# How to transform a NACA airfoil into a near-circle using the inverse Karman-Trefftz?

Is there anyway to find the leading-edge's center of curvature of a NACA Airfoil?

I'm trying to transform a NACA airfoil into a near-circle using the inverse Karman-Trefftz transformation. the transformation has 2 singularity points, s1 which is located on the trailing edge, and s0 is located midway between the leading edge nose and its center of curvature.. now for symmetrical airfoils it's easy to find s0, since the center of curvature is on the x axis, and the leading edge radius formula is known, but for cambered airfoils it's a bit complicated with MATLAB, I've done all the hard work, and now stuck here.

• Possible duplicate of What does leading edge radius mean? Commented Oct 8, 2018 at 0:47
• Leading-edge radius r=1.1019(t), where t is the airfoil thickness.... but how can i find the center of the circle in the leading edge if i only know the radius ??
– Ali
Commented Oct 8, 2018 at 7:40

As you say yourself, you know the radius of curvature on the leading edge. You also probably have the leading-edge coordinate itself. You appear to have the equations for the airfoil, too ($$x(t), z(t)$$). That means you can also compute the tangent at the leading edge:

$$X_{LE}=\begin{pmatrix} x_{LE}\\ z_{LE} \end{pmatrix}$$

$$\dot X_{LE}=\begin{pmatrix} dx/dt\\ dz/dt \end{pmatrix}_{LE}$$

You might also get the tangent directly from the profile parameters, in the same way you got the curvature radius. Note I'm regarding the airfoil in (x,z) space, where x points downstream and z upwards, as is custom in aircraft aerodynamics and flight mechanics (y is the spanwise direction).

The center of the leading-edge circle is on a normal line to the leading-edge tangent, at the leading edge, at a distance of $$r_{LE}$$:

$$X_{cLE} = X_{LE} + \begin{pmatrix} dz/dt\\-dx/dt \end{pmatrix}_{LE} {{r_{LE}}\over{||\dot X_{LE}||}}$$

In the general case, you may not have $$r_{LE}$$ directly, but you can of course compute it from the parametric equations(or after-the-fact numerical analysis of a given shape, using finite differences). Instead of typing the equations for curvature radius out, I'll just refer you to Wikipedia for this: https://en.wikipedia.org/wiki/Curvature#In_terms_of_a_general_parametrization