Kubische Kurve durch zwei Punkte - Matrixnotation

Equation system and matrices

This article resumes what has been introduced under "cubic curve through two points" already.

The equations required to specify the coefficients may be put in a matrix, and the condition values and coefficients are vectors. The matrices are constructed from the conditions.

Ac = b

Given is vector b containing the condition values. In the case of a curve through two points with given slopes b is equal to:  [y₀, y₁, m₀, m₁]

Further, x₀ und x₁ are given, from which the vectors v₀ = [1, x₀, x₀², x₀³] and v₁ = [1, x₁, x₁², x₁³] may be specified, which serve to construct the rows of matrix A:

A₁ = v₀ (row 1)

A₂ = v₁ (row 2)

A₃ = [0, 1v₀₁, 2v₀₂, 3v₀₃] (row 3)

A₄ = [0, 1v₁₁, 2v₁₂, 3v₁₃] (row 4)

By calculating the inverse of this matrix, the coefficient vector c is obtained from the matrix multiplication A^-1b

Example

The image below shows a sample calculation in Excel

The tidbit: Area under the curve

Sometimes, a curve is serched enclosing a given area (F) underneath between the two end points. This condition is obtained by integrating the curve function

F = ax + b/2x² + c/3x³ + d/4x⁴, at the two points:

F = ax + b(x₁² - x₀²)/2 + c(x₁³ - x₀³)/3 + d(x₁⁴ - x₀⁴)/4

The condition is inserted in one row of matrix A, leaving one more condition to be chosen.