[Gzz-commits] gzz/Documentation/Manuscripts/Irregu irregu.tex

[Janne V. Kujala]

CVSROOT:        /cvsroot/gzz
Module name:    gzz
Changes by:     Janne V. Kujala <address@hidden>        02/11/19 12:47:57

Modified files:
        Documentation/Manuscripts/Irregu: irregu.tex 

Log message:
        small changes

CVSWeb URLs:
http://savannah.gnu.org/cgi-bin/viewcvs/gzz/gzz/Documentation/Manuscripts/Irregu/irregu.tex.diff?tr1=1.57&tr2=1.58&r1=text&r2=text

Patches:
Index: gzz/Documentation/Manuscripts/Irregu/irregu.tex
diff -u gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.57 
gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.58
--- gzz/Documentation/Manuscripts/Irregu/irregu.tex:1.57        Tue Nov 19 
12:07:08 2002
+++ gzz/Documentation/Manuscripts/Irregu/irregu.tex     Tue Nov 19 12:47:57 2002
@@ -305,12 +305,11 @@
 envelope, respectively, and the ripples are contained between these two curves.
 The envelope should not intersect itself.
 
-An envelope can be defined with a spine $E(x,1/2)$ and a normal vector
+An envelope can be defined with a \emph{spine} $E(x,1/2)$ and a normal vector
 $N(x)$ so that $E(x,y) = E(x,1/2) + (y-1/2) N(x)$.
 
-For the connected edge, we can obtain the final edge curve $C(x)$ by 
-simply shifting
-the spine of the envelope along its normal direction by a function 
+The edge curve $C(x)$ of a conneccted shape can be obtained by simply 
+shifting in the normal direction of the envelope by a function 
 $0\le f({\bf p})\le 1$ which only depends on the location ${\bf p}$ of the 
spine:
 $C(x) = E(x, f(E(x,1/2)))$.
 
@@ -326,7 +325,7 @@
 
 Although these algorithms seem different and produce different results, there 
is 
 actually a general formulation which yields to a visual explanation.
-Both algorithms can be seen as computing the intersection of 
+Both algorithms can be seen as computing the intersection of a
 \emph{ripple volume}, the volume below the surface $({\bf p}, f({\bf p}))$,
  and a \emph{cutting surface} $(E(x,g(y)), y)$, 
 and then mapping the intersection on the envelope $E(x,y)$ 
@@ -343,7 +342,7 @@
 \begin{equation} \label{eq:inside}
     f(E(x,g(y))) \ge y.
 \end{equation}
-For more variation on the edge shapes, the inequality can be 
+For more variation on edge shapes, the inequality can be 
 generalized to
 \begin{equation} \label{eq:inside2}
    (1-\alpha(y)) f_1(E(x,g_1(y))) + \alpha(y) f_2(E(x,g_2(y)) \ge y,
@@ -372,8 +371,9 @@
 In the following, we shall concentrate on drawing one rectangular section
 of the envelope, in the unit square, with $y=0$ inside the tear-out,
 $y=1$ outside the tear-out, and $x$ along the length of the envelope.
-It assumed that the canvas position $E(x,y)$ inside the section of the 
-envelope depends linearly on the parameters $x$ and $y$.
+It assumed that the canvas location $E(x,y)$ 
+depends linearly on the parameters $x$ and $y$
+inside the section of the envelope.
 Furthermore, without loss of generality, we assume that $E(x,y) = (x,y)$.
 At the end of this section, we consider how to use the rectangular 
 pieces to create a complete tear-out shape.
@@ -654,7 +654,7 @@
 the ripple function for either one of each pair of facing sides.
 But then a 180 degree rotation of a pair of fitting pieces 
 inverts the torn shape between them, breaking the principle of
-tying ripple shape to canvas positions.
+tying ripple shape to canvas location.
 
 The problem can be fully solved with a vector valued ripple function 
 ${\bf F}({\bf p})$, $\Vert{\bf F}({\bf p})\Vert \le 1$,
@@ -665,9 +665,8 @@
 
 The texture shader version can directly use the vector valued ripple function.
 It can also be used with the
-pre-computed borders method by pre-computing the dot product, too,
-when computing the outer surfaces. 
-However, even the connected case then requires a full 360 span
+pre-computed borders method by pre-computing the dot product, too.
+However, even the connected case then requires a full 360 degree span
 of pre-computed outer surfaces.