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105 lines
3.5 KiB
105 lines
3.5 KiB
1 year ago
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\documentclass[a4paper]{article}
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\usepackage[english]{babel}
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\usepackage[utf8]{inputenc}
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\usepackage[fleqn]{amsmath}
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\usepackage{hyperref}
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\begin{document}
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\section{Math stuff}
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Cubic interpolation for one segment $[x_k, x_{k+1}]$ can be described as:
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\begin{equation*}
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\begin{aligned}
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f(t) &= c_{oef1}t^3 + c_{oef2}t^2 + c_{oef3}t + c_{oef4} \hspace{1cm} with \\
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t(x) &= \frac{x - x_k}{x_{k+1} - x_k}
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\end{aligned}
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\end{equation*}
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\hspace{1cm} and
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\begin{equation*}
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\begin{aligned}
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c_{oef1} &= 2p_0 - 2p_1 - m_0 - m_1 \\
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c_{oef2} &= -3p_0 + 3p_1 - 2m_0 - m_1 \\
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c_{oef3} &= m_0 \\
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c_{oef4} &= p_0
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\end{aligned}
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\end{equation*}
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(see Wikipedia-Links below)\\
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\\
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If we rewrite this as function of $d = x - x_k$ we get
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\begin{equation*}
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\begin{aligned}
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f'(d) &= c_{oef1}' d^3 + c_{oef2}' d^2 + c_{oef3}' d + c_{oef4}' \hspace{1cm} with \\
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c_{oef1}' &= \frac{c_{oef1}}{(x_{k+1} - x_k)^3} \\
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c_{oef2}' &= \frac{c_{oef2}}{(x_{k+1} - x_k)^2} \\
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c_{oef3}' &= \frac{c_{oef3}}{x_{k+1} - x_k} \\
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c_{oef4}' &= c_{oef4}
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\end{aligned}
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\end{equation*}
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\\
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The implemented algorithm uses two helper variables to calculate the coefficients of $f'$ efficiently:
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\begin{equation*}
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\begin{aligned}
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common = m_k + m_{k+1} - 2 \frac{p_{k+1} - p_k}{x_{k+1} - x_k} \\
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invLength = \frac{1}{x_{k+1} - x_k}
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\end{aligned}
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\end{equation*}
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\\
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We use $p_0 = p_k$, $p_1 = p_{k+1}$, $m_0 = m_k (x_{k+1} - x_k)$, $m_1 = m_{k+1} (x_{k+1} - x_k)$ and $s = \frac{p_{k+1}-p_k}{x_{k+1}-x_k}$. The tangents are scaled with the length of the segment. \\
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\\
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If we insert this into the equations for the coefficients we get the formulas that are used in the algorithm:
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\begin{equation*}
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\begin{aligned}
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c_{oef1}' &= \frac{c_{oef1}}{(x_{k+1} - x_k)^3} \\
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&= \frac{2p_0 - 2p_1 + m_0 + m_1}{(x_{k+1} - x_k)^3}\\
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&= (2p_k - 2p_{k+1} + m_k (x_{k+1} - x_k) + m_{k+1} (x_{k+1} - x_k) / (x_{k+1} - x_k)^3 \\
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&= \frac {(2p_k - 2p_{k+1} + m_k (x_{k+1} - x_k) + m_{k+1} (x_{k+1} - x_k)}{x_{k+1} - x_k} / (x_{k+1} - x_k)^2 \\
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&= (\frac {2p_k - 2p_{k+1}}{x_{k+1} - x_k} + m_k + m_{k+1} ) * invLength^2 \\
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&= (-2\frac {p_{k+1}- p_k}{x_{k+1} - x_k} + m_k + m_{k+1} ) * invLength^2 \\
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&= common * invLenght^2
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\end{aligned}
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\end{equation*}
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\begin{equation*}
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\begin{aligned}
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c_{oef2}' &= \frac{c_{oef2}}{(x_{k+1} - x_k)^2} \\
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&= (-3p_0 + 3p_1 - 2m_0 - m_1) / (x_{k+1} - x_k)^2 \\
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&= (-3p_k + 3p_{k+1} - 2* m_k (x_{k+1} - x_k) - m_{k+1} (x_{k+1} - x_k)) / (x_{k+1} - x_k)^2 \\
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&= (\frac{-3p_k + 3p_{k+1}}{x_{k+1} - x_k} - 2m_k - m_{k+1}) * invLenght \\
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&= (3\frac{p_{k+1} - p_k}{x_{k+1} - x_k} - 2m_k - m_{k+1}) * invLenght \\
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&=(\frac{p_{k+1} - p_k}{x_{k+1} - x_k} + 2\frac{p_{k+1} - p_k}{x_{k+1} - x_k} - m_k - m_{k+1} - m_k) * invLenght \\
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&= (s - common - m_k) * invLenght
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\end{aligned}
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\end{equation*}
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\begin{equation*}
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\begin{aligned}
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c_{oef3}' &= \frac{c_{oef3}}{x_{k+1} - x_k} \\
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&= \frac{m_0}{x_{k+1} - x_k} \\
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&= \frac{m_k (x_{k+1} - x_k)}{x_{k+1} - x_k} \\
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&= m_k
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\end{aligned}
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\end{equation*}
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\begin{equation*}
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\begin{aligned}
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c_{oef4}' &= c_{oef4} = p_0 = p_k
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\end{aligned}
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\end{equation*}
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\section{Useful Links}
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\url{http://de.wikipedia.org/w/index.php?title=Kubisch_Hermitescher_Spline&oldid=130168003)}\\
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\url{http://en.wikipedia.org/w/index.php?title=Monotone_cubic_interpolation&oldid=622341725}\\
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\url{http://math.stackexchange.com/questions/45218/implementation-of-monotone-cubic-interpolation}\\
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\url{http://math.stackexchange.com/questions/4082/equation-of-a-curve-given-3-points-and-additional-constant-requirements#4104}
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\end{document}
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