fixed curve-def + y-per-x-view

Algebraic_kernel_d/doc_tex/Algebraic_kernel_d_ref/AlgebraicKernelWithAnalysis_d_2.tex

@@ -4,10 +4,23 @@
 
 The \ccc{AlgebraicKernelWithAnalysis_d_2} concept refines
 the \ccc{AlgebraicKernel_d_2} concept by interpreting bivariate polynomials
-as algebraic curves and provides analysis of single curves and pairs of them.
-By an $y$-per-$x$-view 
-these analysis provide easy access to the topology of curves and pairs of
-curves. If not stated otherwise, a model is required to 
+as real algebraic plane curves. That is, for given bivariate polynomial~$f$,
+we consider the curve in the two-dimensional real plane induced by the set of
+vanishing points of~$p$: $V_\mathbb{R}(p) := \{(x,y) \in \mathbb{R}^2
+\mid p(x,y) = 0 \}$. The kernel provides a way to analyse a single
+curves and one to analyse pairs of them. Each such analysis does so in two
+steps: First, critical \ccc{x}-coordinates are detected. Second, for each such
+coordinate and coordinates contained in open
+intervals in between such, status lines are computed. Such a line reflects
+the topology of a curve (or a pair of curves) in \ccc{y}-direction
+over the coordinate (or interval, respectively). A status line always exists at
+a specific \ccc{x}-coordinate (for an interval, a representative might be
+chosen) and thus, a status line is also expected to provide access to the
+\ccc{y}-coordinates of points of the algebraic curve along the line, where the
+topology changes. But mainly, the ``$y$-per-$x$-analyses'' provide easy
+combinatorial access to the topology of curves and pairs of curves. 
+
+If not stated otherwise, a model is required to 
 provide the analysis for algebraic curves of general degree $d$ in $\R^2$.
 
 \ccRefines