= Seidel coma pupil mapping
{c}
{title2=$w=h\rho^3\cos\theta$}
Consider an original, explicitly normalized model of primary <coma>, rather than a prescription for a particular <optical lens>. Let $(u,v)$ be <Cartesian coordinates> in a circular <optical pupil> divided by its radius $a$, with $u^2+v^2\leq1$. Let $h=\alpha/\alpha_0$ be a signed small field <angle> divided by an arbitrary positive reference <angle>. With <polar coordinates> $u=\rho\cos\theta$, $v=\rho\sin\theta$, prescribe the <wavefront error>
$$
W(u,v)=W_0w(u,v),\qquad
w(u,v)=h\rho^3\cos\theta=h\,u(u^2+v^2).
$$
Here $W_0$ is an arbitrary positive <optical path difference> scale, not a measured error. The cubic <optical pupil> dependence and linear field dependence are the classical Seidel form.<Amateur Telescope Optics by Vladimir Sacek/Coma aberration>[Amateur Telescope Optics, §4.2, classical coma formula after equation 15 and equation 13: pupil-cubic and field-linear dependence]{ref}
In the <paraxial approximation>, <light ray> direction errors follow the transverse <gradient> of <wavefront error>. For a reference sphere a distance $L$ from the exit <optical pupil> in a uniform image medium of <refractive index> $n$, the corresponding image-error scale is $s=LW_0/(na)$. Choose image-error coordinate signs so the normalized displacement is $\nabla w$; reversing the convention reverses the plotted orientation. Image coordinates below are errors relative to the ideal image of this particular object point, rather than positions of different objects:
$$
X=\frac{\partial w}{\partial u}=h(3u^2+v^2),\qquad
Y=\frac{\partial w}{\partial v}=2huv.
$$
The <partial derivatives> explain the asymmetry: opposite <optical pupil> points $(u,v)$ and $(-u,-v)$ land together, rather than on opposite sides of the ideal image. For $h>0$, every $X$ is nonnegative, while $Y$ can have either sign. The figures stipulate this single <wavefront error>, omit other <optical aberrations> and <diffraction>, and assign no instrument dimensions.
For a ring of fixed <optical pupil> radius $\rho$, the double-angle formulas for <trigonometric functions> give
$$
X=h\rho^2(2+\cos2\theta),\qquad
Y=h\rho^2\sin2\theta,\qquad
(X-2h\rho^2)^2+Y^2=h^2\rho^4.
$$
Thus each ring maps into a <circle> of radius $|h|\rho^2$ whose centre is displaced by $2h\rho^2$. Doubling $\rho$ quadruples both sizes; for $h=1$, the edge ring has centre $(2,0)$ and radius $1$, while the half-radius ring has centre $(1/2,0)$ and radius $1/4$. Superposing these displaced <circles> produces the comet-like envelope.
\Image[/seidel-coma-pupil-ring-mapping.png]
{title=Primary <coma>: <optical pupil> rings map into displaced image <circles>}
{id=seidel-coma-pupil-ring-mapping}
{description=Original normalized <Seidel coma pupil mapping> at $h=1$. Colours identify equal-radius <optical pupil> rings and their image <circles>; crosses mark circle centres. The opposite points A and B have the same image position. These are <light ray> intersections, with no instrument dimensions or <diffraction> calculation.}
For uniform <irradiance> over the circular <optical pupil>, equal <area> samples give an equal-weight geometrical image. Its <centroid> is $(h,0)$ because $\langle u^2\rangle=\langle v^2\rangle=1/4$ and $\langle uv\rangle=0$. Moving the coordinate origin to that <centroid> translates the image; it does not remove its asymmetric shape. Adding a linear term to $w$ likewise translates all <light rays> equally. These origin choices must be distinguished from changing the <optical aberration> itself.
\Image[/seidel-coma-off-axis-point-image.png]
{title=The normalized geometrical point image grows and reverses with field <angle>}
{id=seidel-coma-off-axis-point-image}
{description=Original equal-<area> sampling of the same circular <optical pupil>, mapped at $h=0,1/2,1,-1$ with one image scale. The orange cross is the geometrical <centroid>. Each blue point represents one equal-weight <light ray>; point density is not a calibrated <light intensity> display or a measured <point spread function>. At $h=0$ all model <light rays> coincide, while a real finite <aperture> still gives <diffraction>.}
The model's full transverse extent is $3|h|$ along $X$ and $2|h|$ along $Y$. Halving the field <angle> halves the blur, and reversing it reverses the pattern about the ideal image. This isolates why <coma> matters for an <optical camera>'s usable <field of view>: collecting more <light> does not ensure that off-axis points remain compact. Near <diffraction-limited resolution>, a <physical optics> calculation of the <point spread function> is needed instead of interpreting the geometrical tip as an infinitely sharp image.<Amateur Telescope Optics by Vladimir Sacek/Coma aberration>[Amateur Telescope Optics, §4.2, Figure 41: geometrical spots versus diffraction images]{ref}
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