viewing surface with at least approxi-mate preservation of orthoscopy.

2. An ophthalmic lens according to claim 1 wherein said principal meridian of said progressive power viewing surface is disposed in a substantially vertical orientation.

3. An ophthalmic lens according to claim 2 wherein said principal meridian of said progressive power viewing surface is inclined from vertical orientation of said lens.

4. An ophthalmic lens according to claim 1 in-cluding a third viewing surface disposed above and ad-joining said first viewing surface, said third surface being of spherical configuration and value of curvature approximately corresponding to said minimum value of said first surface.

5. An ophthalmic lens according to claim 4 where-in upper limits of said first viewing surface are defined by a boundary with said third viewing surface.

6. An ophthalmic lens according to claim 5 where-in said boundary with said third viewing surface is sub-stantially straight.

7. An ophthalmic lens according to claim 5 where-in said boundary with said third viewing surface is at least partially upwardly concave.

8. An ophthalmic lens according to claim 7 wherein the upwardly concave boundary is approximately symmetrical with respect to said principal vertical meri-dian of said first viewing zone.

9. An ophthalmic lens according to claim 1 wherein said boundary of said second viewing surface is approximately circular.

10. An ophthalmic lens according to claim 1 wherein said boundary of said second viewing surface is of generally parabolic configuration.

11. An ophthalmic lens according to claim 1 wherein said progressive power first surface is generated according to the equation:
f(x,y) = ?(u)-{r2(u)-[x-u+r(u)sin.theta.(u)]2-y2}?
wherein u(x,y)=.delta. + sgn(x-.delta.) wherein u represents the x-coordinate of a point Q on the meridian line, the coordinates (?, ?, ? ) of the corresponding point q on the evolute and the radius of curvature qQ being expressed as functions of u:
? = ?(u), ? = O, ? = ?(u) and r = r(u), respectively, x and y represent coordinates for a point of auxiliary function ?(x,y) satisfying:

, ? represents the angle of rotation of the lens, rD represents the radius of curvature of the distance portion sphere, rR represents the radius of curvature of the reading portion sphere, c2 -= 10/3h , c3 -= 0, C4 = -5/h4, c5 = 8/3h5, .delta.= , and w = (h-.delta.) + 2a(h-.delta.); and wherein a = radius of reading portion zone boundary, b = radius of distance portion boundary, and h = length of progressive corridor.

12. An ophthalmic lens according to claim 1 wherein said second viewing surface is defined by the equation:
f(x,y) = ?(h)-{rR2-[x-h+rRsin.theta.(h)]2-y2}?
and wherein each symbol is as defined in claim 11.

13. An ophthalmic lens according to claim 4 wherein said third viewing surface is defined by the equation:
f(x,y) = rD-(rD2-x2-y2)?
and wherein each symbol is as defined in claim 11.

14. An ophthalmic lens according to claim 1 wherein said progressive power viewing surface is approximately geometrically centered on said lens.

15. An ophthalmic lens according to claim 1 wherein said progressive power viewing surface is decentered on said lens.