Structure Theorem Explorer

drag a tip to move it · drag elsewhere to rotate · unit sphere in gray

The set D = {v₁, v₂, v₃, w₀, w₃}

Drag the tips of v₁, v₂, v₃ on the sphere to change the linear basis. The two remaining vectors follow the theorem: w₀ = −(a₁v₁ + a₂v₂) and w₃ = −(b₃v₃ + c₃v₂), so D is always a positive basis.

w₀ between −v₁ and −v₂

hook of the pair towards −v₂

Readout

a₁, a₂
b₃, c₃
pair is
shared vector v₂
critical vector −c₃v₂
det(v₁, v₂, v₃)
cm(D)

Is it critical?

A vector is critical for a positive basis when it cannot replace any element without losing positive spanning. Test a vector against the triangle ∪ pair:

Show

plane of the triangle

critical rays of the triangle: −v₁, −v₂, −w₀

split w₃ = −b₃v₃ − c₃v₂

the test vector (when it is not the hook)

the polar P(D)

Legend

the five vectors of D; drag the tips of v₁, v₂, v₃; v₂ is ringed: the shared vector the hook: dashed split of w₃, and the critical vector −c₃v₂ critical rays of the triangle a test vector other than the hook