From: Michael Orlitzky Date: Mon, 2 May 2011 20:38:03 +0000 (-0400) Subject: Finish adding the coefficients, begin the barycentric stuff. X-Git-Tag: 0.0.1~346 X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=082802206f477fdebc1833ee2a4ced3da5dfb2a4;p=spline3.git Finish adding the coefficients, begin the barycentric stuff. --- diff --git a/doc/spline3.lyx b/doc/spline3.lyx index 0fd7922..3a9f578 100644 --- a/doc/spline3.lyx +++ b/doc/spline3.lyx @@ -1095,6 +1095,264 @@ sage: c1020 = (1/4)*I(1,1,1) + (1/12)*(LT(1,1,1) + FL(1,1,1) + FT(1,1,1)) \end_layout +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1002 = (1/4)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (1/6)*(F(1,1,1) + R(1,1,1) + T(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/12)*(RT(1,1,1) + FR(1,1,1) + FT(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1011 = (1/3)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (5/24)*(F(1,1,1) + T(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/12)*FT(1,1,1) + (1/24)*(L(1,1,1) + R(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/48)*(LT(1,1,1) + RT(1,1,1) + FL(1,1,1) + FR(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1110 = (1/3)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (5/24)*F(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (1/8)*(L(1,1,1) + T(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (5/96)*(FL(1,1,1) + FT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/48)*(D(1,1,1) + R(1,1,1) + LT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/96)*(FD(1,1,1) + LD(1,1,1) + RT(1,1,1) + FR(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1101 = (1/3)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (5/24)*F(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (1/8)*(R(1,1,1) + T(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (5/96)*(FR(1,1,1) + FT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/48)*(D(1,1,1) + L(1,1,1) + RT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/96)*(FD(1,1,1) + LT(1,1,1) + RD(1,1,1) + FL(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1200 = (1/3)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (5/24)*F(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (7/96)*(L(1,1,1) + R(1,1,1) + T(1,1,1) + D(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/32)*(FL(1,1,1) + FR(1,1,1) + FT(1,1,1) + FD(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/96)*(RT(1,1,1) + RD(1,1,1) + LT(1,1,1) + LD(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c2010 = (3/8)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (7/48)*(F(1,1,1) + T(1,1,1) + L(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/48)*(R(1,1,1) + D(1,1,1) + B(1,1,1) + LT(1,1,1) + FL(1,1,1) + + FT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/96)*(RT(1,1,1) + BT(1,1,1) + FR(1,1,1) + FD(1,1,1) + LD(1,1,1) + + BL(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c2001 = (3/8)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (7/48)*(F(1,1,1) + T(1,1,1) + R(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/48)*(L(1,1,1) + D(1,1,1) + B(1,1,1) + RT(1,1,1) + FR(1,1,1) + + FT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/96)*(LT(1,1,1) + BT(1,1,1) + FL(1,1,1) + FD(1,1,1) + RD(1,1,1) + + BR(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c2100 = (3/8)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (1/12)*(T(1,1,1) + R(1,1,1) + L(1,1,1) + D(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/64)*(FT(1,1,1) + FR(1,1,1) + FL(1,1,1) + FD(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (7/48)*F(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (1/48)*B(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (1/96)*(RT(1,1,1) + LD(1,1,1) + LT(1,1,1) + RD(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/192)*(BT(1,1,1) + BR(1,1,1) + BL(1,1,1) + BD(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c3000 = (3/8)*I(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + (1/12)*(T(1,1,1) + F(1,1,1) + L(1,1,1) + R(1,1,1) + D(1,1,1) + + B(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/96)*(LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + RT(1,1,1) + BT(1,1,1) + + FR(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + FD(1,1,1) + LD(1,1,1) + BD(1,1,1) + BR(1,1,1) + RD(1,1,1) + + BL(1,1,1)) +\end_layout + \end_inset @@ -1259,6 +1517,171 @@ sage: c1020 8/3 \end_layout +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1002 +\end_layout + +\begin_layout Plain Layout + +23/6 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1011 +\end_layout + +\begin_layout Plain Layout + +13/4 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1110 +\end_layout + +\begin_layout Plain Layout + +23/8 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1101 +\end_layout + +\begin_layout Plain Layout + +27/8 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c1200 +\end_layout + +\begin_layout Plain Layout + +3 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c2010 +\end_layout + +\begin_layout Plain Layout + +10/3 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c2001 +\end_layout + +\begin_layout Plain Layout + +4 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c2100 +\end_layout + +\begin_layout Plain Layout + +7/2 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c3000 +\end_layout + +\begin_layout Plain Layout + +4 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Example* +Now that we have the coefficients, we'll choose a particular tetrahedron + and compute the polynomial over it. + If we look at the +\begin_inset Quotes eld +\end_inset + +top +\begin_inset Quotes erd +\end_inset + + face of the cube (in the positive +\begin_inset Formula $z$ +\end_inset + + direction), there are only four tetrahedra to choose from. + We'll be consider the +\begin_inset Quotes eld +\end_inset + +right +\begin_inset Quotes erd +\end_inset + + tetrahedron; that is, the one with vertices, +\end_layout + +\begin_layout Example* +\begin_inset Formula \begin{eqnarray*} +v_{0}=\left(0.5,1.5,1.5\right) & & \mbox{at the front-right of the cube}\\ +v_{1}=\left(1.5,1.5,1.5\right) & & \mbox{at the back-right of the cube}\\ +v_{2}=\left(1,1,1.5\right) & & \mbox{at the center of the top face of the cube}\\ +v_{3}=\left(1,1,1\right) & & \mbox{at the center of the cube}\end{eqnarray*} + \end_inset