Projection Fn
Projection Fn is a direct subtype of Real To Real Function
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TYPE INCLUSION RELATIONSHIPS
AVAILABLE FUNCTIONS
AVAILABLE CREATE FUNCTION KEYS
TYPICAL OBJECTS OF TYPE Projection Fn
This type represents a function ƒ that maps an array of u real variable numbers x₁, x₂, ..., xᵤ to an array of v numbers xₑ₍₁₎, xₑ₍₂₎, ..., xₑ₍ᵥ₎ the elements of which already exist in the input array.
Formally:
ƒ(x₁,x₂,...,xᵤ) = (xₑ₍₁₎, xₑ₍₂₎, ..., xₑ₍ᵥ₎)
where e: i -> e(i) is a given integer mapping that prescribes that the e(i)ᵗʰ input coordinate xₑ₍ᵢ₎ appears in the returned array in the iᵗʰ position.
In typical usage, v = 1 and e(1) = r, where r is some fixed integer between 1 and u, so that the formula becomes:
ƒ(x₁,x₂,...,xᵤ) = xᵣ
, which represents the projection along the rᵗʰ axis.
Projection over a multi-dimensional subspace is possible by setting v > 1
By letting v = u and e representing a permulation of the integers 1, 2, ..., u, the projection assumes the characteristics of permutation.
For example, when v = u = 3 and e is such that e(1) = 2, e(2) = 3, e(3) = 1, we get:
ƒ(x₁,x₂,x₃) = (x₂,x₃,x₁)
Extrapolation into higher dimensions is possible by setting v > u and defining e so that certain coordinates are repeated.
For example, when u = 1, v = 3 and e is such that e(1) = e(2) = e(3) = 1, we get:
ƒ(x₁) = (x₁,x₁,x₁)
, which effectively forks the input variable x₁ into the triplet (x₁,x₁,x₁), thus mapping a one-dimensional space to a three-dimensional space.