Finding IFS Rules from Images of Points

Background: Presence of Reflection

To determine if the transformation T involves a reflection, consider the initial points

p₁ = (x₁, y₁), p₂ = (x₂, y₂), and p₃ = (x₃, y₃),

and their images

q₁ = T(p₁) = (u₁, v₁), q₂ = T(p₂) = (u₂, v₂), and q₃ = T(p₃) = (u₃, v₃).

Viewing these as points in the xy-plane in 3-dimensional space, we form the cross-products

(p₂ - p₁) x (p₃ - p₁) = 0i - 0j + ((x₂ - x₁)(y₃ - y₁) - (x₃ - x₁)(y₂ - y₁)k

(q₂ - q₁) x (q₃ - q₁) = 0i - 0j + ((u₂ - u₁)(v₃ - v₁) - u₃ - u₁)(v₂ - v₁)k

If both vectors point in the same direction, the orientation of the triple of image points is the same as that of the triple of initial points, so T does not involve a reflection.

If the vectors point in opposite directions, T does inolve a reflection.

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