Calculating sag height from an ellipse

Sadly, some topography software doesn’t offer the option to calculate sag at a measurement chord, in which case the next best option is to use the ellipse model that, as far as I am aware, is provided by all corneal topography software. This isn’t quite as accurate a determinant for sag corneal height at the peripheral lens bearing point as you can’t be sure over how large a range the fitted ellipse has been matched to the cornea. The sag height of the ellipse model at the designated peripheral lens bearing chord can be easily calculated using the formula shown below.

Sag from ellipse equation

Topography software fits an elliptical model using the process of least square error over a predetermined chord – essentially the parameters of the ellipse are altered until the error (gaps) between the cornea and the ellipse path along the measurement chord are minimized. If measured over a small chord the fitted ellipse will be close to circular in shape, because the central cornea is close to being spherical. As the chord is increased the shape becomes progressively more elliptical and generally more accurate model for OrthoK fitting purposes. The problem is you typically won’t know the measurement chord used, hence reiterating it being better to use a measure of corneal sag height if available to you. That said, any elliptical model is almost certainly going to be more accurate to work from than simulated keratometry.

About Paul

Dr Paul Gifford is a co-founder of Eyefit, an information resource to assist contact lens practitioners in all modes of practice. Learn more about him here.