The authors discussed about the problem of special form in astigmatism classification. This special type of astigmatism is the form of obliquely crossed astigmatism. In which the meridians, major and minor, are not right angles. In this astigmatism is not possible to prescribing for cylindrical (toric) spectacle lens. Authors describe the Thompson formula for oblique crossed cylinder and observe that this formula is to complicate for calculation new cylinder power. In this reason, the authors create the new formula and simple procedure for this calculation. This simple formula based on vector analysis and read: DM3 = DM2 ´ cos2 b

Božo Vojniković

Iva Dekaris

Nikica Gabrić

English

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Coll. Antropol. 37 (2013) Suppl. 1: 103–106Original scientific paperHow to Convert the Obliquely Crossed toNon-Crossed Astigmatism? A Simple MethodUsing Vector AnalysisBo`o Vojnikovi}1,2, Nikica Gabri}1 and Iva Dekaris11 »Svjetlost« University Eye Clinic, Zagreb, Croatia2 Velika Gorica University of Applied Sciences, Velika Gorica, CroatiaA B S T R A C TThe authors discussed about the problem of special form in astigmatism classification. This special type of astigma-tism is the form of obliquely crossed astigmatism. In which the meridians, major and minor, are not right angles. In thisastigmatism is not possible to prescribing for cylindrical (toric) spectacle lens. Authors describe the Thompson formulafor oblique crossed cylinder and observe that this formula is to complicate for calculation new cylinder power. In this rea-son, the authors create the new formula and simple procedure for this calculation. This simple formula based on vectoranalysis and read: DM3 = DM2 ´ cos2 b.Key words: obliquely crossed astigmatism, principal meridians, vector analysisIntroductionIn the classification of astigmatisms, special type ispresented as the oblique lycrossed astigmatism1. Thistype of astigmatism is regular, but the principal meridi-ans, major and minor, do not close 90*, they are not rightangles. This optical conditions is also called: bi-obliqueastigmatism, the first determine by Roure, 18962. I pre-fer the latin term: astigmatismus obliquus decussatus(This term – decussatus, decusso = crossed in form of let-ter »X«, which exactly represent those crossed meridians.In this way, calculation of sphero-cylindrical power is»not the function« of equation D ´ sin2 a and is not possi-ble prescribing for cylindrical spectacle lens. The value oftwo principal meridians, with the crossed meridian pla-nes, must be calculated with the special equations.One of this formulas is Thompson’s formula:C C C C C12122222 2= + + cos γgives the amount of the new cylinder power.S = S SC C C1 21 2+ + + −2gives the amount of the new sphere power.tansincos222θ γγ= CC +C21 2gives q, the amount to be added to a1 to give the newaxes.This Thompson’s formula is objectively very complex,confusion and no clarity. This was the motive to createthe new formula with the clarity of meridians power, me-ridians plane and direction cylinders axis. Also, must besimple for calculation and convert to usualy sphero--cylindical power for cylindrical spectacle lense. But, thiscalculation is also necessary in refractive surgery3, be-cause the front surface and back surface of the cornea in100 percent have obliquely crossed cilinders.Material and MethodsBy 25 patients we measured (apparatus: Oculus-Pen-tacam) anterior and posterior corneal surface astigma-tism, estemated the obliquely crossed cylinders in 100%.The means of major meridian power was from ±0.25 to103Received for publication September 10, 2012U:\coll-antropolo\coll-antro-(Suppl 1)-2013\19 Vojnikovic Gabric.vp2. travanj 2013 15:02:48Color profile: Generic CMYK printer profileComposite  150 lpi at 45 degrees±1.50 diopters. In Figure 1. is presented the sine func-tion of major meridian angle: D ´ sin2 a. All cases ofobliquely crossed cylinders we calculated with the vectoranalysis and convert to the not crossed cylinders.Description the methodsProcedure: If we take the first major meridian M1ax160*. than the second major meridian M2ax 20*, wemust transposing in M3ax 70* (alfa3 – alfa2 = cos beta).In this new position the angle between two major merid-ian is ax = 90 degrees. In this picture, we must cath sightof arrow direction of the sine and cosine. In this reason,the angle of alfa3 – alfa2 is not sine function, than cosinebeta.Example: (»easier calculation«)1. Major meridian: M1 @1 ax 160* + 3.0D2. Major meridian: M2 @2 ax 20* + 2.0D3. M3 @3 = 70* – alfa2 = cos beta 50*4. DM3 = + 2.0D ´ cos2 (50) = 2 ´ 0.593= 1.186 dioptriesNew combination for prescription:DM1 = +3.0 D, major meridian 160*DM3 = +1.25 D, major meridian 70*New prescription ophthalmic lens:I. +3.0 = –1.75 ax 160*II. +1.25 = +1.75 ax 70*ResultsIn the basic of our vectors method for convert theobliquely crossed cylinders to not crossed, we constructthe trigonometric vectors tables for simple and quicklyestimated the new major meridian »DM3« for prescrip-tion ophthalmic lens. This table consist the means of co-sine from 0 degree to 90 degrees and corresponding newvalues of diopters in the new direction of second majormeridian.The simplification of our formula:DM3 = DM2 ´ cos2 bDiscussion and ConclusionsOur clinical analyses of anterior et posterior cornealastigmatism estimate that the astigmatism of back cor-B. Vojnikovi} et al.: How to Convert the Obliquely Crossed to non Crossed Astigmatism?, Coll. Antropol. 37 (2013) Suppl. 1: 103–106104Fig. 1. Present the new method Oculus-Pentacam with possibility the meassured of front and back corneal surface. It’s visiblethat major meridians anterior and back surface make the crossed cylinders. This method was used in our examination.Fig. 2. Represent the diopters changeable as usual sine function(D ´ sin2 a of angle the major (cardinal) meridian. The cardinalmeridian M1-M3 and M2-M4 are obliquely but not crossed, theaxis is 90 degrees. Meridian M1-M2 are obliquely and crossed,angle alfa is 160–20=140 degrees, and this case must be convert,with the special formula, before prescription cylindrical spectaclelens.Fig. 3. Represent the vector’s method for calculation of obliquelycrossed cylinder to non crossed. In our example it’s visible thatthe angle between two major meridians (M1 ax160* = +3.D andM2 ax20* = +2.0D) is 140 degrees.U:\coll-antropolo\coll-antro-(Suppl 1)-2013\19 Vojnikovic Gabric.vp2. travanj 2013 15:02:52Color profile: Generic CMYK printer profileComposite  150 lpi at 45 degreesB. Vojnikovi} et al.: How to Convert the Obliquely Crossed to non Crossed Astigmatism?, Coll. Antropol. 37 (2013) Suppl. 1: 103–106105TABLE 1.VECTOR ANALYSIS TABLES FOR CONVERT THE OBLIQUELY CROSSED CYLINDERS: BI – OBLIQUE ASTIGMATISMAnglea3–a2DM2 x (cosb)2 = DM3Anglea3–a2DM2 x (cosb)2 = DM3b (cosb)2 0,25 0,50 0,75 1,00 1,25 1,50 b (cosb)2 0,25 0,50 0,75 1,00 1,25 1,500 1.0 0.25 0.50 0.75 1.0 1,25 1,50 46 0.49 0.12 0.24 0.36 0.49 0.61 0.731 0.98 0.24 0.49 0.73 0.98 1,22 1,47 47 0.46 0.11 0.23 0.35 0.46 0.58 0.692 0.98 0.24 0.49 0.73 0.98 1,22 1,47 48 0.45 0.11 0.22 0.34 0.45 0.56 0.673 0.98 0.24 0.49 0.73 0.98 1,22 1,47 49 0.44 0.11 0.22 0.33 0.44 0.55 0.664 0.98 0.24 0.49 0.73 0.98 1,22 1,47 50 0.41 0.10 0.20 0.33 0.41 0.52 0.625 0.98 0.24 0.49 0.73 0.98 1,22 1,47 51 0.40 0.1 0.2 0.3 0.40 0.50 0.606 0.98 0.24 0.49 0.73 0.98 1,22 1,47 52 0.37 0.09 0.18 0.28 0.37 0.46 0.567 0.98 0.24 0.49 0.73 0.98 1,22 1,47 53 0.36 0.09 0.18 0.27 0.36 0.45 0.548 0.98 0.24 0.49 0.73 0.98 1,22 1,47 54 0.35 0.08 0.17 0.26 0.35 0.44 0.529 0.96 0.24 0.48 0.72 0.96 1,20 1,44 55 0.33 0.08 0.16 0.25 0.33 0.41 0.5010 0.96 0.24 0.48 0.72 0.96 1,20 1,44 56 0.31 0.08 0.15 0.23 0.31 0.39 0.4711 0.96 0.24 0.48 0.72 0.96 1,20 1,44 57 0.29 0.07 0.14 0.22 0.29 0.36 0.4412 0.94 0.23 0.47 0.70 0.94 1,18 1,41 58 0.28 0.07 0.14 0.21 0.28 0.35 0.4213 0.94 0.23 0.47 0.70 0.94 1,18 1,41 59 0.26 0.06 0.13 0.20 0.26 0.33 0.4014 0.92 0.23 0.46 0.69 0.92 1,15 1,38 60 0.25 0.06 0.12 0.19 0.25 0.31 0.3815 0.92 0.23 0.46 0.69 0.92 1,15 1,38 61 0.24 0.06 0.12 0.18 0.24 0.30 0.3616 0.90 0.22 0.45 0.67 0.90 1,13 1,35 62 0.22 0.05 0.11 0.17 0.22 0.27 0.3317 0.90 0.22 0.45 0.67 0.90 1,13 1,35 63 0.20 0.05 0.10 0.15 0.20 0.25 0.3018 0.88 0.22 0.44 0.66 0.88 1,10 1,32 64 0.19 0.04 0.09 0.14 0.19 0.24 0.2919 0.88 0.22 0.44 0.66 0.88 1,10 1,32 65 0.18 0.04 0.09 0.14 0.18 0.23 0.2720 0.86 0.21 0.43 0.64 0.86 1,08 1,29 66 0.17 0.04 0.08 0.13 0.17 0.21 0.2621 0.86 0.21 0.43 0.64 0.86 1,08 1,29 67 0.15 0.03 0.07 0.11 0.15 0.19 0.2322 0.84 0.21 0.42 0.63 0.84 1,05 1,26 68 0.14 0.03 0.07 0.10 0.14 0.18 0.2123 0.83 0.20 0.41 0.62 0.83 1,04 1,25 69 0.13 0.03 0.06 0.09 0.13 0.16 0.2024 0.81 0.20 0.40 0.61 0.81 1,01 1,22 70 0.12 0.03 0.06 0.09 0.12 0.15 0.1825 0.81 0.20 0.40 0.61 0.81 1,01 1,22 71 0.10 0.02 0.05 0.08 0.10 0.13 0.1526 0.79 0.20 0.39 0.59 0.79 0.99 1,19 72 0.09 0.02 0.04 0.07 0.09 0.11 0.1327 0.77 0.19 0.38 0.58 0.77 0.96 1,16 73 0.08 0.02 0.04 0.06 0.08 0.10 0.1228 0.76 0.19 0.38 0.57 0.76 0.95 1,14 74 0.07 0.02 0.04 0.05 0.07 0.09 0.1029 0.75 0.19 0.37 0.56 0.75 0.94 1,13 75 0.06 0.02 0.03 0.05 0.06 0.08 0.1030 0.74 0.18 0.37 0.56 0.74 0.93 1,11 76 0.06 0.02 0.03 0.05 0.06 0.08 0.0931 0.72 0.18 0.36 0.54 0.72 0.90 1,08 77 0.05 0.01 0.03 0.04 0.05 0.06 0.0832 0.71 0.18 0.36 0.53 0.71 0.89 1,06 78 0.04 0.01 0.02 0.03 0.04 0.05 0.0633 0.70 0.18 0.36 0.53 0.70 0.88 1,05 79 0.04 0.01 0.02 0.03 0.04 0.05 0.0634 0.69 0.17 0.35 0.52 0.69 0.86 1,04 80 0.03 0.01 0.02 0.02 0.03 0.04 0.0535 0.67 0.17 0.34 0.50 0.67 0.84 1,01 81 0.02 0.01 0.01 0.02 0.02 0.03 0.0436 0.66 0.16 0.33 0.49 0.66 0.83 1.00 82 0.02 0.01 0.01 0.02 0.02 0.02 0.0337 0.64 0.16 0.32 0.48 0.64 0.80 0.96 83 0.01 0.003 0.01 0.01 0.01 0.01 0.0238 0.61 0.15 0.31 0.46 0.61 0.76 0.92 84 0.01 0.003 0.01 0.01 0.01 0.01 0.0239 0.59 0.15 0.30 0.44 0.59 0.74 0.89 85 0.008 0.002 0.002 0.01 0.01 0.01 0.0140 0.58 0.14 0.29 0.43 0.58 0.73 0.87 86 0.005 0.001 0.002 0.004 0.005 0.01 0.0141 0.56 0.14 0.28 0.42 0.56 0.70 0.84 87 0.003 0.001 0.001 0.002 0.003 0.004 0.0142 0.55 0.13 0.27 0.41 0.55 0.69 0.82 88 0.002 0.001 0.001 0.002 0.002 0.003 0.00343 0.53 0.13 0.26 0.40 0.53 0.66 0.80 89 0.001 0.000 0.000 0.001 0.001 0.002 0.00244 0.51 0.12 0.25 0.38 0.51 0.64 0.77 90 0.000 0.000 0.000 0.000 0.000 0.000 0.00045 0.50 0.12 0.25 0.37 0.50 0.63 0.75By Vojnikovi}-Gabri}-Dekaris: Vector Convert Crossed Cylinders – VGD formulasU:\coll-antropolo\coll-antro-(Suppl 1)-2013\19 Vojnikovic Gabric.vp2. travanj 2013 15:02:52Color profile: Generic CMYK printer profileComposite  150 lpi at 45 degreesneal surface is constantly present3–5, and always obli-quely and crossed in correlation to the astigmatism offront corneal surface. Astigmatism anterior et posteriorcorneal surface both of them tends to increase the powerof major meridians. Astigmatism of posterior cornealsurface tended to partially compensate the astigmatismof anterior corneal surface3,4. Anterior corneal astigma-tism is much changeable in aging than posterior astigma-tism,and tend the vertical major meridian transpositio-ned to horizontal2.Today, in refractive surgery procedure, IOL implanta-tion, laser corneal refractive treatment, can’t ignoringthe astigmatism of posterior corneal surface, than it’snecessary estimated the total corneal astigmatism.R E F E R E N C E S1. STONER ED, PERKINS P, Optical Formulas Tutorial (Butter-warth-Heinemann, Boston-Oxford, 1997). — 2. DUKE-ELDER S, Systemof Ophthalmology, Ophthalmic Optics and Refraction (London, HenryKimpton, 1970). — 3. WEIKERT M P, Baylor Collegue of Medicine-BCM,October 23, 2011. — 4. MAS D, ESPINOSA J, DAMENECH B, PEREZ J,KASPRZAK H, ILLVECA C, Ophthal. Physiol, Opt, 29 (2008) 1. — 5. AT-CHISON D A, SMITH G, Optics of the Human Eye, (Butterworth Heine-mann, Oxford, 2002).B. Vojnikovi}»Svjetlost« University Eye Clinic, Heinzelova 39, 10 000 Zagreb, Croatiae-mail: decv@decv.comKAKO KONVERTIRATI KOSI UKR[TENI ASTIGMATIZAM U NEUKR[TENI?PRIMJENA JEDNOSTAVNE VEKTORSKE METODES A @ E T A KPojava ukr{tenih ili ukri`anih (u formi slova X) cilindara, kao refrakcijska gre{ka, tvore posebnu formu astigma-tizma: ukr{teni astigmatizam, kod kojeg dva glavna meridijana ne tvore me|usobno pravi kut. Autori tako|er rasprav-ljaju sa semanti~kog aspekta, terminolo{ku ispravnost naziva »ukri`anost«, gdje u latinskoj osnovi to stanje ukri`anostiodre|uje naziv »decussatio«, {to upravo zna~i ukri`anost u formi slova »X«. Najraniji naziv, za tako ukri`ane cilindre,poti~e od autora Roure, iz 1894. koji ih imenuje, kao: bioblique astigmatism. Iz tih razloga njihov odnos se vi{e ne mo`etretirati kvadratom sin funkcije: D = sin2 a. Ta se pojava refrakcijske gre{ke pojavljuje, kako kod direktnog, tako i kodkosog astigmatizma, koji zajedno spadaju u grupu regularnog astigmatizma, s obzirom da su preslikavanja po ~itavommeridijanu pravilna i identi~na. U klini~koj refrakciji takav astigmatizam ima naziv: Astigmatism crossed i pojavljuje sena obje plohe ro`nice. Autori uvode vlastitu vektorsku metodu konverzije, dovode}i ukr{tene cilindre u neukr{tene, sanovim vrijednostima opti~ke snage meridijana i nove osovine. Ova korekcija se danas mora sve vi{e upotrebljavati,posebice u implantacijskoj refrakcijskoj kirurgiji, kada se implantiraju tori~ke le}e. To je iz razloga, {to je konstatirano,da astigmatizam stra`nje plohe ro`nice igra va`nu ulogu u kalkulaciji implantacijske le}e, i {to prakti~ki u 100% slu-~ajeva, glavni meridijani prednje i stra`nje plohe ro`nice tvore ukr{tene meridijane, odnosno cilindre.B. Vojnikovi} et al.: How to Convert the Obliquely Crossed to non Crossed Astigmatism?, Coll. Antropol. 37 (2013) Suppl. 1: 103–106106U:\coll-antropolo\coll-antro-(Suppl 1)-2013\19 Vojnikovic Gabric.vp2. travanj 2013 15:02:52Color profile: Generic CMYK printer profileComposite  150 lpi at 45 degrees

How to Convert the Obliquely Crossed to Non-Crossed Astigmatism? A Simple Method Using Vector Analysis

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