Doc: improve description of geometric multiplication/division.

David Johnston reminded me that the per-point calculations being done
by these operators are equivalent to complex multiplication/division.
(Once I would've recognized that immediately, but it's been too long
since I did any of that sort of math.)

Also put in a footnote mentioning that "rotation" of a box doesn't do
what you might expect, as I'd griped about in the referenced thread.

Discussion: https://postgr.es/m/158110996889.1089.4224139874633222837@wrigleys.postgresql.org

diff --git a/doc/src/sgml/func.sgml b/doc/src/sgml/func.sgml
index 90a2582dfd1d4fe2dafda5b5ca9b27036532529b..63b870de04898731fccc15d4122d999c79e35c73 100644 (file)
--- a/doc/src/sgml/func.sgml
+++ b/doc/src/sgml/func.sgml
@@ -9506,21 +9506,19 @@ CREATE TYPE rainbow AS ENUM ('red', 'orange', 'yellow', 'green', 'blue', 'purple
         <returnvalue><replaceable>geometric_type</replaceable></returnvalue>
        <?br?>
         Multiplies each point of the first argument by the second
-        <type>point</type><footnote><para>For this purpose, the
-        product of two points
-        (<replaceable>x1</replaceable>,<replaceable>y1</replaceable>) and
-        (<replaceable>x2</replaceable>,<replaceable>y2</replaceable>) is
-        defined as
-        (<replaceable>x1</replaceable>*<replaceable>x2</replaceable> -
-        <replaceable>y1</replaceable>*<replaceable>y2</replaceable>,
-        <replaceable>x1</replaceable>*<replaceable>y2</replaceable> +
-        <replaceable>y1</replaceable>*<replaceable>x2</replaceable>).</para></footnote>.
-        Interpreting the <type>point</type> as a vector, this is equivalent to
+        <type>point</type> (treating a point as being a complex number
+        represented by real and imaginary parts, and performing standard
+        complex multiplication).  If one interprets
+        the second <type>point</type> as a vector, this is equivalent to
         scaling the object's size and distance from the origin by the length
         of the vector, and rotating it counterclockwise around the origin by
         the vector's angle from the <replaceable>x</replaceable> axis.
-        Available for <type>point</type>, <type>box</type>, <type>path</type>,
-        <type>circle</type>.
+        Available for <type>point</type>, <type>box</type>,<footnote
+        id="functions-geometry-rotation-fn"><para><quote>Rotating</quote> a
+        box with these operators only moves its corner points: the box is
+        still considered to have sides parallel to the axes.  Hence the box's
+        size is not preserved, as a true rotation would do.</para></footnote>
+        <type>path</type>, <type>circle</type>.
        <?br?>
         <literal>path '((0,0),(1,0),(1,1))' * point '(3.0,0)'</literal>
         <returnvalue>((0,0),(3,0),(3,3))</returnvalue>
@@ -9535,26 +9533,16 @@ CREATE TYPE rainbow AS ENUM ('red', 'orange', 'yellow', 'green', 'blue', 'purple
         <replaceable>geometric_type</replaceable> <literal>/</literal> <type>point</type>
         <returnvalue><replaceable>geometric_type</replaceable></returnvalue>
        <?br?>
-        Divides each point of the first argument by the
-        second <type>point</type><footnote><para>For this purpose, the
-        quotient of two points
-        (<replaceable>x1</replaceable>,<replaceable>y1</replaceable>) and
-        (<replaceable>x2</replaceable>,<replaceable>y2</replaceable>) is
-        defined as
-        ((<replaceable>x1</replaceable>*<replaceable>x2</replaceable> +
-        <replaceable>y1</replaceable>*<replaceable>y2</replaceable>) /
-        <replaceable>L</replaceable>,
-        (<replaceable>y1</replaceable>*<replaceable>x2</replaceable> -
-        <replaceable>x1</replaceable>*<replaceable>y2</replaceable>) /
-        <replaceable>L</replaceable>),
-        where <replaceable>L</replaceable> =
-        <replaceable>x2</replaceable>*<replaceable>x2</replaceable> +
-        <replaceable>y2</replaceable>*<replaceable>y2</replaceable>.</para></footnote>.
-        Interpreting the <type>point</type> as a vector, this is equivalent to
+        Divides each point of the first argument by the second
+        <type>point</type> (treating a point as being a complex number
+        represented by real and imaginary parts, and performing standard
+        complex division).  If one interprets
+        the second <type>point</type> as a vector, this is equivalent to
         scaling the object's size and distance from the origin down by the
         length of the vector, and rotating it clockwise around the origin by
         the vector's angle from the <replaceable>x</replaceable> axis.
-        Available for <type>point</type>, <type>box</type>, <type>path</type>,
+        Available for <type>point</type>, <type>box</type>,<footnoteref
+        linkend="functions-geometry-rotation-fn"/> <type>path</type>,
         <type>circle</type>.
        <?br?>
         <literal>path '((0,0),(1,0),(1,1))' / point '(2.0,0)'</literal>