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Cut a symmetric three-ray cone, ask which triangle shapes it can produce, or reverse the question and trace the possible apex positions. Start with the concepts, glossary and guided investigations →
Every point is a triangle shape. In the angle chart its weights are A/π, B/π, C/π; in the complex chart its third vertex is C = (ξ, η), with A = (−1, 0), B = (1, 0). The same classifications appear in two coordinate systems. E/P/H counts distinguish elliptic, parabolic and hyperbolic inverse shapes at the chosen cone height.
Gold curves mark μ = 1, dark curves mark physical folds, and purple dashed curves mark escape to the boundary of the physical domain. Zoom recomputes detail; use “Enlarge” or export a 4096-pixel PNG. A finely sampled image is still numerical evidence. Read the boundary guide · Understand all five classifications · Paper Figures 3–4.
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The cone apex is S = (0, 0, h), its base circumradius is 1, and the cutting plane is z = −hμ(x + 1), with μ = tan(α)/h. The regular height is √2. The circumsection is elliptic for μ < 1, parabolic for μ = 1, and hyperbolic for μ > 1. Only intersections with all three forward rays are accepted.
Triangle-space coordinates use A = (−1, 0), B = (1, 0), C = (ξ, η), or angle-barycentric weights θᵢ/π. E/P/H counts identify unordered section shapes; symmetric targets can have several labeled spatial apices for one inverse shape. The axial equilateral family is shown together with its off-axis solutions.
These are numerical explorations. Counts can be uncertain very close to folds, symmetry seams, or degenerate triangles. Atlas colors use adaptive sampling with event-curve refinement and do not certify exact cell topology. Curves are sampled; the view clips distant branches and centers at infinity. Use the paper’s exact certificates for proof-level claims; the Julia and browser numerical implementations have finite precision.
The browser runs locally on your device; it requires no Mathematica license or calculation server. The Julia package provides the mathematical API. Download the source · Porting and verification notes · MIT license for this port.