G: Yes. What you are imagining does exist, but not as one standard “umbrella equation.” It is scattered across several branches of engineering and applied math.

The closest thing I found to your full question is surprisingly direct: a paper titled “In the Rain with and without an Umbrella? The Reynolds Transport Theorem to the Rescue.” It explicitly models a person plus a circular umbrella and mathematically optimizes the umbrella’s position and angle as the rain direction changes. The authors calculate rain accumulation and solve for placement variables that minimize how much water reaches the person. In their model, the optimum umbrella moves forward and downward as the apparent rain becomes more horizontal. citeturn763785search2

There is also a very simple underlying equation for one component of the problem. The rain you experience is governed by relative velocity:

[ \mathbf v_{\rm apparent\ rain}

\mathbf v_{\rm rain}

\mathbf v_{\rm person}. ]

For vertical rain without wind, that reduces to the familiar

[ \tan\theta=\frac{v_{\rm walking}}{v_{\rm rain}}, ]

where (\theta) is the ideal tilt from vertical. With wind, you use the full horizontal and vertical components of the rain velocity instead. That's essentially the first-order equation for where the umbrella ought to point. citeturn763785search3turn763785search5

But once you ask, as you did, about shape + coverage + material + placement, it becomes a genuine multi-variable engineering optimization problem.

A fairly complete umbrella model could be expressed schematically as

[ \underset{x}{\operatorname{minimize}}\quad J(x)

w_1 W(x) +w_2 F_{\rm wind}(x) +w_3 M_{\rm handle}(x) +w_4 m(x) +w_5 \sigma_{\max}(x) ]

subject to constraints such as

[ \sigma(x)<\sigma_{\rm allowable}, \qquad \delta(x)<\delta_{\rm allowable}, ]

where (x) contains things like diameter, canopy curvature, rib count, rib cross-section, fabric permeability, handle position, umbrella height and tilt.

Here, roughly:

(W) = rain reaching the user
(F_{\rm wind}) = aerodynamic force
(M_{\rm handle}) = torque you have to resist with your wrist
(m) = umbrella mass
(\sigma) = structural stress
(\delta) = deformation

And every one of those subproblems already has established equations.

For example, wind loading starts with the standard aerodynamic relationship

[ F=\frac12\rho V^2 C_D A, ]

but an umbrella is awkward because its (C_D), projected area (A), pressure distribution and shape change as the flexible canopy deforms. So serious work uses wind-tunnel testing or computational fluid dynamics rather than treating it as a rigid disk. There are published wind-load studies measuring simultaneous pressure on both surfaces of canopy structures and deriving force and moment coefficients from those measurements. citeturn577407search5turn577407search8

And people really do run umbrellas through wind tunnels and CFD. This isn't just an analogy to aircraft engineering. Umbrella-like canopies are explicitly modeled for surface pressure, velocity fields and structural load. citeturn577407search2

Material properties enter too. One recent fluid-structure-interaction study treats an umbrella canopy as a permeable flexible membrane. It uses a pressure-loss relationship of the form

[ \frac{\Delta P}{L}=aV+bV^2, ]

where (a) represents viscous resistance and (b) inertial resistance of the fabric. Changing canopy permeability substantially changed canopy inflation, stress distribution and line loads. citeturn577407search2

The actual geometry of an ordinary umbrella canopy has even received its own mathematical treatment. A 2024 Computer-Aided Design paper models the curved fabric between adjacent ribs as trimmed bilinear surfaces, derives its differential geometry, and works out how that 3-D surface can be flattened to manufacture the fabric panels. citeturn577407search3

And structural optimization exists separately. Researchers have optimized umbrella-type ribs for minimum mass under stiffness and loading constraints; one published design cut calculated rib mass by about 50% while satisfying its specified mechanical requirements. citeturn577407search0

So I think the thing you were trying to determine exists can fairly be described as:

a coupled geometric–aerodynamic–structural optimization problem for an umbrella.

And there is a particularly interesting wrinkle: “maximum coverage” isn't necessarily maximum diameter. Once wind-driven rain is included, what matters is the projection of the canopy perpendicular to the rain's apparent trajectory, the position of the person's body inside that projected shadow, and how much aerodynamic torque the resulting arrangement creates. A smaller canopy in the right location and orientation can therefore outperform a larger badly positioned one for a specified rain/wind condition. The explicit umbrella/rain optimization paper actually demonstrates this placement effect. citeturn763785search2

So, yes—the mathematical machinery you're looking for absolutely exists, and there is even literature explicitly applying it to umbrellas. What I haven't yet found is one canonical paper saying, effectively, “Here is the globally optimal everyday rain umbrella.” The literature seems fractured into rain interception, canopy geometry, aerodynamics, flexible membranes, and rib optimization.

I suspect that fractured state may actually make your question more interesting, because those pieces are sufficient to construct a fairly rigorous “ideal umbrella” objective function and then ask what shape falls out of it rather than beginning with the assumption that the traditional round umbrella is optimal.