Refraction Parameters

🔵 Initial Refraction

🟢 Refraction to Add

🔴 Final Refraction

-2.00 (-1.00 × 0.0°)
Initial Refraction
-2.00 (-2.00 × 180°)
J0: 1.00, J45: -0.00
+
Added Refraction
1.00 (-1.00 × 90°)
J0: -0.50, J45: 0.00
=
Final Refraction
-2.00 (-1.00 × 0.0°)
J0: 0.50, J45: -0.00

Exploded View of Refractive Components

Visual representation of spherical and cylindrical components of each refraction. The circle represents the sphere, the rectangle represents the cylinder with its axis.

0°30°60°90°120°150°180°Initial RefractionS: -2.00 D180°C: -2.00 D0°30°60°90°120°150°180°Added RefractionS: 1.00 D90°C: -1.00 D0°30°60°90°120°150°180°Final RefractionS: -2.00 D0°C: -1.00 D+==(c) D. Gatinel – www.gatinel.com

Pure Cylinder Profiles

Power profiles of pure cylinders (spherical equivalent = 0). Vectors show the astigmatism amplitude (|C|/2) at principal axes.

0306090120150180210240270300330360−1−0.500.51
Initial: (-2.00×180°)Added: (-1.00×90°)Final: (-1.00×0.0°)SE = 0Angle (degrees)Power (D)|C|/2 = 1.00180°1.00D-|C|/2 = -1.0090°-1.00D|C|/2 = 1.000°1.00D-|C|/2 = -1.0090°-1.00D|C|/2 = 0.5090°0.50D-|C|/2 = -0.500°-0.50D|C|/2 = 0.50270°0.50D-|C|/2 = -0.500°-0.50D|C|/2 = 0.500°0.50D-|C|/2 = -0.5090°-0.50D|C|/2 = 0.50180°0.50D-|C|/2 = -0.50270°-0.50D(c) D. Gatinel – www.gatinel.com

Sinusoidal Power Profiles

Sinusoidal variation of optical power as a function of angle. Horizontal lines indicate spherical equivalents.

-2.00 D180°-4.00 D270°1.00 D90°0.00 D180°-2.00 D0°-3.00 D90°−300306090120150180210240270300330360−4−3−2−101
Initial : -2.00 (-2.00×180°)Added   : 1.00 (-1.00×90°)Final   : -2.00 (-1.00×0.0°)Angle (°)Power (D)SE=-3.00SE=0.50SE=-2.50(c) D. Gatinel – www.gatinel.com
145Initial : -2... : −2.65798Added   : 1.... : 0.3289899Final   : -2... : −2.328990 D : 0trace 4 : −3trace 5 : 0.5trace 6 : −2.5

Clinical Astigmatism Diagram

Vector representation of astigmatism in the J0-J45 plane. Components are doubled (×2) for better visibility.

−6−4−20246−2−1.5−1−0.500.511.52
Initial: -2.00×180°Ajouté: -1.00×90°Final: -1.00×0.0°J0 Component (×2)J45 Component (×2)1 D2 D0°30°60°90°120°150°-2.00 (-2.00 × 180°)1.00 (-1.00 × 90°)-2.00 (-1.00 × 0.0°)(c) D. Gatinel – www.gatinel.com
Initial Refraction
J0 = 1.00
J45 = -0.00
SE = -3.00
Added Refraction
J0 = -0.50
J45 = 0.00
SE = 0.50
Final refraction
J0 = 0.50
J45 = -0.00
SE = -2.50

3D Vector Addition

Three-dimensional visualization of vector addition including spherical equivalent. Projections show components on each plane.

Initial RefractionAdded RefractionFinal Refraction(c) D. Gatinel – www.gatinel.com
Initial Refraction
J0 = 1.00
J45 = -0.00
SE = -3.00
Added Refraction
J0 = -0.50
J45 = 0.00
SE = 0.50
Final Refraction
J0 = 0.50
J45 = -0.00
SE = -2.50

Polar Power Profiles

Polar representation of refractive power. Distance from center indicates power at each meridian. Dotted circle corresponds to emmetropia.

-2.00D 180°-4.00D 90°1.00D 90°0.00D 0°-2.00D 0°-3.00D 90°0°45°90°135°180°225°270°315°−4−3−2−101
Initial : -2.00 (-2.00×180°)Added : 1.00 (-1.00×90°)Final : -2.00 (-1.00×0.0°)Plan 0 DPower (D)(c) D. Gatinel – www.gatinel.com

3D Tube Profiles

Visualization of power profiles on a tubular surface. Height represents optical power at each meridian. Dotted circle corresponds to emmetropia.

Plan 0 DInitial : -2.00 (-2.00×180°)Added : 1.00 (-1.00×90°)Final : -2.00 (-1.00×0.0°)(c) D. Gatinel – www.gatinel.com-2.00 D180°-4.00 D90°1.00 D90°0.00 D0°-2.00 D0°-3.00 D90°

Calculation Details

Detailed mathematical analysis of astigmatism addition with formulas and calculation steps.

📚 Step by step Analysis

🔹 Step 1 : Converting refractions into vectors

Initial Refraction : -2.00 (-2.00 × 180°)

J0₁ = -C₁/2 × cos(2×A₁) = --2.00/2 × cos(2×180°) = 1.00
J45₁ = -C₁/2 × sin(2×A₁) = --2.00/2 × sin(2×180°) = -0.00
SE₁ = S₁ + C₁/2 = -2.00 + -2.00/2 = -3.00

Added refraction : 1.00 (-1.00 × 90°)

J0₂ = --1.00/2 × cos(2×90°) = -0.50
J45₂ = --1.00/2 × sin(2×90°) = 0.00
SE₂ = 1.00 + -1.00/2 = 0.50
🔹 Étape 2 : Vector Addition
J0_final = J0₁ + J0₂ = 1.00 + -0.50 = 0.50
J45_final = J45₁ + J45₂ = -0.00 + 0.00 = -0.00
SE_final = SE₁ + SE₂ = -3.00 + 0.50 = -2.50
🔹 Étape 3 : Recombination in spherocylindrical notation

Cylinder magnitude :

|C| = 2 × √(J0² + J45²)
= 2 × √(0.50² + -0.00²)
= 2 × 0.50 = 1.00

Cylinder Axis :

A = 0.5 × arctan2(J45, J0)
= 0.5 × arctan2(-0.00, 0.50)
= 0.0°

Final Sphere :

S = SE - C/2 = -2.50 - -1.00/2 = -2.00
✅ Final Result
-2.00 (-1.00 × 0.0°)

Calculating...

© D. Gatinel, www.gatinel.com