= NACA four-digit airfoil
{c}
A <NACA four-digit airfoil> encodes three quantities: maximum <airfoil camber> $m$ in hundredths of the <airfoil chord>, its position $p$ in tenths of the <airfoil chord>, and nominal maximum <airfoil thickness-to-chord ratio> $\eta$ in hundredths. Thus <NACA 2412> means $m=0.02$, $p=0.4$ and $\eta=0.12$; <NACA 0012> has zero <airfoil camber> and the same nominal thickness ratio.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A.1, four-digit definition]{ref}
With $\xi=x/c$, the <airfoil mean camber line> is
$$
\frac{y_c}{c}=\begin{cases}\dfrac{m}{p^2}(2p\xi-\xi^2),&0\le\xi\le p,\\[4pt]\dfrac{m}{(1-p)^2}(1-2p+2p\xi-\xi^2),&p\le\xi\le1.\end{cases}
$$
For a <symmetric airfoil>, take $y_c=0$ directly; do not divide by the zero position parameter. The half-thickness $y_t$ follows the polynomial in <trailing edge>. If $\theta=\arctan(dy_c/dx)$, the upper/lower coordinates are
$$
x_u=x-y_t\sin\theta,\quad y_u=y_c+y_t\cos\theta,\qquad x_l=x+y_t\sin\theta,\quad y_l=y_c-y_t\cos\theta.
$$
Thickness is placed along a <normal vector> to the mean line, rather than simply added vertically.<Applied Computational Aerodynamics by W. H. Mason>[Appendix A, equations (A-1)–(A-4) and (A-7)–(A-8)]{ref}
For an original <NACA 2412> calculation at $\xi=0.4$, the mean-line slope is zero, $y_c/c=0.02$ and the standard finite-edge polynomial gives $y_t/c\simeq0.0580301$. Hence the surface points are approximately $(0.4,0.0780301)$ and $(0.4,-0.0380301)$ in chord-normalized coordinates. A 12-percent designation does not mean the section is 12 percent thick at every chordwise position.
\Image[/naca-2412-normal-offset-geometry.png]
{title=NACA 2412 mean camber line and surfaces constructed by normal offsets. Original evaluation of the stated four-digit equations.}
The trailing-edge convention must also be specified. The usual coefficient $-0.1015$ leaves a small finite thickness; the modified $-0.1036$ closes the endpoint. With a sloping mean line, the upper and lower endpoint coordinates can differ in both components, so <directional measurement of trailing-edge thickness> must state which separation it reports. These constructions define geometry, not a universal aerodynamic performance curve.
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