Five-Body Kite Central Configuration Collapse

Kite with Three Degrees of Freedom

Five-Body Kite Central Configuration Collapse

This simulator lets you explore the collapse of a five-body kite central configuration by changing the key parameters and viewing the resulting animation. You can adjust the angles θ, φ, and δ to see how they affect the geometry and the admissible domain, and you can choose different collapse centers and powers to compare different collapse behaviors. Angle markers are drawn directly on the configuration panels.

Model

Five masses lie at positions q₁, q₂, q₃, q₄, q₅, with q₂ and q₄ placed symmetrically about the x-axis. The default normalization is m₂ = m₄ = 1.

r13 = 1 r12 = sin(δ) / sin(δ + θ) r23 = sin(θ) / sin(δ + θ) r35 = [sin(θ)/sin(δ+θ)] · [sin(φ)/sin(π-δ+φ)] α = 1 + r35 m = P⁻¹ Q [γ, β, β + 2α - 2]ᵀ

Animation rules

Each frame is uniformly scaled by ρ(t) = (1 - t)^p. You can collapse toward the center of mass (com) or the geometric centroid (centroid). You can also switch the complete body-body graph on or off and adjust the playback speed and number of frames.

τ = k / (N - 1) ρ = (1 - τ)^p pts(τ) = center + ρ · (finalPts - center)

geometry: –

admissible: –

center: –

progress: –

Animation panels

Kite collapse animation

Final configuration

Collapse scale ρ(τ)

Diagnostics

angles

–

alpha / beta / gamma

–

det(P)

–

masses

–

current rho

–

distances

–

Legend

Body-body edges

Symmetry axis y = 0

Collapse center

Positive mass

Negative mass

Near-zero mass

Angle markers: θ, φ, δ

Parameter notes

Parameter	Main effect	Related condition	Animation impact
`θ`	Controls the angle that `q₂` and `q₄` make with the x-axis, which affects `r23` and the upper/lower vertex positions.	Must satisfy `θ > 0` and `δ + θ < π`.	Changes the kite height and the left-right opening of the shape.
`φ`	Controls the right-side geometry and `r35`, so it directly affects `α` and `m₅`.	Must satisfy `0 < φ < δ`.	Changes the position of `q₅` and the collapse shape on the right side.
`δ`	Controls the left-half geometry and the admissible-domain test.	Chapter 3 also requires `π/6 < δ < π/2` inside the admissible domain.	Changes the overall configuration and the admissibility flag.
`collapse center`	Chooses either the center of mass or the geometric centroid.	If the total mass is nearly zero and `com` is selected, the code automatically falls back to the centroid.	Determines the point toward which the animation collapses.
`collapse power p`	Controls the shrinking rate in `ρ = (1 - τ)^p`.	`p > 0`	Smaller `p` slows the early stage; larger `p` accelerates the later stage.