Module shapes 

Nose cone and transition profiles: radius and slope along the axis for every shape.

A profile gives the outer radius r(x) at distance x aft of its forward end, over 0 ≤ x ≤ L. Every shape is defined by a normalized curve g(ξ) with g(0) = 0 at the tip and g(1) = 1 at the base, where ξ runs from tip to base. The formulas, with R the base radius and L the length, are from G. A. Crowell Sr., The Descriptive Geometry of Nose Cones, 1996 (pp. 1–6), and S. Niskanen, OpenRocket technical documentation v13.05, 2013, appendix A (pp. 102–105):

conical            g = ξ
ogive              y = √(ρ² − (Lξ − ρ cos α)²) + ρ sin α,   α = atan(R/L) − acos(√(L² + R²) / 2ρ)
elliptical         g = √(1 − (1 − ξ)²)
power series       g = ξⁿ,                            0.05 ≤ n ≤ 1
parabolic series   g = (2ξ − K′ξ²) / (2 − K′),        0 ≤ K′ ≤ 1
Haack series       g = √((θ − sin 2θ / 2 + C sin³θ) / π),   θ = acos(1 − 2ξ),   0 ≤ C ≤ 2/3

The ogive is Crowell’s secant ogive: a circular arc of radius ρ through the tip and the base rim. ρ is given as a multiple of the tangent-ogive radius ρ_t = (R² + L²) / 2R (NoseShape::Ogive::radius_ratio): 1 is the tangent ogive, larger values are secant ogives that meet the base at an angle, and values below 1 bulge beyond R before the base. The arc passes through the tip only while its center is not above the axis, which needs ρ ≥ (L² + R²) / 2L, that is radius_ratio ≥ R/L. A cone is the limit of infinite ρ. The Haack series is monotone for C ≤ 2/3 (d(g²)/dθ ∝ sin²θ (2 + 3C cos θ)); C = 0 is the LD-Haack (von Kármán) ogive and C = 1/3 the LV-Haack.

Transitions join a fore radius R_f to an aft radius R_a over length L. The shape’s tip lies at the smaller end, so a transition that grows aft has r = R_f + (R_a − R_f) g(x/L) and one that shrinks aft (a boattail) is its mirror image, r = R_a + (R_f − R_a) g(1 − x/L). A clipped transition instead takes a whole nose cone of base radius max(R_f, R_a) and length L_n ≥ L, and cuts it where its radius is min(R_f, R_a); L_n is chosen so that the piece left is L long (OpenRocket technical documentation, §A.7, p. 105). A conical or tangent-ogive transition is the same clipped or not.

See docs/physics/shapes.md.

Structs

Profile

An axisymmetric profile: a nose cone (fore radius zero) or a transition, as radius against distance aft of its forward end.

Enums

NoseShape

The shape of a nose cone, or of a transition’s profile.

Constants

MIN_POWER_EXPONENT

The smallest power-series exponent accepted. Blunter profiles approach a flat face whose area the integrals can’t resolve: at n = 1e-9 a nose’s wetted area misses the face’s πR², a nose fails to converge between 1e-8 and 0.01, and an unclipped transition up to about 0.038, where the surface integrand ∝ ξ^(2n−1) drives bisection to subnormal stations. 0.05 is the bluntest checked, as a nose and as transitions both ways, against closed-form volumes. Model a flat face as a tube and a bulkhead.