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Shader/math ref, curvilinear coords: 2D (x, y) cartesian -> polar coordinates: > (distance ρ, angle θ); 3D (x, y, z) world -> spherical coordinates: > (distance ρ, elevation φ, azimuth θ) ρ = length(pos); φ = acos(pos.z / ρ); θ = atan2(pos.y, pos.x); #gamedev #realtimevfx

Mirza Beig

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51,960 görüntüleme • 2 yıl önce •via X (Twitter)

#gamedev #realtimevfx

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1. Spatial Constraints & Magnitude: K(u) = A₁ * cos(u / λ₁) E(v) = (v / λ₂) - δ_offset Distance Driver: D(u,v) = √( K(u)² + E(v)² ) 2. Pulsation Kinematics: Q(u,v,t) = αsin(2K) + (β/K) + sin(v/λ₃) * K * [γ + εsin(ω₁E - ω₂D + ω₃t)] ↳ The (β/K) term creates the asymptotic stretch of the tentacles, while the nested sine waves act as a damped harmonic oscillator for the contraction of the bell. pause - push - pause - push (action) 3. Polar-to-Cartesian Mapping Rotational Phase: θ(u,v,t) = D(u,v) - t X(u,v,t) = Q * cos(θ) * S_x + C_x Y(u,v,t) = Q * sin(θ) + S_y * D - C_y 4. Particle Sizing Filter: Radius = K(u)² > τ ? 2 : 1 Alpha-Blended Buffer: C_final = (1 - ρ) * C_new + ρ * C_buffer (u,v = particle indices, t = continuous time, ρ = retention rate for trails)

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This is points P_k at polar coords (radius = k·r, angle = k·θ) as θ spins. Connect sequentially: • small θ → smooth Archimedean spiral • θ = 2π·p/q → perfect n-gons & stars emerge!

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ρ(x,t) = |Ψ(x,t)|² 𝘪𝘯𝘥𝘦𝘵𝘦𝘳𝘮𝘪𝘯𝘢𝘵𝘦 𝘴𝘵𝘢𝘵𝘦𝘴

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$A mathematical construction in which one strip completes exactly ten turns. The polar angle of the generating curve is controlled by the exponential φ(θ) = (π/2) e^{-θ/8π}. Successive windings therefore nest inside one another, forming a compact layered rose with an exponentially tapering profile.$

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