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
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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
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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
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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
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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
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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
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Curvature maps with “Move-Axis”
Advantages of Gaussian Maps
Are indipendent from the VK axis Locate corneal apex Possible integration with indices (CLMI)
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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)
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…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)
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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
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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
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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
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Irregular astigmatism or keratoconus?
Local Curvature Map:
Irregular astigmatism:
Local Curvature Maps:
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Irregular astigmatism:
Local Curvature Maps:
Gaussian Maps:
Irregular astigmatism:
Local Curvature Maps:
Gaussian Maps:
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Subclinical keratoconus
Local Curvature Maps:
Subclinical keratoconus
Local Curvature Maps:
Gaussian Maps:
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Pellucida
Local Curvature Maps:
Gaussian Maps:
PKP – suture removal
PRE Local Curvature Maps: POST
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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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