Added: Documents with NUBRS derivatives and first Jerk matrix
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Arne Morten Kvarving
Knut Morten Okstad
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712a848874
commit
00e4da0822
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doc/jerk_matrix.pdf
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doc/jerk_matrix.pdf
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doc/jerk_matrix.tex
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doc/jerk_matrix.tex
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\documentclass[twoside, 11pt, a4paper]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath,amsfonts,amssymb,graphicx,parskip}
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\usepackage{mathpazo}
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\usepackage{color}
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\usepackage[margin=1in]{geometry}
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%\usepackage{fourierx} % eller lmodern
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% for debug utskrift
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\newcommand{\debug}[1]{\texttt{#1}}
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% for ingen utskrift av kommentarer/debug
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%\newcommand{\debug}[1]{}
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\DeclareMathOperator{\erf}{erf}
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\DeclareMathOperator{\erfc}{erfc}
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\DeclareMathOperator{\eps}{\epsilon}
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\newcommand{\dee}{\mathrm{d}}
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\begin{document}
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Jacobian:
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\[
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\begin{split}
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\frac{\partial u}{\partial x} &= \frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x} \\
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\frac{\partial u}{\partial y} &= \frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial y}
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\end{split}
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\]
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Hessian:
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\[
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\begin{split}
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\frac{\partial^2 u}{\partial x^2} &= \frac{\partial}{\partial\xi}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\xi}{\partial x} \\
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&+ \frac{\partial}{\partial\eta}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\eta}{\partial x} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x}+\frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial x\partial\xi} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x}+\frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial x\partial\xi}\right)\frac{\partial\xi}{\partial x} \\
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&+\left(\frac{\partial^2 u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x} + \frac{\partial u}{\partial \xi}\frac{\partial^2\xi}{\partial\xi\partial\eta} + \frac{\partial^2 u}{\partial\eta^2}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial x\partial\eta}\right)\frac{\partial\eta}{\partial x} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x}+\frac{\partial u}{\partial\xi}\frac{\partial\xi}{\partial x} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x}\right)\frac{\partial\xi}{\partial x} \\
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&+\left(\frac{\partial^2 u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x} + \frac{\partial^2 u}{\partial\eta^2}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial\eta}{\partial x}\right)\frac{\partial\eta}{\partial x} \\
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&= \frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}
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\end{split}
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\]
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\[
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\begin{split}
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\frac{\partial^2 u}{\partial y^2} &= \frac{\partial}{\partial\xi}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial y}\right)\frac{\partial\xi}{\partial y} \\
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&+ \frac{\partial}{\partial\eta}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial y}\right)\frac{\partial\eta}{\partial y} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial y} + \frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial\xi\partial y} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial y}+\frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\xi\partial y}\right)\frac{\partial\xi}{\partial y} \\
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&+\left(\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y} + \frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial \eta\partial y} + \frac{\partial^2 u}{\partial\eta^2}\frac{\partial\eta}{\partial y} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\eta\partial y}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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\newpage
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\[
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\begin{split}
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\frac{\partial^2 u}{\partial x\partial y} &= \frac{\partial}{\partial\xi}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\xi}{\partial y} \\
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&+ \frac{\partial}{\partial\eta}\left(\frac{\partial u}{\partial\xi}\frac{\partial \xi}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial \eta}{\partial x}\right)\frac{\partial\eta}{\partial y} \\
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&= \left(\frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x} + \frac{\partial u}{\partial \xi}\frac{\partial^2\xi}{\partial\xi\partial x} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\xi\partial x}\right)\frac{\partial\xi}{\partial y} \\
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&+ \left(\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}+\frac{\partial u}{\partial\xi}\frac{\partial^2\xi}{\partial\eta\partial x} + \frac{\partial^2u}{\partial\eta^2}\frac{\partial\eta}{\partial x} + \frac{\partial u}{\partial\eta}\frac{\partial^2\eta}{\partial\eta\partial x}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^2u}{\partial\xi^2}\frac{\partial\xi}{\partial x}\frac{\partial\xi}{\partial y} + \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\eta}{\partial x}\frac{\partial\xi}{\partial y}
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+ \frac{\partial^2u}{\partial\eta^2}\frac{\partial\xi}{\partial x}\frac{\partial \eta}{\partial y} + \frac{\partial^2 u}{\partial \xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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Jerk matrix:
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\[
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\begin{split}
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\frac{\partial^3u}{\partial x^3} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\xi}{\partial x} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\eta}{\partial x} \\
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&= \frac{\partial^3u}{\partial\xi^3}\left(\frac{\partial\xi}{\partial x}\right)^3 + \frac{\partial^3u}{\partial \xi\partial\eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2\frac{\partial\xi}{\partial x} + 2\frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\eta}{\partial x} \\
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&+\frac{\partial^3u}{\partial\eta^3}\left(\frac{\partial\eta}{\partial x}\right)^3 + \frac{\partial^3u}{\partial \xi^2\partial\eta}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\eta}{\partial x} + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial x}\left(\frac{\partial\eta}{\partial x}\right)^2
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\end{split}
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\]
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\[
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\begin{split}
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\frac{\partial^3u}{\partial y^3} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\xi}{\partial y} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^3u}{\partial\xi^3}\left(\frac{\partial\xi}{\partial y}\right)^3 + \frac{\partial u^3}{\partial\xi\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2\frac{\partial\xi}{\partial y} + 2 \frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial y}\right)^2\frac{\partial\eta}{\partial y} \\
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&+ \frac{\partial^3 u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial y}\right)^2\frac{\partial\eta}{\partial y} + \frac{\partial^3 u}{\partial\eta^3}\left(\frac{\partial \xi}{\partial y}\right)^3 + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial y}\left(\frac{\partial\eta}{\partial y}\right)^2
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\end{split}
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\]
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\newpage
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\[
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\begin{split}
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\frac{\partial^3u}{\partial x^2\partial y} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\xi}{\partial y} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2u}{\partial \xi^2}\left(\frac{\partial\xi}{\partial x}\right)^2 + \frac{\partial^2u}{\partial \eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2 + 2 \frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\right)\frac{\partial\eta}{\partial y} \\
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&= \frac{\partial^3u}{\partial\xi^3}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\xi}{\partial y} + \frac{\partial^3u}{\partial\xi\partial\eta^2}\left(\frac{\partial\eta}{\partial x}\right)^2\frac{\partial\xi}{\partial y} + 2\frac{\partial^3u}{\partial\xi^2\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial x} \\
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&+ \frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial x}\right)^2\frac{\partial\eta}{\partial y}+\frac{\partial^3u}{\partial\eta^3}\left(\frac{\partial\eta}{\partial x}\right)^2\frac{\partial\eta}{\partial y} + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial x}\frac{\partial\eta}{\partial x}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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\[
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\begin{split}
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\frac{\partial^3u}{\partial x\partial y^2} &= \frac{\partial}{\partial\xi}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\xi}{\partial x} \\
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&+\frac{\partial}{\partial\eta}\left(\frac{\partial^2 u}{\partial\xi^2}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^2u}{\partial\eta^2}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^2u}{\partial\xi\partial\eta}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y}\right)\frac{\partial\eta}{\partial x} \\
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&= \frac{\partial^3u}{\partial\xi^3}\frac{\partial\xi}{\partial x}\left(\frac{\partial\xi}{\partial y}\right)^2 + \frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial \xi}{\partial x}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^3u}{\partial\xi^2\partial\eta}\frac{\partial\xi}{\partial x}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial y} \\
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&+ \frac{\partial^3u}{\partial\xi^2\partial\eta}\left(\frac{\partial\xi}{\partial y}\right)^2\frac{\partial\eta}{\partial x} + \frac{\partial^3u}{\partial\eta^3}\frac{\partial\eta}{\partial x}\left(\frac{\partial\eta}{\partial y}\right)^2 + 2\frac{\partial^3u}{\partial\xi\partial\eta^2}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial x}\frac{\partial\eta}{\partial y}
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\end{split}
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\]
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\end{document}
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doc/nurbs_deriv.pdf
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doc/nurbs_deriv.pdf
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doc/nurbs_deriv.tex
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doc/nurbs_deriv.tex
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\documentclass[twoside, 11pt, a4paper]{article}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath,amsfonts,amssymb,graphicx,parskip}
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\usepackage{mathpazo}
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\usepackage{color}
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\usepackage[margin=1in]{geometry}
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%\usepackage{fourierx} % eller lmodern
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% for debug utskrift
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\newcommand{\debug}[1]{\texttt{#1}}
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% for ingen utskrift av kommentarer/debug
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%\newcommand{\debug}[1]{}
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\DeclareMathOperator{\erf}{erf}
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\DeclareMathOperator{\erfc}{erfc}
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\DeclareMathOperator{\eps}{\epsilon}
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\newcommand{\dee}{\mathrm{d}}
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\begin{document}
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Nurbs function:
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\[
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R_i(x) = \frac{N_i(x)}{\sum_{j=1}^N w_jN_j(x)}w_i
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\]
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Let
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\[
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W = \sum_{j=1}^Nw_jN_j(x)
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\]
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Nurbs derivative:
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\[
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R_i'(x) = \frac{N_i'W - N_iW'}{W^2}w_i = \frac{H}{W^2}w_i
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\]
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Nurbs second derivative:
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\[
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R_i''(x) = \frac{H'W - 2HW'}{W^3} = \frac{G}{W^3}w_i
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\]
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where
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\[
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H' = N_i''W - NW''
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\]
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Third derivative:
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\[
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R_i'''(x) = \frac{G'W - 3GW'}{W^4}w_i
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\]
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where
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\[
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G' = H''W - 2HW'' - H'W'
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\]
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and
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\[
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H'' = N_i'''W + N_i''W' - N'W'' -NW'''.
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\]
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\newpage
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Nurbs 2D function:
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\[
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R_{ij}(x,y) = \frac{N_i(x)N_j(y)}{\sum_k\sum_l N_k(x)N_l(y)w_{kl}}w_{ij} = R_{ij}(x,y) = \frac{N_i(x)N_j(y)}{W}w_{ij}
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\]
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First derivative:
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\[
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\begin{split}
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\frac{\partial R_{ij}}{\partial x} &= \frac{N_i'(x)N_j(y)W - N_i(x)N_j(y)\frac{\partial W}{\partial x}}{W^2}w_{ij} = \frac{H_1}{W^2}w_{ij}\\
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\frac{\partial R_{ij}}{\partial y} &= \frac{N_i(x)N_j'(y)W - N_i(x)N_j(y)\frac{\partial W}{\partial y}}{W^2}w_{ij} = \frac{H_2}{W^2}w_{ij}
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\end{split}
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\]
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Second derivative:
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\[
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\begin{split}
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\frac{\partial^2 R_{ij}}{\partial x^2} &= \frac{\frac{\partial H_1}{\partial x}W - 2H_1\frac{\partial W}{\partial x}}{W^3}w_{ij} = \frac{G_1}{W^3}w_{ij} \\
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\frac{\partial^2 R_{ij}}{\partial y^2} &= \frac{\frac{\partial H_2}{\partial y}W - 2H_2\frac{\partial W}{\partial y}}{W^3}w_{ij} = \frac{G_2}{W^3}w_{ij} \\
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\frac{\partial^2 R_{ij}}{\partial x\partial y} &= \frac{\frac{\partial H_1}{\partial y}W - 2H_1\frac{\partial W}{\partial y}}{W^3}w_{ij}
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\end{split}
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\]
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where
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\[
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\begin{split}
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\frac{\partial H_1}{\partial x} &= N_i''(x)N_j(y)W - N_i(x)N_j(y)\frac{\partial^2W}{\partial x^2} \\
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\frac{\partial H_1}{\partial y} &= N_i'(x)N_j'(y)W + N_i'(x)N_j(y)\frac{\partial W}{\partial y} - N_i(x)N_j'(y)\frac{\partial W}{\partial x} - N_i(x)N_j(y)\frac{\partial^2W}{\partial x\partial y} \\
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\frac{\partial H_2}{\partial y} &= N_i(x)N_j''(y)W - N_i(x)N_j(y)\frac{\partial^2W}{\partial y^2}
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\end{split}
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\]
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\newpage
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Third derivative:
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\[
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\begin{split}
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\frac{\partial^3 R_{ij}}{\partial x^3} &= \frac{\frac{\partial G_1}{\partial x}W - 3G_1\frac{\partial W}{\partial x}}{W^4}w_{ij} \\
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\frac{\partial^3 R_{ij}}{\partial y^3} &= \frac{\frac{G_2}{\partial y}W - 3G_2\frac{\partial W}{\partial y}}{W^4}w_{ij} \\
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\frac{\partial^3 R_{ij}}{\partial x^2\partial y} &= \frac{\frac{\partial G_1}{\partial y}W - 3G_1\frac{\partial W}{\partial y}}{W^4}w_{ij} \\
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\frac{\partial^3 R_{ij}}{\partial x\partial y^2} &= \frac{\frac{\partial G_2}{\partial x}W - 3G_2\frac{\partial W}{\partial x}}{W^4}w_{ij}
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\end{split}
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\]
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where
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\[
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\begin{split}
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\frac{\partial G_1}{\partial x} &= \frac{\partial^2H_1}{\partial x^2}W + \frac{\partial H_1}{\partial x}\frac{\partial W}{\partial x} - 2\frac{\partial H_1}{\partial x}\frac{\partial W}{\partial x} - 2 H_1\frac{\partial^2W}{\partial x^2} \\
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\frac{\partial G_1}{\partial y} &= \frac{\partial^2H_1}{\partial x\partial y}W + \frac{\partial H_1}{\partial x}\frac{\partial W}{\partial y} - 2\frac{\partial H_1}{\partial y}\frac{\partial W}{\partial x} - 2 H_1\frac{\partial^2W}{\partial x\partial y} \\
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\frac{\partial G_2}{\partial x} &= \frac{\partial^2H_2}{\partial x\partial y}W + \frac{\partial H_2}{\partial y}\frac{\partial W}{\partial x} - 2\frac{\partial H_2}{\partial x}\frac{\partial W}{\partial y} - 2 H_2\frac{\partial^2W}{\partial x\partial y} \\
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\frac{\partial G_2}{\partial y} &= \frac{\partial^2H_2}{\partial y^2}W + \frac{\partial H_2}{\partial y}\frac{\partial W}{\partial y} - 2\frac{\partial H_2}{\partial y}\frac{\partial W}{\partial y} - 2 H_2\frac{\partial^2W}{\partial y^2}
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\end{split}
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\]
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and
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\[
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\begin{split}
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\frac{\partial^2H_1}{\partial x^2} &= N_i'''N_jW + N_i''N_j\frac{\partial W}{\partial x} - N_i'N_j\frac{\partial^2W}{\partial x^2} - N_iN_j\frac{\partial^3W}{\partial x^3} \\
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\frac{\partial^2H_1}{\partial x\partial y} &= N_i''N_j'W + N_i''N_j\frac{\partial W}{\partial y} - N_iN_j'\frac{\partial^2W}{\partial x^2} - N_iN_j\frac{\partial^3W}{\partial x^2\partial y} \\
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\frac{\partial^2H_2}{\partial y^2} &= N_iN_j'''W + N_iN_j''\frac{\partial W}{\partial y} - N_iN_j'\frac{\partial^2W}{\partial y^2} - N_iN_j\frac{\partial^3W}{\partial y^3} \\
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\frac{\partial^2H_2}{\partial x\partial y} &= N_i'N_j''W + N_iN_j''\frac{\partial W}{\partial x} - N_i'N_j\frac{\partial^2W}{\partial y^2} - N_iN_j\frac{\partial^3W}{\partial x\partial y^2}
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\end{split}
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\]
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\end{document}
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