Editing Cubic Hermite spline (section)

===Uniqueness===
The formula specified above provides the unique third-degree polynomial path between the two points with the given tangents.

'''Proof.''' Let <math>P, Q</math> be two third-degree polynomials satisfying the given boundary conditions. Define <math>R = Q - P,</math> then:
: <math>R(0) = Q(0)-P(0) = 0,</math>
: <math>R(1) = Q(1) - P(1) = 0.</math>
Since both <math>Q</math> and <math>P</math> are third-degree polynomials, <math>R</math> is at most a third-degree polynomial. So <math>R</math> must be of the form
<math display="block">R(x) = ax(x - 1)(x - r).</math>
Calculating the derivative gives
<math display="block">R'(x) = ax(x - 1) + ax(x - r) + a(x - 1)(x - r).</math>

We know furthermore that

: <math>R'(0) = Q'(0) - P'(0) = 0,</math>
{{NumBlk|:|<math>R'(0) = 0 = ar,</math>|{{EquationRef|1}}}}

: <math>R'(1) = Q'(1) - P'(1) = 0,</math>
{{NumBlk|:|<math>R'(1) = 0 = a(1 - r).</math>|{{EquationRef|2}}}}

Putting ({{EquationNote|1}}) and ({{EquationNote|2}}) together, we deduce that <math>a = 0</math>, and therefore <math>R = 0,</math> thus <math>P = Q.</math>