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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional+edit//EN" "xhtml1-transitional+edit.dtd">
<html lang="en" xmlns="http://www.w3.org/1999/xhtml" xmlns:edit="http://xmlns.grorg.org/SVGT12NG/">
<head>
<title>Implementation Notes</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"/>
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"/>
<!-- Style sheets for local dev. Will be standardized in processing.
Add attribute data-keep="" to any extra stylesheet link you do not want removed
(or use <style>), and include it before here. -->
<link rel="stylesheet" href="style/svg.css" type="text/css" />
<link rel="stylesheet" href="style/W3C-ED.css" type="text/css" />
</head>
<body>
<h1>Implementation Notes</h1>
<h2 id="Introduction">Introduction</h2>
<p>The following notes describe algorithms and other strategies
which can be used by software developers when converting content
to and from the formats required by features in the SVG language.</p>
<h2 id="ArcImplementationNotes">Elliptical arc parameter conversion</h2>
<p>
To be consistent with other path segment notation,
arcs in SVG paths are defined in terms of start and end points on the curve.
</p>
<p>
This parameterization of elliptical arcs will be referred to
as <em>endpoint parameterization</em>. One of the
advantages of endpoint parameterization is that it permits a
consistent path syntax in which all path commands end in the
coordinates of the new "current point".
</p>
<p>
However, this is not the only way of describing arc geometry
used in software or mathematics.
This section describes the alternative center parameterization,
and how to convert it from and to SVG's endpoint parameterization.
</p>
<div class='math'>
<h3 id="ArcSyntax">Elliptical arc endpoint syntax</h3>
<p>An elliptical arc, as represented in the SVG path command,
is described by the following parameters in order:</p>
<p>(<var>x</var><sub>1</sub>, <var>y</var><sub>1</sub>) are the absolute coordinates of the
current point on the path, obtained from the last two
parameters of the previous path command.</p>
<p><var>r<sub>x</sub></var>
and <var>r<sub>y</sub></var>
are the radii of the ellipse (also known as its semi-major and
semi-minor axes).</p>
<p><var>φ</var> is the angle from the x-axis of the current coordinate
system to the x-axis of the ellipse.</p>
<p><var>f<sub>A</sub></var> is
the large arc flag, and is 0
if an arc spanning less than or equal to 180 degrees is chosen, or 1
if an arc spanning greater than 180 degrees is chosen.</p>
<p><var>f<sub>S</sub></var> is
the sweep flag, and is 0 if
the line joining center to arc sweeps through decreasing
angles, or 1 if it sweeps
through increasing angles.</p>
<p>(<var>x</var><sub>2</sub>, <var>y</var><sub>2</sub>) are the absolute coordinates of the
final point of the arc.</p>
<h3 id="ArcParameterizationAlternatives">Parameterization alternatives</h3>
<p>An arbitrary point (<var>x</var>, <var>y</var>) on the elliptical
arc can be described by the 2-dimensional matrix equation:</p>
(eq. 3.1)
<div role="math" aria-describedby="math-arbitrary-point-on-elliptical-arc">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mo>(</mo>
<mtable>
<mtr><mtd><mi>x</mi></mtd></mtr>
<mtr><mtd><mi>y</mi></mtd></mtr>
</mtable>
<mo>)</mo>
<mo>=</mo>
<mo>(</mo>
<mtable>
<mtr>
<mtd><mrow><mi>cos</mi><mo>⁡</mo><mi>φ</mi></mrow></mtd>
<mtd><mrow><mo>-</mo><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow></mtd>
</mtr>
<mtr>
<mtd><mrow><mi>sin</mi><mo>⁡</mo><mi>φ</mi></mrow></mtd>
<mtd><mrow><mi>cos</mi><mo>⁡</mo><mi>φ</mi></mrow></mtd>
</mtr>
</mtable>
<mo>)</mo>
<mo>·</mo>
<mo>(</mo>
<mtable>
<mtr>
<mtd><mrow><msub><mi>r</mi><mi>x</mi></msub><mo>⁢</mo><mi>cos</mi><mo>⁡</mo><mi>θ</mi></mrow></mtd>
</mtr>
<mtr>
<mtd><mrow><msub><mi>r</mi><mi>y</mi></msub><mo>⁢</mo><mi>sin</mi><mo>⁡</mo><mi>θ</mi></mrow></mtd>
</mtr>
</mtable>
<mo>)</mo>
<mo>+</mo>
<mo>(</mo>
<mtable>
<mtr><mtd><msub><mi>c</mi><mi>x</mi></msub></mtd></mtr>
<mtr><mtd><msub><mi>c</mi><mi>y</mi></msub></mtd></mtr>
</mtable>
<mo>)</mo>
</mrow>
</math>
<pre id="math-arbitrary-point-on-elliptical-arc">
x = rx*cos(θ)*cos(φ) - ry*sin(θ)*sin(φ) + cx
y = rx*cos(θ)*sin(φ) + ry*sin(θ)*cos(φ) + cy
</pre>
</div>
<p>(<var>c<sub>x</sub></var>, <var>c<sub>y</sub></var>) are
the coordinates of the center of the ellipse.</p>
<p><var>r<sub>x</sub></var> and <var>r<sub>y</sub></var>
are the radii of the ellipse (also known as its semi-major and
semi-minor axes).</p>
<div class="ready-for-wider-review">
<p><var>φ</var> is the angle from the
x-axis of the current coordinate system to the x-axis of the
ellipse.</p>
<p><var>θ</var> is the angle around the arc that the point
(<var>x</var>, <var>y</var>) lies at, and ranges from:</p>
<ul>
<li><var>θ</var><sub>1</sub> which is
the start angle of the elliptical arc prior to the stretch and
rotate operations.</li>
<li><var>θ</var><sub>2</sub> which is
the end angle of the elliptical arc prior to the stretch and
rotate operations.</li>
<li>Δ<var>θ</var> which is the difference between these
two angles.</li>
</ul>
</div>
<p>If one thinks of an ellipse as a circle that has been
stretched and then rotated, then <var>θ</var><sub>1</sub>,
<var>θ</var><sub>2</sub> and Δ<var>θ</var>
are the start angle, end angle and sweep angle, respectively
of the arc prior to the stretch and rotate operations.
This leads to an alternate parameterization which is common
among graphics APIs, which will be referred to as <em>center
parameterization</em>. In the next sections, formulas are
given for mapping in both directions between center
parameterization and endpoint parameterization.</p>
<h3 id="ArcConversionCenterToEndpoint">Conversion from center to endpoint parameterization</h3>
<p>Given the following variables:</p>
<p class='indented separated'>
<var>c<sub>x</sub></var>
<var>c<sub>y</sub></var>
<var>r<sub>x</sub></var>
<var>r<sub>y</sub></var>
<var>φ</var>
<var>θ</var><sub>1</sub>
Δ<var>θ</var>
</p>
<p>the task is to find:</p>
<p class='indented separated'>
<var>x</var><sub>1</sub>
<var>y</var><sub>1</sub>
<var>x</var><sub>2</sub>
<var>y</var><sub>2</sub>
<var>f<sub>A</sub></var>
<var>f<sub>S</sub></var>
</p>
<p>This can be achieved using the following formulas:</p>
<table style="width: 90%">
<tr>
<td>
<!--<img
alt="Equation 4.1"
src="images/implnote/arcs/image004.png" />-->
<div role="math">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<msub>
<mi>x</mi>
<mi>1</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>y</mi>
<mi>1</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<mn>cos</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo lspace='2px' rspace='2px'>-</mo>
<mn>sin</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>sin</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>cos</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo lspace='2px' rspace='2px'>⋅</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<msub>
<mi>r</mi>
<mi>x</mi>
</msub>
<mi> </mi>
<mn>cos</mn>
<msub>
<mi mathvariant='italic'>θ</mi>
<mi mathvariant='italic'>1</mi>
</msub>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>r</mi>
<mi>y</mi>
</msub>
<mi> </mi>
<mn>sin</mn>
<msub>
<mi mathvariant='italic'>θ</mi>
<mi mathvariant='italic'>1</mi>
</msub>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo lspace='2px' rspace='2px'>+</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<msub>
<mi>c</mi>
<mi>x</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>c</mi>
<mi>y</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
</math>
</div>
</td>
<td style="text-align: right">(eq. 4.1)</td>
</tr>
<tr>
<td>
<!--<img
alt="Equation 4.2"
src="images/implnote/arcs/image006.png" />-->
<div role="math">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<msub>
<mi>x</mi>
<mi mathvariant='italic'>2</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>y</mi>
<mi mathvariant='italic'>2</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<mn>cos</mn>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>φ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mo lspace='2px' rspace='2px'>-</mo>
<mn>sin</mn>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>φ</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mn>sin</mn>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>φ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>cos</mn>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>φ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo lspace='2px' rspace='2px'>⋅</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<msub>
<mi>r</mi>
<mi>x</mi>
</msub>
<mi> </mi>
<mn>cos</mn>
<mfenced open = '(' close = ')'>
<mrow>
<msub>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>θ</mi>
<mi mathvariant='italic'>1</mi>
</msub>
<mo lspace='2px' rspace='2px'>+</mo>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>Δ</mi>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>θ</mi>
</mrow>
</mfenced>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<msub>
<mi>r</mi>
<mi>y</mi>
</msub>
<mi> </mi>
<mn>sin</mn>
<mfenced open = '(' close = ')'>
<mrow>
<msub>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>θ</mi>
<mi mathvariant='italic'>1</mi>
</msub>
<mo lspace='2px' rspace='2px'>+</mo>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>Δ</mi>
<mi mathvariant='italic' fontfamily='Lucida Sans Unicode'>θ</mi>
</mrow>
</mfenced>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo lspace='2px' rspace='2px'>+</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<msub>
<mi>c</mi>
<mi>x</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>c</mi>
<mi>y</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
</math>
</div>
</td>
<td style="text-align: right">(eq. 4.2)</td>
</tr>
<tr>
<td>
<!--<img
alt="Equation 4.3"
src="images/implnote/arcs/image008.png" />-->
<div role="math">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub>
<mi>f</mi>
<mi>A</mi>
</msub>
<mo>=</mo>
<mfenced open = '{' close = ''>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mi mathvariant='italic' > </mi>
<mi mathvariant='italic' > </mi>
<mtext>if</mtext>
<mi mathvariant='italic' > </mi>
<mi mathvariant='italic' > </mi>
<mfenced open = '|' close = '|'>
<mrow>
<mi>Δ</mi>
<mi mathvariant='italic' >θ</mi>
</mrow>
</mfenced>
<mo mathvariant='italic' >></mo>
<mn>180</mn>
<mi mathvariant='normal' >°</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtext>0</mtext>
<mi mathvariant='italic' > </mi>
<mi mathvariant='italic' > </mi>
<mtext>if</mtext>
<mi mathvariant='italic' > </mi>
<mi mathvariant='italic' > </mi>
<mfenced open = '|' close = '|'>
<mrow>
<mi>Δ</mi>
<mi mathvariant='italic' >θ</mi>
</mrow>
</mfenced>
<mo lspace='3px' rspace='3px'>≤</mo>
<mn>180</mn>
<mi mathvariant='normal' >°</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
</math>
</div>
</td>
<td style="text-align: right">(eq. 4.3)</td>
</tr>
<tr>
<td>
<!--<img
alt="Equation 4.4"
src="images/implnote/arcs/image010.png" />-->
<div role="math">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msub>
<mi>f</mi>
<mi>S</mi>
</msub>
<mo>=</mo>
<mfenced open = '{' close = ''>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<mn>1</mn>
<mi mathvariant='italic' > </mi>
<mtext>if</mtext>
<mi mathvariant='italic' > </mi>
<mi>Δ</mi>
<mi mathvariant='italic' >θ</mi>
<mn > </mn>
<mo lspace='0px' rspace='0px'>></mo>
<mtext > </mtext>
<mn>0</mn>
<mi mathvariant='normal' >°</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mtext>0</mtext>
<mi mathvariant='italic' > </mi>
<mtext>if</mtext>
<mi mathvariant='italic' > </mi>
<mi>Δ</mi>
<mi mathvariant='italic' >θ</mi>
<mn > </mn>
<mo lspace='0px' rspace='0px'><</mo>
<mtext > </mtext>
<mn>0</mn>
<mi mathvariant='normal' >°</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
</math>
</div>
</td>
<td style="text-align: right">(eq. 4.4)</td>
</tr>
</table>
<h3 id="ArcConversionEndpointToCenter">Conversion from endpoint to center parameterization</h3>
<p>Given the following variables:</p>
<p class='indented separated'>
<var>x</var><sub>1</sub>
<var>y</var><sub>1</sub>
<var>x</var><sub>2</sub>
<var>y</var><sub>2</sub>
<var>f<sub>A</sub></var>
<var>f<sub>S</sub></var>
<var>r<sub>x</sub></var>
<var>r<sub>y</sub></var>
<var>φ</var>
</p>
<p>the task is to find:</p>
<p class='indented separated'>
<var>c<sub>x</sub></var>
<var>c<sub>y</sub></var>
<var>θ</var><sub>1</sub>
Δ<var>θ</var>
</p>
<p>The equations simplify after a translation which
places the origin at the midpoint of the line joining
(<var>x</var><sub>1</sub>, <var>y</var><sub>1</sub>) to
(<var>x</var><sub>2</sub>, <var>y</var><sub>2</sub>), followed by
a rotation to line up the coordinate axes with the axes of the ellipse.
All transformed coordinates will be written with primes. They are
computed as intermediate values on the way toward finding the required
center parameterization variables. This procedure consists of the
following steps:</p>
<ul>
<li>
<p><em>Step 1: Compute</em> (<var>x</var><sub>1</sub>′, <var>y</var><sub>1</sub>′)</p>
<table style="width: 90%"
>
<tr>
<td><!--<img
alt="Equation 5.1"
src="images/implnote/arcs/image012.png" />-->
<div role="math">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<msup>
<msub>
<mi>x</mi>
<mi mathvariant='italic'>1</mi>
</msub>
<mo mathvariant='italic' fontfamily='Lucida Sans Unicode'>′</mo>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<msub>
<mi>y</mi>
<mi mathvariant='italic'>1</mi>
</msub>
<mo mathvariant='italic' fontfamily='Lucida Sans Unicode'>′</mo>
</msup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<mn>cos</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>sin</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mn>sin</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
<mtd>
<mrow>
<mn>cos</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo lspace='2px' rspace='2px'>⋅</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>x</mi>
<mi mathvariant='italic'>1</mi>
</msub>
<mo mathvariant='italic'>-</mo>
<msub>
<mi>x</mi>
<mi mathvariant='italic'>2</mi>
</msub>
</mrow>
<mi mathvariant='italic'>2</mi>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>y</mi>
<mi mathvariant='italic'>1</mi>
</msub>
<mo mathvariant='italic'>-</mo>
<msub>
<mi>y</mi>
<mi mathvariant='italic'>2</mi>
</msub>
</mrow>
<mi mathvariant='italic'>2</mi>
</mfrac>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
</math>
</div>
</td>
<td style="text-align: right">(eq. 5.1)</td>
</tr>
</table>
</li>
<li>
<p><em>Step 2: Compute</em> (<var>c<sub>x</sub></var>′, <var>c<sub>y</sub></var>′)</p>
<table style="width: 90%"
>
<tr>
<td><!--<img
alt="Equation 5.2"
src="images/implnote/arcs/image014.png" />-->
<div role="math">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<msup>
<msub>
<mi>c</mi>
<mi>x</mi>
</msub>
<mo>′</mo>
</msup>
</mtd>
</mtr>
<mtr>
<mtd>
<msup>
<msub>
<mi>c</mi>
<mi>y</mi>
</msub>
<mo>′</mo>
</msup>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mo lspace='2px' rspace='2px'>±</mo>
<msqrt>
<mfrac>
<mrow>
<msup>
<msub>
<mi>r</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<mtext> </mtext>
<msup>
<msub>
<mi>r</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>r</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<msup>
<mfenced open = '(' close = ')'>
<msup>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>′</mo>
</msup>
</mfenced>
<mn>2</mn>
</msup>
<mo>-</mo>
<msup>
<msub>
<mi>r</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
<msup>
<mfenced open = '(' close = ')'>
<msup>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>′</mo>
</msup>
</mfenced>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<msub>
<mi>r</mi>
<mi>x</mi>
</msub>
<mn>2</mn>
</msup>
<msup>
<mfenced open = '(' close = ')'>
<msup>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>′</mo>
</msup>
</mfenced>
<mn>2</mn>
</msup>
<mo>+</mo>
<msup>
<msub>
<mi>r</mi>
<mi>y</mi>
</msub>
<mn>2</mn>
</msup>
<msup>
<mfenced open = '(' close = ')'>
<msup>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>′</mo>
</msup>
</mfenced>
<mn>2</mn>
</msup>
</mrow>
</mfrac>
</msqrt>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mfrac>
<mrow>
<msub>
<mi>r</mi>
<mi>x</mi>
</msub>
<mtext> </mtext>
<msup>
<msub>
<mi>y</mi>
<mn>1</mn>
</msub>
<mo>′</mo>
</msup>
</mrow>
<msub>
<mi>r</mi>
<mi>y</mi>
</msub>
</mfrac>
</mtd>
</mtr>
<mtr>
<mtd>
<mrow>
<mo>-</mo>
<mfrac>
<mrow>
<msub>
<mi>r</mi>
<mi>y</mi>
</msub>
<mtext> </mtext>
<msup>
<msub>
<mi>x</mi>
<mn>1</mn>
</msub>
<mo>′</mo>
</msup>
</mrow>
<msub>
<mi>r</mi>
<mi>x</mi>
</msub>
</mfrac>
</mrow>
</mtd>
</mtr>
</mtable>
</mfenced>
</mrow>
</math>
</div>
</td>
<td style="text-align: right">(eq. 5.2)</td>
</tr>
</table>
<p>where the + sign is chosen if <var>f<sub>A</sub></var> ≠ <var>f<sub>S</sub></var>, and
the − sign is chosen if <var>f<sub>A</sub></var> = <var>f<sub>S</sub></var>.</p>
</li>
<li>
<p><em>Step 3: Compute</em> (<var>c<sub>x</sub></var>, <var>c<sub>y</sub></var>)
<em>from</em> (<var>c<sub>x</sub></var>′, <var>c<sub>y</sub></var>′)</p>
<table style="width: 90%"
>
<tr>
<td><!--<img
alt="Equation 5.3"
src="images/implnote/arcs/image016.png" />-->
<div role="math">
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<msub>
<mi>c</mi>
<mi>x</mi>
</msub>
</mtd>
</mtr>
<mtr>
<mtd>
<msub>
<mi>c</mi>
<mi>y</mi>
</msub>
</mtd>
</mtr>
</mtable>
</mfenced>
<mo>=</mo>
<mfenced open = '(' close = ')'>
<mtable rowalign='center'>
<mtr>
<mtd>
<mrow>
<mn>cos</mn>
<mi mathvariant='italic'>φ</mi>
</mrow>
</mtd>
<mtd>