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Geometry of the cornea

Geometric and mathematical of the cornea description

Topographic and optical corneal properties are intertwined, and corneal geometry-dependent. The cornea is the most powerful of the diopters of the eye. She has this property to the significant difference in index between air and the tear film to the level of its anterior. This page is a general and simplified geometric figures and analytical functions that are used to model the profile and the relief of the cornea. The properties of these mathematical tools are widely used in corneal topography to set some constants and indices.

General information

The cornea is a spherical Dome, and its anterior surface is the first interface that meets the light captured by the eye. This surface is covered with the tear film and it acts as a first approximation as a spherical diopter. The important change of index of refraction between air and the tear film explains that the cornea is the most powerful of the eye dioptres, but to also act as a convex mirror, reflecting part of the incident light (originally from the existence of a corneal reflection. The morpho-functional study of the cornea ideally requires an accuracy of the order of a micron because variations that order can induce significant changes in its optical properties: photoablative corneal refractive surgery. LASIK, PKR - is based on this property)1).

The mathematical formulation offers simultaneously a schematic representation (modeling) and a tool to explore the physico-optiques characteristics of the cornea)2-4).

Reported descriptions refer to the average cornea of the adult. The study of the forms is a science in itself that relates to geometry. The description of the corneal relief which it is a branch requires the use of a specific vocabulary, whose terms are borrowed from the geometric study of curved surfaces. In order to avoid misinterpretation and to help visualization of sometimes complex concepts for the no surveyor, it is particularly important to know the meaning of the terms used. Pure surfaces used modeling have a more or less complex analytical formulation, which will be reduced or simplified.

The principal features of the corneal geometry, and their variations to procedures such as the corneal transplant, keratoplasty or LASIK and the to sight refractive photoablation will be studied.

We will start by recalling the definitions of the terms most frequently used before to approach the study of the geometric features of the cornea which its curvature. This knowledge allows access to the understanding and interpretation of the different maps (topographic)5).



Definition of terms used for the mathematical description of the corneal surface and their curvature

The precise changes in corneal curvature appealed to various geometric tools of varying complexity.

 General description

The curvature is mathematical object defined in a particular space. In our study, the cornea is similar to a body whose volume is bounded by two surfaces curved in a three-dimensional space.  The study of the geometric properties of the corneal surface curvature can be defined as the curvature of the sphere who marries the corneal surface at a point "at best" given)6). The radius of this sphere is low, the more the curvature of the surface studied at the point, and vice versa. A corneal surface profile can be studied "Cup", depending on the direction of its meridians, which cut the horny in as many intersection who spend his Summit or apex. the curvature at a point of each Meridian is equal to that of the circle says "osculating" at considered.

representation of the osculating circle at the apex of the Meridian

The meridian is shown in green. The osculating circle is the one who marries the top (apex). In the apical region, the two curves are almost confused (she actually are at the top of the O Meridian). Ro is the radius of the osculating circle, whose value can be used to calculate the curvature and deduce, knowing the change of index of refraction, the apical optical power of the Meridian.

Each Meridian corneal can be defined by its azimuth (e.g. the azimuth of the vertical Meridian is equal to 90 °, the horizontal Meridian equal to 0 °). A Meridian is made up of two hemimeridiens, whose curvature is the same in the absence of asymmetry.

The curvature of a surface varies according to the location and direction that we measure, with two exceptions:

-If all the points studied is located on a purely spherical surface

-If the point is located at the top of a surface to be symmetrical (which all meridians have the same profile)

Apart from these particular case, the actual curvature at a point of a surface any (non-spherical) is not equal to the one that you would measure along different directions through that point.

for analysis of the curvature cutting plans

Representation of different cutting plans according to which it is possible to measure the curvature of a convex surface (ex: anterior surface of the cornea) at a given point. These plans are locally perpendicular to the surface at the point of measurement. The orange plan corresponds to the direction South, or tangential or instant. The plan in full purple corresponds to axial and sagittal direction. The radius of curvature measured in the axial direction at the point of curvature has its origin located on the axis of symmetry of the surface (this applies to the ellipsoid type surfaces, free from deformations asymmetric or irregular)

When we study the curvature changes according to directions from a point on a surface locally not spherical, observed that the curvature varies always between two extreme, located perpendicular directions. ' Ophthalmologists sometimes use the term "'astigmatism local"to designate this feature, although the term of"local toricite"is more appropriate. This local toricite is responsible for "astigmatism of the oblique beams", optical aberration that can affect the refraction of light beams to impact slash)7).

Thus, the existence of a local toricite not null in every point is a feature of non-spherical surfaces. The variation of the curvature at the same point of a surface depending on the direction of measurement is at the origin of the different modes of topographical representation (ex: axial mode, southern fashion, etc...). The value of the average curvature ("mean curvature") at a point of a given surface is an average calculated from the measured values according to each of the branches passing through this point: we can measure the way arithmetic or geometric average (Gaussian curvature).


In ophthalmology, the use of the term toricite is often implicitly the existence of a southern variation of the curvature at the level of the corneal top (apical curvature according to the Meridian considered variations). In this case, the difference of apical curvature measured between the more curved Meridian and the Meridian less curved is estimated clinically by the simulated keratometry measurement: sim - K, after use of a formula paraxial to convert the difference in curvature in k power. This measure can be used to estimate astigmatism caused by the cornea.

If the apical toricite is a change in curvature between the meridians at the top of the cornea, the Asphericity characterized as the variation of the curvature along a Meridian (from the Center to the corneal periphery) (8). Indeed, the curvature of the meridians of an aspheric surface do not marry one of a circle but varies in each of the considered meridian points.

asphericity of a Meridian

Aspheric Meridian: the curvature varies between the Center and the periphery (here the curvature is represented atop C and a peripheral point P). The curvature at the top of the meridian is called "apical" curvature


These concepts are vital to the study of the relief of the corneal surface.


Descriptors of the corneal curvature


The before and after corneal surface geometry is dictated by the combination of four basic properties: apical curvature, Asphericity, toricite and asymmetry. These properties are seeking to describe changes in corneal curvature between the meridians or along these. The use of these properties to build a semiology simple and relevant for the interpretation of topographic maps.




-the apical curvature determines the value of the paraxial k power.

-the Asphericity depend on the rate and the sign of the spherical aberration of corneal origin,

-the toricite (heard here in the sense of "apical toricite") is the magnitude of corneal astigmatism.

-asymmetry governs the rate of irregular astigmatism (optical aberrations of high degree of origin existant)... Detect and quantify the degree of asymmetry of the corneal surface is essential for early diagnosis of Keratoconus.

 Apical curvature

The average apical curvature RADIUS is defined as the sphere who "marries" the curvature of the top of the cornea (osculatrice sphere). The corneal meridians are individualized by sagittales cuts through the center of the cornea; the apical curvature of a corneal Meridian can be likened to that of its osculating circle, which is "tangent" to the top of the Meridian. In case important toricite, apical curvature varies according to the meridians between two extreme values.

For a healthy and Virgin of refractive surgery cornea, the value of the previous average apical curvature is generally similar to that provided by the keratometry average, although the latter is estimated from measurements performed light remote part and other corneal Summit (1.5 mm). This is due to the low gradient curvature within the region close to the corneal apex (apical region or paraxial) under physiological conditions.

The apical posterior curvature is physiologically more marked than the anterior apical curvature (its apical curvature RADIUS is smaller). If the curvature of its posterior than the anterior, unlike index with aqueous humor about ten times lower and sign objected. Most of the power of the corneal diopter so depends on its anterior surface, and is slightly mitigated by his posterior. The calculation of the corneal vergence (power of the cornea paraxial) shows that we can assimilate the cornea to a spherical diopter.


The Table  According to information on the values of apical curvatures reported by various authors)9-16).

(Table n ° 1).


S/YAnteriorPosteriorTotal power (D)
R (mm)F (D)R (mm)F (D)
Lowe and Clark (1973)46/927, 65±0, 2649,26, 46±0, 26-6,243,2
Kiely and al. (1982)88/1767, 72±0, 2748,7
Edmund and Sjontorft (1985)40/807, 76±0, 2548,5
Gold et al (1986)110/2207, 78±0, 2548,3
Koretz et al (1989) FemmesHommes68 /-32.7, 69±0, 237, 78±0, 2448,948,3
Dunne et al (1992) FemmesHommes40/4040/407, 93±0, 208, 08±0, 1648,047,16, 53±0, 206, 65±0, 16-0 6, 1-6,42,041,2
Patel et al20/207, 68±0, 4049,05, 81±0, 41-6,942,2
Read and coll100/2007.77 ± 0.248.4


Table n ° 1: the apical radius of curvature R and F diopter powers corresponding values) D: diopters - S: number of threads - Y: number of eyes.


The use of the average apical keratometry enough in common clinical situations such as biometric implant power calculation or the choice of a ring of microkeratome.

In contactology, measure k is naturally expressed in milimtres.  In other clinical situations, the keratometry is often expressed in diopters, which is a unit of optical power, and no curvature. This 'transgression' highlights both the interest in the estimation of the optical power of the corneal diopter, and the close link between the form and function of the cornea.

The Keratoconuswhich is characterized by a progressive deformation and continues to the cornea, often features in his advanced by a high central keratometry (49 D above) forms

The conversion of the keratometry in power requires the use of a refractive index value that is usually close to 1.333. The relationship between curvature and keratometry is then expressed by the following relationship: K = (1.333 - 1) / R, or K is keratometry (diopter), and R the radius of curvature apical (in meter).

This selected index value does not have a physical value (close to 1,376 to the stroma). However, this slight reduction allows the expressed in diopter keratometry to take account of the effect 'average' of the posterior of the cornea, which has an interest for some applications such as biometric calculation, at least for free history of corneal surgery eyes;

Reduce the assessment of corneal curvature to a simple value k must not be done remotely from the top of the cornea. The contrary would reflect the misperception that the previous profile of the cornea wife simply that of a sphere; This approximation is actually valid that the area of the Central 3 mm and for corneas healthy and weakly o-rings. In addition, it is imprecise at the waning of diseases or surgical acts that caused a significant reworking of the corneal profile prior.



An optical surface is defined as aspheric when its curvature varies from the Summit to the periphery. There is a difference between the apical curvature and the curvature of the peripheral. In fact, an aspheric surface do not marry the shape of a sphere. The human cornea provides an appropriate example of aspheric surface. It has two naturally aspheric convex surfaces, that is, outside the area located immediately near the top (in the Central 3 mm) they do not correspond to a spherical surface.  If it were so, the representation of the corneal topography of our patients with a corneal curvature at any point constant would be uniformly monochrome!

Each of the meridians of the cornea aspherical presents a curve that varies from the Center to the edges. This variation of the curvature along a Meridian is explored by the instantaneous curvature (still called tangential, or South Bend).

The family of Conic sections mathematical curves provide a good approximation of the corneal profile. As their name suggests, they are generated by the section of the water of a cone by a plan. Depending on the angle of cut, you get a circle, an ellipse, a parabola or a Hyperbola. To analytically describe each of these curves, two parameters are sufficient.

Conic Sections

Conic sections belong to a family of curve which are generated by the section of the water of a cone by a plan.

To represent the aspheric reality of the profile of a lens or a cornea, the choice is naturally directed towards a family of simple figures called Conic sections because they can be created as their name by the section of a simple cone by a plan of the space. The ellipse, circle, the parabola and Hyperbola are the building blocks of this family of curves and is is distinguished by the value and/or the sign of their Asphericity. Remarkably, they were discovered by a Greek scientist Appolonius, and this discovery was driven only by intellectual curiosity, because outside of any application context (well before it is discovered that these curves are used to describe the movement of the planets and of some comets, the shape of some light interference fringes, some State of polarization of the light (, ect.)


Includes the families of Conic sections: ellipses (oblate and prolate), circle the parabola, and the Hyperbola. All except the circle have a variable curvature. Two parameters are sufficient to their description: the apical curvature RADIUS and Asphericity factor. All Conic sections have a common mathematical equation (equation of Baker) containing these two parameters)17)

THERE2= 2Ro X - X (1-Q)2

Ro corresponds to the radius of curvature at the apex of the conical: it is the radius of the circle tangent to the top of the conical also osculating circle. The Asphericity variable is commonly designated by the letter «» Q ». This setting follows from the geometrical properties of the aspheric curves like the ellipse. Its sign determines the way in which which the apical curve varies to the periphery: negative, it reflects the reduction of the curvature of the Center to the periphery (the local radius of curvature increases): called asphericity of type prolate. When Q is a positive value, the curvature increases from the Center to the periphery: the Asphericity is oblate. In the two case, the absolute value of Q is proportional to the difference in curvature between centre and periphery.

vertices of the ellipse oblate and prolate

Summits prolate and oblate d ' an ellipse, and geometric definition of asphericity descriptor. Ro is the radius of the osculating circle at the top of the ellipse. The corneal s profile ' apparent in general atop prolate d ' an ellipse.

The value of Q determines the type of Conic section

Q <-1: curve is a Hyperbola

Q = - 1: curve is a parabola


Q = 0: the curve is a circle

Q > 0: curve is an oblate ellipse

Other descriptors of the Asphericity, known as p, e, are found in the literature. They can all be calculated using one of them as Q = p - 1 and p = 1 - e2


To adjust to a corneal Meridian, we can choose these settings as the apical radius of curvature and a parameter that describes the degree of asphericity. The apical radius of curvature fits the curvature at the top, and the Asphericity parameter is one that allows to marry the peripheral changes in corneal curvature. Therefore, a model both simple, and "intelligible" clinically)18-22), which can be used to generally describe the aspheric curve formed by a Meridian anterior or posterior of the cornea.


The average value of the Q of the anterior corneal Asphericity factor is close to-0.2)15-16, 20-26). The average profile of a corneal section can be approximated by that of an ellipse prolate (Q factor is between 0 and -1) and by assimilation, the cornea is then referred to as prolate.  The local radius of curvature increases as you move away from the corneal apex (peripheral Flattening). The following figure represents different profiles prolates, who share the same apical curvature and differ only by the value of their Asphericity.

effect of asphericity on corneal profile

Representation at l ' scale of an optical surface profiles with the same apical curvature (paraxial power), but different asphericites of the circle (Q = 0) to the parable (Q = - 1). These profiles are almost confused in the paraxial region, where the refraction d ' a ray of light would be little different whatever l ' Asphericity.
However, beyond the paraxial region, the difference d ' Asphericity between the s profiles ' expressed in different variations of curvature (gradients of curvature) term and so power refractive local.



The following table shows the different values found in the literature for the Asphericity factor.


Table n ° 2: asphericity of the anterior face of the cornea.


Mandell and St Helen (1971)8/8-0,23-0,04 to 0.72
Kiely et al (1982)88/176-0,260,18
Edmund and Sjontorft (1985)40/80-0: 280,13
Gold et al. (1986)110/220-0,180,15
Patel et al. (1993)20/20-0,010,25
Lam and Douthwaite (1997)60/60-0,300,13
Read et al. (2006)100/200-0.190.1



It is however a great interpersonal variability for the topographical characteristics of the cornea, and the Asphericity which is often reported by the topographical software as an 'average value', in fact varies according to the meridians of the cornea. The value of the asphericity of the main meridians can also be reported.

Asphericity and corneal topography

Example of calculation of the eccentricite and the Asphericity (Q) for the main meridians (steep / flat) of the anterior face of the cornea by topographic software.

The search for a possible link between previous corneal Asphericity and ametropia underwent several publications whose results are contradictory; lack of correlation for some (16,19), while Carney et al. have recently highlighted a trend at least peripheral flattening of the cornea at short-sighted (Q approaching 0))21). We did not find difference of corneal Asphericity between myopic and normalsighted subjects)21). We have recently measured the corneal Asphericity after simple refractive desepithelialisation in operated subjects of procedures and found an Asphericity on average more prolate to level the layer of Bowman)25).

A more functional consequence of the prolate Asphericity is to allow the reduction of the aberrations of sphericity-or spherical aberrations.

spherical aberration and Asphericity: relationships

Schematic representation of changes in the variable spherical aberration for curved surfaces of asphericity Q between two media of different and homogeneous indexes (the index of refraction of the diopter to the highest value). For better viewing, the contour surface is extended dotted. The paraxial rays are shown in red, the marginal rays in green. Surfaces have the same curvature paraxial and therefore the same apical power (paraxial rays) converge in the same way for each surface and differ by their type of asphericity (value and sign of Q): oblate (a), spherical (b); prolate (c), (d) parabolic and hyperbolic (e). The spherical aberration (AS) is positive and decreases in a, b, c, d, e negative.
The Asphericity value that cancels the spherical aberrations (in d) depends on the value of the indices of refraction.

Unfortunately, we sometimes meet some confusion between Asphericity and spherical aberration. Corneal spherical aberration reflects the variation between the central power (paraxial: close to the optical axis) and the peripheral power (no paraxial, at distance from the optical axis) of the cornea. A spherical lens (whose sides have a circular profile) presents a greater refractive power at its periphery to the Center, and the rays refracted on the outskirts are therefore more quickly focused.  The impact of the aberrations of sphericity, partially offset by the corneal Asphericity on vision quality is therefore all the more important that the diameter of the pupil of entry is important (letting more peripheral rays).

Peripheral corneal flattening (profile prolate, reduction of the radius of curvature at the periphery) is part of the radius of curvature selected when a contact lens adaptation is not exactly equal to that measured near the center of the cornea with a keratometer.  In certain clinical circumstances (after refractive surgery, transplant, etc.) the strongly aspherical character of corneas imposes rules of special adaptations. The radiaire keratotomii is generally responsible for an Asphericity positive (oblate) very marked, the Q factor values that can reach 3 or 4.

The Keratoconus induces a camber marked in the central region of the cornea; it comes with a peripheral flattening, which sometimes increases significantly the corneal profile prolate character. An accentuation of the negative Asphericity (hyperprolaticite) can be an evocative of sign infra clinical Keratoconus.



The posterior corneal Asphericity is physiologically more negative (more prolate) than the Asphericity corneal anterior)14,27,28). The back is more arched at the top (apical curvature accentuated), and these characteristics explain that topography of elevation, the posterior corneal elevation with respect to its sphere of reference is on average higher than that of the anterior surface of the cornea. In other words, the posterior corneal surface is simply 'furthest' from the sphere of reference, the previous profile. This topographical separation is accentuated during the evolution of Keratoconus: in its evolved forms, this condition is accompanied by an exaggeration character prolate of the corneal Asphericity not only before, but also later)28).

The earlier corneal Asphericity was reversed in a vast majority of case after refractive surgery, corneal demyopisante and corneal transplant: of prolate (Q<0), elle="" devient="" oblate="" (q="">0) ( )26). Conversely, a hyperprolate Asphericity is usually measured after corneal surgery for hyperopia (LASIK).


corneal topography: Asphericity oblate

Postoperative axial mode specular topography of a surgery cornea LASIK for myopia. Note the increase in the curvature of the Center towards the edges, which is represented in cutting down (vertical Meridian). The average value of the asphericity of corneal meridians in the 6 mm Central is 0.25.

The study of the posterior face of the cornea is more difficult because it must be done through the front and the corneal stroma. The characteristics of the posterior face of the cornea have initially estimated by mathematical models based on knowledge of the curvature of the anterior face of the cornea and the product in some points. Patel and al. have estimated the average apical posterior curvature RADIUS to 5.8 mm, and the posterior corneal Asphericity average –0,42 (prolate); the earlier corneal Asphericity used for this calculation was quasi-spherique (-0,01))29). The choice of different previous asphericites led other authors to propose values more prolates for the back; It is clear that ray-tracing procedures for the calculation of the radius of curvature and the posterior corneal Asphericity assume a single corneal index, and that any inaccuracy of measurement at the level of the front of the cornea is postponed to the level of the back. The average ratio of the radii of curvature anterior and posterior has been estimated at 1,210 according to some studies)9,14). The assimilation of the cornea to a single refractive surface in the simplified eye models assume a constant relationship between the medium bends the anterior and the posterior face of the cornea; This explains the value generally close to 1.33 for the choice of an index of refraction "minus" (or k). This choice results in a systematic reduction of about 10% from the corneal power earlier 'true '.

The average ratio of curvature is changed after previous surgical corneal reshaping (ex: LASIK), and the use of the k index results in a smaller reduction of corneal power earlier real (it declined through the myopic LASIK, while the back remains unchanged). This partly explains the imprecision for the biometric calculations of the power of the implant for the cataract surgery in operated patients of corneal refractive surgery.



In corneal topography, the earlier corneal toricite mainly refers to the existence of a variation of the curvature between the meridians. She is said to be regular (the more arched hemimeridiens and flattest respectively aligned and perpendicular between them) or irregular when these two conditions are not met (Keratoconus, scar, etc.).

The toricite of the apical region of the cornea characterizes the lack of symmetry of revolution, the curvature of the main meridians varies between two value: the a minimum (Meridian less arched), the other maximum (the more arched Meridian).

toricite of a surface

A spherical surface has an identical curvature at each point, whatever the considered Meridian.
An o-ring surface has a variation of apical curvature (Apex: has) between the different meridians. At each point of this type of surface (not only the apex), there are an infinite number of curves according to reporting management. These curves vary between two extremes. For this surface (apex) Summit, these two extremes are the respective curves as flat Meridian and the Meridian the camber.

case regular toricite, these meridians are perpendicular (ex: 0 ° / 90 °). A difference of a few hundredths of mm for the apical curvature RADIUS is purveyor to refractive astigmatism because apical refractive power of the analyzed meridians varies in conjunction with their curvature. There is a physiological corneal regular toricite; in most cases, the vertical radius of corneal curvature is slightly below the horizontal corneal curvature RADIUS (in accordance with toricite).


case of congenital marked corneal astigmatism, corneal sides (front and back) are o-rings (the appreciation of the posterior toricite is performed by topography to sweep by slot or rotary type camera Scheimpflug, specular measure being impossible).

When the corneal toricite is excessive, or improper axis, or even not compensated for by the other eye dioptres, it engenders a refractive astigmatism, which corresponds to the result of the original corneal astigmatism and astigmatism from the other eye dioptres (cristallinien for the most part).


The corneal toricite is originally of corneal astigmatism.


The curvature of the corneal surface thus varies:

-the way of a hemimeridien (corneal Asphericity)

-between the apical meridians (corneal toricite).


The combination of the toricite and the earlier corneal Asphericity induces an aspect in "Hourglass" or "bowtie" at the level of the topography specular (previous)2-5). This hourglass is warm colors for prolates corneas, and cold to the Oblate corneas.

effect of the toricite and the Asphericity

The combined toricite and Asphericity properties explain the appearance in "Hourglass" of corneal topography of axial and tangential warp.

Using a biconical surface allows to model a cornea o-ring and aspherical. It is constructed from the values in the rays of apical curvature and the respective asphericites of the two main meridians for reporting surface.

representation of a biconical surface

Representation in elevation compared to a sphere of reference of a corneal surface shaped by a "biconical" surface. This surface is generated by interpolation between two Conic sections (r1, r2 - Q1, Q2). The color scale varies in steps of 5 microns.

The o-ring ellipsoid is a model widely used to model a cornea aspherical and o-ring, and can be seen as a sphere having suffered an elongation and / or compression along two of its main meridians. At each point of the surface of this type, the main radii of curvature are tangential and sagittal rays factored into the algorithms of topography specular. The tangential radius of curvature at a point of this surface corresponds to the radius of curvature measured in the direction of the meridian passing through this point. The sagittal (or axial) radius of curvature is perpendicular to it, and its center located on the axis of revolution of the surface. The tangential Ray is equal to the axial RADIUS to the top of an ellispoide of revolution (the top of the ellipsoid of revolution is the only point where the curvature is constant regardless of the direction of measurement). All points located at a similar distance from the top have a same sagittal curvature RADIUS and a same tangential turning radius.



In addition to being more or less-rings (change in curvature between the meridians at the apex) and aspherical (variation of the curvature along the meridians), corneal surface are also slightly asymmetrical. This corneal asymmetry can be characterized by a particular axis, and differs in that the irregularity, which does not own orientation. This axis then delineates meridians hemi opposites whose difference of curvature is the most important.

The asymmetry can be objectified in a qualitative way by a skewed distribution of the colors used to represent changes in curvature or elevation. Alone, the presence of a asymmetry marked must evoke the presence of degenerative pathology of type Keratoconus subclinical or pellucid marginal degeneration. Patients who tend to rub regularly the eyes (atopy, long on screen, etc.) often more or less minor corneal deformities characterized by an asymmetry of curvature (most often less hypercambrure). Many clues made from curvature information can be used to quantify the asymmetry. From the values calculated for these indices, some authors have suggested a diagnostic classification of corneal conditions such as Keratoconus)30,31).


The modeling of the corneal irregularity is usually performed by a family of surfaces with special properties: Zernike polynomials. These are functions of two variables)x and There in Cartesian coordinates, ρ and θ in polar coordinates) defined on the disk of unit RADIUS (the pupil is standard). The coordinate system used by convention for the characterization of these polynomials is represented in the following figure.

coordinates Zernile polynomials

The polar coordinate system used for the representation by polynomials of Zernike is by the same convention for both eyes. It is centered on the center of the pupil. The horizontal category axis is directed to the right, the vertical axis is oriented upwards. The elevation Z axis is directed to the outside of the eye. The coordinate of azimuth and the radial coordinate are represented.
In the Cartesian coordinate system can convert polar coordinates to Cartesian coordinates:
x = r sinθ, and y = r cosθ, r = (x 2 + y2) 1/2

The linear combination of these functions allows to model a wave front or the surface (corneal)32): each function is multiplied by a coefficient (coefficient of Zernike said) then the results added together. This approach is similar to the decomposition of a function Fourier series.

Each Zernike polynomial noted Znm is the product between a polynomial of degree n and a trig function of frequency Azimuthal m.  (more details on the)use of these polynomials in the context of the study of wave-front)


analytic expression of the Zernike polynomials

Analytic expression of the Zernike polynomials



The general equation of a Zernike polynomial is therefore the product between a normalization factor, a polynomial of degree n, and a trig function of frequency m.


The three-dimensional structure of a polynomial depends on the values of n and m

n is a positive integer

For each value of n, there are possible values for m varying between - n and -n in steps of 2. For example, if n = 4, m takes the following values:-4, -2, 0, 2 and 4. Thus, for an order radial n given, there are (n + 1) this order polynomials which differ in their frequency m value.

The normalization factor allows the variance of the polynomial to be equal to 1 (the sum of the squares of deviations from the average of the elevation of each point of the polynomial is equal to 1).


 Properties of the decomposition of a surface in Zernike polynomials:


A spherical outside surface can be decomposed into a linear combination of polynomials of Zernike Znm affected by a particular factor cnm:

These polynomial functions are orthogonal, which means in practice that there is only a single possible decomposition for a corneal surface given (in addition, adding or subtracting a term given to the surface does not change the value of the remaining terms).

The elevation of the points located at a distance radiaire ρ from the center of the cornea is represented by a linear combination of trigonometric functions of varying frequencies. Each of these functions is assigned a constant equal to the product of the coefficient of the polynomial of Zernike Cnm and (ρ function) value of the polynomial Rnm(Ρ) decomposition of the front wave or the corneal surface in Zernike polynomials is so similar to the principles that govern the Fourier series decomposition.

The corneal surface by polynomials of Zernike modeling allows to quickly calculate the effect printed on the light wave front (aberrations induced by corneal surface under review), if you know (or accurately predict) a spherical wave front created by a reference giving the cornea corneal surface optical power 'ideal '.

Thus, any surface of circular perimeter can be decomposed into a combination of basic surfaces represented by different polynomials of Zernike and assigned a coefficient whose absolute value reflects 'the weight' of each elementary surface present in the decomposition. Three-dimensional morphology of the first Zernike polynomials allows to specify simple shapes to characterize certain optical aberrations, or geometric characteristics of the corneal surface.

Indeed, the first modes can be connected to the main optical aberrations: Z11 and Z1-1 correspond to the tilt, Z20 to the defocus, Z22 and  Z2-2 in astigmatism,...

The polynomials Zn0 (m = 0) are invariant by rotation (symmetry of revolution: their expressions depend on ρ). The other polynomials have a symmetry axis equal to m and are present in two "best" oriented one compared to the other at an angle 90 ° divided by Mr. for example, Z22etZ2-2  (degree astigmatism 2) have two axes of symmetry and are turning to other 45 °. This property of symmetry allows to reproduce, by linear combination of the two polynomials Z22etZ2-2an astigmatism of any magnitude and oriented according to any axis.

the asigmatisme in polynomials of Zernike decomposition

Astigmatism is expressed as a linear combination of polynomials Z2-2 and Z22, whose result is an aberration also shaped Saddle (shown here in purple). The orientation of these polynomials is fixed (vertical for Z22, oblique for Z2 - 2). The respective values of the c2-2 coefficients and c22 determine the magnitude and direction of astigmatism expressed by this combination. In case of astigmatism in conformity or non-conformity of vertical (90 °), the value of the c2-2 coefficient is zero (Z22 orientation is vertical and sufficient to the characterization of this type of astigmatism).

Astigmatism expressed by these polynomials is a cylindrical power Jacksonian (the phase shift is measured against the average defocus: analogy with sphero-cylindrical expression in crossed cylinders). Unlike the rating (cylinder x axis), where any change in the cylinder results in a change of the spherical equivalent, the use of the Zernike polynomials performs an implicit distinction between spherical and cylindrical purely of the cylindrical ametropia components.

Coefficient RMS (Root Mean Square)

The RMS ("Root Mean Square") coefficient is calculated on a standard pupil (RADIUS equal to unity). It represents the 'weight' of the considered polynomial. It is expressed in microns.

For the analyzed corneal surface, the value of the RMS of each polynomials of Zernike coefficients depends on:

-analyzed corneal diameter

-the surface or future reference plan against which the polynomials of Zernike decomposition is carried out.

The following figure shows a decomposition into polynomials of Zernike of an anterior corneal surface represented compared to a reference ellipsoid. The difference between the perfectly even surface of the ellipsoid and the measured cornea is decomposed into a sum of terms of Zernike. The values of each of the Zernike coefficients is reported in percentage screw screw of a normal cornea of reference.

Corneal elevation and Zernike polynomials

Representation of the elevation relative to a reference ellipsoid by decomposition in Zernike polynomials (topographer Pentacam) and corneal on a diameter of 8 mm

The calculation of the wave front corneal ("corneal wavefront") depends on him to assumptions relating to a reference wave front, which would be that of a cornea optically perfect (not inducing any aberration of high degree) and a 'physical' stromal corneal refractive index. The following figure shows an example of theoretical calculation of optical aberrations induced by corneal surface for a pupil diameter of 6 mm

corneal aberrations after radial keratotomy

On the left, the map of refractive power reveals a central area of lesser power to the rhombic perimeter. This conformation, generated by the realization of 4 radial keratotomy incisions, involves the induction of a high rate of optical aberrations of high degree, dominated by the tetrafoil (T.4 foil). The orientation of this aberration is stackable to the incisions of keratotomii.


The first 28 Zernike polynomials are represented in grayscale here:

pyramid of Zernike

Representation in gray scale corresponding to the vertical rise of the 28 first Zernike polynomials based on their frequency of Azimuthal and radial order.

Equations in polar coordinates (without the normalization factors) of 36 first Zernike polynomials are listed in the following table:

j =n =m =
11-12 r sin θ
2112 r cos θ
32-2r2 Sin 2 θ
522r2 COS 2 θ
63-3r3 Sin θ 3
73-1(3r3(- 2r) sin θ
831(3r3(- 2r) cos θ
933r3 COS θ 3
104-4r4 4 θ SIN
114-2(4r4-3r2) sin 2 θ
1342(4r4-3r2) cos 2 θ
1444r4 4 θ COS
155-5r5 Sin θ 5
165-3(5r5-4r3) sin θ 3
175-1(10r5-12r3+ 3r) sin θ
1851(10r5-12r3 + 3r) cos θ
1953(5r5-4r3) cos θ 3
2055r5 COS θ 5
216-6r6 6 θ SIN
226-4(6r6-5r4) sin θ 4
236-2(15r6-20r4+6r2) sin 2 θ
2562(15r6-20r4+6r2) cos 2 θ
2664(6r6-5r4) cos θ 4
2766r6 6 θ COS
287-74 r7 7 θ SIN
297-54 (7r7-6r5) sin θ 5
307-34 (21r7-30r5+10r3) sin θ 3
317-14 (35r7-60r5+30r3(- 4r) sin θ
32714 (35r7-60r5+30r3(- 4r) cos θ
33734 (21r7-30r5+10r3) cos θ 3
34754 (7r7-6r5) cos θ 5
35774 r7 COS 7 θ



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