Start working on simplified expressions for terms of d bounds

main.tex

@@ -229,6 +229,31 @@ This can be rearranged to express a bound on $d$ as follows:
 	\sage{bgmlv1_d_ineq}
 \end{dmath}
 
+\begin{sagesilent}
+	# Seperate out the terms of the lower bound for d
+	bgmlv1_d_lowerbound_exp_term = (bgmlv1_d_lowerbound*(R-2*r)).expand().subs(r==2*R)/(R-2*r)
+	bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0).subs(r==0)
+	bgmlv1_d_lowerbound_linear_term = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0).subs(r==1)*r
+
+	# Verify the simplified forms of the terms that will be mentioned in text
+	assert bgmlv1_d_lowerbound_const_term == v.twist(beta_min).ch[2]/2 + beta_min*q, "fail"
+	assert bgmlv1_d_lowerbound_linear_term == (v.twist(beta_min).ch[2]/2 + beta_min^2/2 + beta_min*q)*r, "fail"
+
+	foo = (
+		- 2*R*v.twist(beta_min).ch[2]
+		- 3*R^2*beta_min^2
+		- 4*R*beta_min*q
+		+ C*q
+		- q^2
+	).expand()/(R-2*r)
+\end{sagesilent}
+
+\begin{equation}
+	\sage{bgmlv1_d_lowerbound_exp_term}
+\end{equation}
+\begin{equation}
+	\sage{foo}
+\end{equation}
 
 \noindent
 In the case $\beta = \beta_{-}$ (or $\beta_{+}$) this can be simplified.
@@ -276,10 +301,23 @@ This can be rearranged to express a bound on $d$ as follows:
 	\sage{bgmlv2_d_ineq}
 \end{equation}
 
+\begin{sagesilent}
+	# Seperate out the terms of the lower bound for d
+	bgmlv2_d_lowerbound_exp_term = (bgmlv2_d_lowerbound*r).expand().subs(r==0)/r
+	bgmlv2_d_lowerbound_const_term = bgmlv2_d_lowerbound.subs(1/r == 0).subs(r==0)
+	bgmlv2_d_lowerbound_linear_term = bgmlv2_d_lowerbound.subs(1/r == 0).subs(r==1)*r
+\end{sagesilent}
 
-\noindent
-In the case $\beta = \beta_{-}$ (or $\beta_{+}$) this can be simplified.
+\begin{equation}
+	\sage{bgmlv2_d_lowerbound_exp_term}
+\end{equation}
 
+\begin{equation}
+	\sage{bgmlv2_d_lowerbound_const_term}
+\end{equation}
+\begin{equation}
+	\sage{bgmlv2_d_lowerbound_linear_term}
+\end{equation}
 
 
 \section{Conclusion}