The Gaussian Curvature 13/09/2007 Renzo Mattioli The Gaussian Curvature Map Renzo Mattioli (Optikon 2000, Roma*) *Author is a full time employee of Optikon 2000 Spa (Int.: C3) Historical background: Theorema Egregium by Carl Friderich Gauss Refractive On Line 2007 1 The Gaussian Curvature 13/09/2007 Renzo Mattioli Historical background: C.F. Gauss (1777-1855) was professor of mathematics in Göttingen (D). Has developed theories adopted in – Analysis – Number theory – Differential Geometry – Statistic (Guaussian distribution) – Physics (magnetism, electrostatics, optics, Astronomy) "Almost everything, which the mathematics of our century has brought forth in the way of original scientific ideas, attaches to the name of Gauss.“ Kronecker, L. Gaussian Curvature of a surface (Definition) Gaussian curvature is the product of steepest and flattest local curvatures, at each point. Negative GC Zero GC Positive GC Refractive On Line 2007 2 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium (1828) Gaussian curvature (C1 x C2) of a flexible surface is invariant. (non-elastic isometric distortion) C1=0 C1=0 C2>0 C2=0 C1 x C2 = 0 C1 x C2 = 0 Gauss’ Theorema Egregium Gaussian curvature is an intrinsic property of surfaces C1 C2 C1 C2 Curvature Gaussian Curvature Gaussian C2 C2 C1 x C2 C1 x C2 C1 C1 Refractive On Line 2007 3 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gauss’ Theorema Egregium • “A consequence of the Theorema Egregium is that the Earth cannot be displayed on a map without distortion. The Mercator projection… preserves angles but fails to preserve area.” (Wikipedia) The Gaussian Map in Corneal Topography Refractive On Line 2007 4 The Gaussian Curvature 13/09/2007 Renzo Mattioli Proposed by B.Barsky et al. in 1997 “Gaussian power with cylinder vector field representation for corneal topography maps”. Optom.Vis.Sci. 74-1997 Other proposals T.Turner (Orbscan, 1995) – Mean curvature maps M.Tang et alt (AJO Dec. 2005) – Mean curvature maps R.Mattioli (Keratron, Refractive-on-line 2007) - Gaussian maps Mean is local curvature arithmetic average (D). Gaussian is curvature geometric average (D): _______ Mean= (C1+C2) / 2 Gaussian= C1 x C2 It can be shown that: (Gaussian)² = (Mean)² + |Astigmatism|² Refractive On Line 2007 5 The Gaussian Curvature 13/09/2007 Renzo Mattioli Gaussian maps: Cancel astigmatisms …but preserves ectasia Local Curvature Maps: Gaussian Maps: Why “Gaussian” differ from “Curvature” maps? Instantaneous Curvatures are Gaussian are the product of main measured along sections from the curvatures at each point and do not VK axis. depend from VK axis Refractive On Line 2007 6 The Gaussian Curvature 13/09/2007 Renzo Mattioli Curvature maps with “Move-Axis” Advantages of Gaussian Maps Are indipendent from the VK axis Locate corneal apex Possible integration with indices (CLMI) Refractive On Line 2007 7 The Gaussian Curvature 13/09/2007 Renzo Mattioli Refractive On Line 2007 8 The Gaussian Curvature 13/09/2007 Renzo Mattioli Refractive On Line 2007 9 The Gaussian Curvature 13/09/2007 Renzo Mattioli Advantages of Gaussian Maps Are indipendent from the VK axis Locate corneal apex (objectively) Possible integration with indices (CLMI) Cone location, with 3 different maps CLMI calculated on Gaussian Gaussian map (Inst.) Curvature Sherical Offset map (Best-Fit sphere) Refractive On Line 2007 10 The Gaussian Curvature 13/09/2007 Renzo Mattioli …same maps, VK-axis-moved to the Gaussian CLMI apex Gaussian map Curvature (+ Sherical Offset Move Axis ) (“Tangent to point ”) Advantages of Gaussian Maps Are indipendent from the VK axis Locate corneal apex objectively Possible integration with indices (CLMI) Refractive On Line 2007 11 The Gaussian Curvature 13/09/2007 Renzo Mattioli Follow-up of a keratoconus after 5 years CLMI on Local CURVATURE RC (29y) OS Mc=50.1 2M Mc=50.5 4y Mc=52.5 5M Mc=53.5 Refractive On Line 2007 12 The Gaussian Curvature 13/09/2007 Renzo Mattioli CLMI on GAUSSIAN RC (29y) OS Mc=4590.1 2M Mc=4590.75 4y Mc=52.35 5M Mc=53.25 Curvature+Move-Axis Sim-K-Avg RC (29y) OS Avg=48.5 2M Avg=48.7 4y Avg=45.4 5M Avg=46.1 Refractive On Line 2007 13 The Gaussian Curvature 13/09/2007 Renzo Mattioli Follow up of RC (29y) OD/OS 65 60 Mc-OS 55 Avg-OS Mg-OS 50 45 19/04/2001 01/09/2002 14/01/2004 28/05/2005 10/10/2006 22/02/2008 Other clinical applications Refractive On Line 2007 14 The Gaussian Curvature 13/09/2007 Renzo Mattioli Irregular astigmatism or keratoconus? Local Curvature Map: Irregular astigmatism: Local Curvature Maps: Refractive On Line 2007 15 The Gaussian Curvature 13/09/2007 Renzo Mattioli Irregular astigmatism: Local Curvature Maps: Gaussian Maps: Irregular astigmatism: Local Curvature Maps: Gaussian Maps: Refractive On Line 2007 16 The Gaussian Curvature 13/09/2007 Renzo Mattioli Subclinical keratoconus Local Curvature Maps: Subclinical keratoconus Local Curvature Maps: Gaussian Maps: Refractive On Line 2007 17 The Gaussian Curvature 13/09/2007 Renzo Mattioli Pellucida Local Curvature Maps: Gaussian Maps: PKP – suture removal PRE Local Curvature Maps: POST Refractive On Line 2007 18 The Gaussian Curvature 13/09/2007 Renzo Mattioli PKP – suture removal PRE Local Curvature Maps: POST PRE Gaussian Maps: POST Conclusions, Gaussian Curvatures: Represent an intrinsic property of surfaces Are “Axis independent” Potentially useful in clinical evaluations, and along with CLMI indices CAUTION: Do NOT represent shape univoquely (hide astigmatism, show ectasia) Recommended together with Instantaneous Curvature and “Move-Axis” THANK YOU ! 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