diff --git a/Partition_2/doc_tex/Partition_2/partition_2_intro.tex b/Partition_2/doc_tex/Partition_2/partition_2_intro.tex index e406c8d7045..f27ea7da617 100644 --- a/Partition_2/doc_tex/Partition_2/partition_2_intro.tex +++ b/Partition_2/doc_tex/Partition_2/partition_2_intro.tex @@ -15,7 +15,7 @@ and \ccc{beyond}, an output iterator \ccc{result}, and a traits class to define a simple polygon whose vertices are in counterclockwise order. The computed partition polygons, whose vertices are also oriented counterclockwise, are written to the sequence starting at position -\ccc{result} and the past-the-end interator for the resulting sequence of +\ccc{result} and the past-the-end iterator for the resulting sequence of polygons is returned. The traits classes for the functions specify the types of the input points and output polygons as well as a few other types and function objects that are required by the various algorithms. diff --git a/Partition_2/doc_tex/Partition_2/partition_2_y_monotone.tex b/Partition_2/doc_tex/Partition_2/partition_2_y_monotone.tex index db14836e413..ab4c9ee4a50 100644 --- a/Partition_2/doc_tex/Partition_2/partition_2_y_monotone.tex +++ b/Partition_2/doc_tex/Partition_2/partition_2_y_monotone.tex @@ -8,7 +8,7 @@ presented in \cite{bkos-cgaa-97} is implemented by the function \ccc{y_monotone_partition_2}\ccIndexGlobalFunction{y_monotone_partition_2}. This algorithm runs in $O(n \log n)$ time and requires $O(n)$ space. This algorithm does not guarantee a bound on the number of polygons -produced with respect to the opitmal number. +produced with respect to the optimal number. For checking the validity of the partitions produced by \ccc{y_monotone_partition_2}, we provide a function \ccc{is_y_monotone_2}, diff --git a/Partition_2/doc_tex/Partition_2_ref/Is_convex_2.tex b/Partition_2/doc_tex/Partition_2_ref/Is_convex_2.tex index cea8e79fca4..d04ec7ccee5 100644 --- a/Partition_2/doc_tex/Partition_2_ref/Is_convex_2.tex +++ b/Partition_2/doc_tex/Partition_2_ref/Is_convex_2.tex @@ -26,7 +26,7 @@ a convex polygon or not. \ccCreationVariable{f} %% choose variable name \ccConstructor{Is_convex_2(const Traits& t);}{\ccc{Traits} satisfies the -requriements of the function \ccc{is_convex_2}} +requirements of the function \ccc{is_convex_2}} \ccOperations diff --git a/Partition_2/doc_tex/Partition_2_ref/OptimalConvexPartitionTraits_2.tex b/Partition_2/doc_tex/Partition_2_ref/OptimalConvexPartitionTraits_2.tex index 9974907f399..5ebf277792e 100644 --- a/Partition_2/doc_tex/Partition_2_ref/OptimalConvexPartitionTraits_2.tex +++ b/Partition_2/doc_tex/Partition_2_ref/OptimalConvexPartitionTraits_2.tex @@ -41,7 +41,7 @@ the ray from point $p$ through point $q$.} determines orderings of \ccc{Point_2}s on a line. Must provide \ccc{bool operator()(Point_2 p, Point_2 q, Point_2 r)} that returns \ccStyle{true}, iff \ccStyle{q} lies between \ccStyle{p} -and \ccStyle{r} and \ccc{p}, \ccc{q}, and \ccc{r} statisfy the precondition +and \ccStyle{r} and \ccc{p}, \ccc{q}, and \ccc{r} satisfy the precondition that they are collinear.} \ccNestedType{Are_stritcly_ordered_along_line_2}{Predicate object type that diff --git a/Partition_2/doc_tex/Partition_2_ref/convex_partition_is_valid_2.tex b/Partition_2/doc_tex/Partition_2_ref/convex_partition_is_valid_2.tex index f4214081451..4be19ee3278 100644 --- a/Partition_2/doc_tex/Partition_2_ref/convex_partition_is_valid_2.tex +++ b/Partition_2/doc_tex/Partition_2_ref/convex_partition_is_valid_2.tex @@ -63,7 +63,7 @@ with the representation type determined by \ccc{InputIterator::value_type}. This function calls \ccc{partition_is_valid_2} using the function object \ccc{Is_convex_2} to determine the convexity of each partition polygon. Thus the time required by this function is $O(n \log n + e \log e)$ where -$n$ is the total number of vertices in the partition polgons and $e$ the +$n$ is the total number of vertices in the partition polygons and $e$ the total number of edges. \ccExample diff --git a/Partition_2/doc_tex/Partition_2_ref/intro.tex b/Partition_2/doc_tex/Partition_2_ref/intro.tex index 160f56c5bf3..e8484c56b86 100644 --- a/Partition_2/doc_tex/Partition_2_ref/intro.tex +++ b/Partition_2/doc_tex/Partition_2_ref/intro.tex @@ -32,7 +32,7 @@ algorithm presented in \cite{bkos-cgaa-97} which requires $O(n \log n)$ time and $O(n)$ space for a polygon with $n$ vertices and guarantees nothing about the number of polygons produced with respect to the optimal number. Three functions are provided for producing -convex partitionings. Two of these functions produce approximately optimal +convex partitions. Two of these functions produce approximately optimal partitions and one results in an optimal partition, where ``optimal'' is defined in terms of the number of partition polygons. The two functions that implement approximation algorithms are guaranteed to produce no more