From: Michael Orlitzky Date: Wed, 22 Aug 2012 14:35:27 +0000 (-0400) Subject: Remove the unused example1.py/spline3.lyx paper that I never finished. X-Git-Tag: 0.0.1~44 X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=cde212ef8213829473d9e03b1fa1f78cfc7dcbe7;p=spline3.git Remove the unused example1.py/spline3.lyx paper that I never finished. --- diff --git a/doc/example1.py b/doc/example1.py deleted file mode 100644 index 9dbe06f..0000000 --- a/doc/example1.py +++ /dev/null @@ -1,100 +0,0 @@ -#!/home/mjo/bin/sage - -# example1.py -# -# In the paper, we display a lot of sage code to be entered at the -# prompt. It's not much fun re-entering all of these formulas every -# time you'd like to work on the paper, so a quick load('example1.py') -# from the sage prompt will pull all of this stuff in for you. -# - -x,y,z = var('x,y,z') - -f = 1 + x + x*y + x*y*z - -g = [[[ f(x=a, y=b, z=c) for a in range(0,3) ] for b in range(0,3) ] for c in range(0,3) ] - -def grid(x,y,z): - return g[z][y][x] - -def I(x,y,z): - return grid(x,y,z) - -def F(x,y,z): - return grid(x-1, y, z) - -def B(x,y,z): - return grid(x+1, y, z) - -def L(x,y,z): - return grid(x, y-1, z) - -def R(x,y,z): - return grid(x, y+1, z) - -def T(x,y,z): - return grid(x, y, z+1) - -def D(x,y,z): - return grid(x, y, z-1) - -def FL(x,y,z): - return grid(x-1, y-1, z) - -def FR(x,y,z): - return grid(x-1, y+1, z) - -def FD(x,y,z): - return grid(x-1, y, z-1) - -def FT(x,y,z): - return grid(x-1, y, z+1) - -def BL(x,y,z): - return grid(x+1, y-1, z) - -def BR(x,y,z): - return grid(x+1, y+1, z) - -def BD(x,y,z): - return grid(x+1, y, z-1) - -def BT(x,y,z): - return grid(x+1, y, z+1) - -def LD(x,y,z): - return grid(x, y-1, z-1) - -def LT(x,y,z): - return grid(x, y-1, z+1) - -def RD(x,y,z): - return grid(x, y+1, z-1) - -def RT(x,y,z): - return grid(x, y+1, z+1) - -def FLD(x,y,z): - return grid(x-1, y-1, z-1) - -def FLT(x,y,z): - return grid(x-1, y-1, z+1) - -def FRD(x,y,z): - return grid(x-1, y+1, z-1) - -def FRT(x,y,z): - return grid(x-1, y+1, z+1) - -def BLD(x,y,z): - return grid(x+1, y-1, z-1) - -def BLT(x,y,z): - return grid(x+1, y-1, z+1) - -def BRD(x,y,z): - return grid(x+1, y+1, z-1) - -def BRT(x,y,z): - return grid(x+1, y+1, z+1) - diff --git a/doc/spline3.lyx b/doc/spline3.lyx deleted file mode 100644 index 3a9f578..0000000 --- a/doc/spline3.lyx +++ /dev/null @@ -1,1691 +0,0 @@ -#LyX 1.6.8 created this file. For more info see http://www.lyx.org/ -\lyxformat 345 -\begin_document -\begin_header -\textclass amsart -\use_default_options false -\begin_modules -theorems-ams -eqs-within-sections -figs-within-sections -\end_modules -\language english -\inputencoding auto -\font_roman ae -\font_sans default -\font_typewriter default -\font_default_family default -\font_sc false -\font_osf false -\font_sf_scale 100 -\font_tt_scale 100 - -\graphics default -\paperfontsize default -\spacing single -\use_hyperref false -\papersize letterpaper -\use_geometry false -\use_amsmath 1 -\use_esint 1 -\cite_engine basic -\use_bibtopic false -\paperorientation portrait -\secnumdepth 3 -\tocdepth 3 -\paragraph_separation skip -\defskip bigskip -\quotes_language english -\papercolumns 1 -\papersides 1 -\paperpagestyle default -\tracking_changes false -\output_changes false -\author "" -\author "" -\end_header - -\begin_body - -\begin_layout Title -Spline3 -\end_layout - -\begin_layout Author -Michael Orlitzky -\end_layout - -\begin_layout Example* -First, will will define our grid -\begin_inset Formula $G$ -\end_inset - - to be over a -\begin_inset Formula $3\times3\times3$ -\end_inset - - cube having grid size -\begin_inset Formula $h=1$ -\end_inset - -. -\end_layout - -\begin_layout Example* -To test the reproduction of trilinears, we will take the function (per Sorokina - and Zeilfelder, p. - 88), -\end_layout - -\begin_layout Example* -\begin_inset Formula \[ -f\left(x,y,z\right)=1+x+xy+xyz\] - -\end_inset - - -\end_layout - -\begin_layout Example* -and compute its value at the 27 points, -\begin_inset Formula $\left(0,0,0\right),\left(0,0,1\right),\dots\left(2,2,2\right)$ -\end_inset - -. -\end_layout - -\begin_layout Example* -\begin_inset listings -inline false -status open - -\begin_layout Plain Layout - -sage: x,y,z = var('x,y,z') -\end_layout - -\begin_layout Plain Layout - -sage: f = 1 + x + x*y + x*y*z -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=0, z=0) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=0, z=0) -\end_layout - -\begin_layout Plain Layout - -2 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=0, z=0) -\end_layout - -\begin_layout Plain Layout - -3 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=1, z=0) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=1, z=0) -\end_layout - -\begin_layout Plain Layout - -3 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=1, z=0) -\end_layout - -\begin_layout Plain Layout - -5 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=2, z=0) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=2, z=0) -\end_layout - -\begin_layout Plain Layout - -4 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=2, z=0) -\end_layout - -\begin_layout Plain Layout - -7 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=0, z=1) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=0, z=1) -\end_layout - -\begin_layout Plain Layout - -2 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=0, z=1) -\end_layout - -\begin_layout Plain Layout - -3 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=1, z=1) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=1, z=1) -\end_layout - -\begin_layout Plain Layout - -4 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=1, z=1) -\end_layout - -\begin_layout Plain Layout - -7 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=2, z=1) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=2, z=1) -\end_layout - -\begin_layout Plain Layout - -6 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=2, z=1) -\end_layout - -\begin_layout Plain Layout - -11 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=0, z=2) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=0, z=2) -\end_layout - -\begin_layout Plain Layout - -2 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=0, z=2) -\end_layout - -\begin_layout Plain Layout - -3 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=1, z=2) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=1, z=2) -\end_layout - -\begin_layout Plain Layout - -5 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=1, z=2) -\end_layout - -\begin_layout Plain Layout - -9 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=0, y=2, z=2) -\end_layout - -\begin_layout Plain Layout - -1 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=1, y=2, z=2) -\end_layout - -\begin_layout Plain Layout - -8 -\end_layout - -\begin_layout Plain Layout - -sage: f(x=2, y=2, z=2) -\end_layout - -\begin_layout Plain Layout - -15 -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Example* -We are most interested in the -\begin_inset Quotes eld -\end_inset - -interior -\begin_inset Quotes erd -\end_inset - - cube centered at -\begin_inset Formula $\left(1,1,1\right)$ -\end_inset - - with datum -\begin_inset Formula $4$ -\end_inset - -. - We can enter the data above into a list, -\end_layout - -\begin_layout Example* -\begin_inset listings -inline false -status open - -\begin_layout Plain Layout - -sage: g = [[[ f(x=a, y=b, z=c) for a in range(0,3) ] for b in range(0,3) - ] for c in range(0,3) ] -\end_layout - -\begin_layout Plain Layout - -sage: g -\end_layout - -\begin_layout Plain Layout - -[[[1, 2, 3], [1, 3, 5], [1, 4, 7]], -\end_layout - -\begin_layout Plain Layout - - [[1, 2, 3], [1, 4, 7], [1, 6, 11]], -\end_layout - -\begin_layout Plain Layout - - [[1, 2, 3], [1, 5, 9], [1, 8, 15]]] -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Example* -although the list will be indexed by -\begin_inset Formula $\left(z,y,x\right)$ -\end_inset - - so we define a function to access it by -\begin_inset Formula $\left(x,y,z\right)$ -\end_inset - -, -\end_layout - -\begin_layout Example* -\begin_inset listings -inline false -status open - -\begin_layout Plain Layout - -sage: def grid(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return g[z][y][x] -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Example* -and define directional functions according to Sorokina and Zeilfelder, p. - 81. -\end_layout - -\begin_layout Example* -\begin_inset listings -inline false -status open - -\begin_layout Plain Layout - -sage: def I(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x,y,z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def F(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def B(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def L(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y-1, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def R(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y+1, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def T(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def D(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FL(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y-1, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FR(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y+1, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BL(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y-1, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BR(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y+1, z) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def LD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y-1, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def LT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y-1, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def RD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y+1, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def RT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x, y+1, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FLD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y-1, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FLT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y-1, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FRD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y+1, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def FRT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x-1, y+1, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BLD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y-1, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BLT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y-1, z+1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BRD(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y+1, z-1) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: def BRT(x,y,z): -\end_layout - -\begin_layout Plain Layout - -....: return grid(x+1, y+1, z+1) -\end_layout - -\end_inset - - -\end_layout - -\begin_layout Example* -Next, we define the coefficients for the cube centered on -\begin_inset Formula $\left(1,1,1\right)$ -\end_inset - - based on these directional functions. -\end_layout - -\begin_layout Example* -\begin_inset listings -inline false -status open - -\begin_layout Plain Layout - -sage: c0030 = (1/8)*( I(1,1,1) + F(1,1,1) + L(1,1,1) + T(1,1,1) + -\end_layout - -\begin_layout Plain Layout - - LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + FLT(1,1,1) ) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0003 = (1/8)*( I(1,1,1) + F(1,1,1) + R(1,1,1) + T(1,1,1) + -\end_layout - -\begin_layout Plain Layout - - RT(1,1,1) + FR(1,1,1) + FT(1,1,1) + FRT(1,1,1) ) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0021 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/24)*(L(1,1,1) + FL(1,1,1) + LT(1,1,1) + FLT(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0012 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/24)*(R(1,1,1) + FR(1,1,1) + RT(1,1,1) + FRT(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0120 = (5/24)*(I(1,1,1) + F(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/8)*(L(1,1,1) + T(1,1,1) + FL(1,1,1) + FT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/24)*(LT(1,1,1) + FLT(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0102 = (5/24)*(I(1,1,1) + F(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/8)*(R(1,1,1) + T(1,1,1) + FR(1,1,1) + FT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/24)*(RT(1,1,1) + FRT(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0111 = (13/48)*(I(1,1,1) + F(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (7/48)*(T(1,1,1) + FT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/32)*(L(1,1,1) + R(1,1,1) + FL(1,1,1) + FR(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/96)*(LT(1,1,1) + RT(1,1,1) + FLT(1,1,1) + FRT(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0210 = (13/48)*(I(1,1,1) + F(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (17/192)*(L(1,1,1) + T(1,1,1) + FL(1,1,1) + FT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/96)*(LT(1,1,1) + FLT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/64)*(R(1,1,1) + D(1,1,1) + FR(1,1,1) + FD(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/192)*(RT(1,1,1) + LD(1,1,1) + FRT(1,1,1) + FLD(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0201 = (13/48)*(I(1,1,1) + F(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (17/192)*(R(1,1,1) + T(1,1,1) + FR(1,1,1) + FT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/96)*(RT(1,1,1) + FRT(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/64)*(L(1,1,1) + D(1,1,1) + FL(1,1,1) + FD(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/192)*(RD(1,1,1) + LT(1,1,1) + FLT(1,1,1) + FRD(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0300 = (13/48)*(I(1,1,1) + F(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (5/96)*(L(1,1,1) + R(1,1,1) + T(1,1,1) + D(1,1,1) + -\end_layout - -\begin_layout Plain Layout - - FL(1,1,1) + FR(1,1,1) + FT(1,1,1) + FD(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (1/192)*(RT(1,1,1) + RD(1,1,1) + LT(1,1,1) + LD(1,1,1) + -\end_layout - -\begin_layout Plain Layout - - FRT(1,1,1) + FRD(1,1,1) + FLT(1,1,1) + FLD(1,1,1)) -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c1020 = (1/4)*I(1,1,1) + -\end_layout - -\begin_layout Plain Layout - - (1/6)*(F(1,1,1) + L(1,1,1) + T(1,1,1)) + -\end_layout - -\begin_layout Plain Layout - - (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 - - -\end_layout - -\begin_layout Example* -We can see what the constant values are now: -\end_layout - -\begin_layout Example* -\begin_inset listings -inline false -status open - -\begin_layout Plain Layout - -sage: c0030 -\end_layout - -\begin_layout Plain Layout - -17/8 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0003 -\end_layout - -\begin_layout Plain Layout - -27/8 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0021 -\end_layout - -\begin_layout Plain Layout - -61/24 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0012 -\end_layout - -\begin_layout Plain Layout - -71/24 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0120 -\end_layout - -\begin_layout Plain Layout - -55/24 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0102 -\end_layout - -\begin_layout Plain Layout - -73/24 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0111 -\end_layout - -\begin_layout Plain Layout - -8/3 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0210 -\end_layout - -\begin_layout Plain Layout - -29/12 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0201 -\end_layout - -\begin_layout Plain Layout - -11/4 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c0300 -\end_layout - -\begin_layout Plain Layout - -5/2 -\end_layout - -\begin_layout Plain Layout - -\end_layout - -\begin_layout Plain Layout - -sage: c1020 -\end_layout - -\begin_layout Plain Layout - -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 - - -\end_layout - -\end_body -\end_document