Finish adding the coefficients, begin the barycentric stuff.

diff --git a/doc/spline3.lyx b/doc/spline3.lyx
index 0fd79228b491b2d485b346a7d597a5a74df92933..3a9f578d88540b7aee5ff375d9d7ece3a7c0a8e1 100644 (file)
--- 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