<?xml version="1.0" encoding="UTF-8"?>
<!--

Documentation for LCL (Lazarus Component Library) and LazUtils (Lazarus 
Utilities) are published under the Creative Commons Attribution-ShareAlike 4.0 
International public license.

https://creativecommons.org/licenses/by-sa/4.0/legalcode.txt
https://gitlab.com/freepascal.org/lazarus/lazarus/-/blob/main/docs/cc-by-sa-4-0.txt

Copyright (c) 1997-2025, by the Lazarus Development Team.

-->
<fpdoc-descriptions>
<package name="lazutils">
<!--
====================================================================
GraphMath
====================================================================
-->
<module name="GraphMath">
<short>
A set of mathematical helper routines to simply cross-platform canvas
drawing.
</short>
<descr>
<p>
<file>graphmath.pp</file> contains math helper routines for use for
graphics drawing. It is used to simply cross-platform canvas drawing
operations. <file>graphmath.pp</file> is part of the <file>LazUtils</file> 
package.
</p>
</descr>

<!-- unresolved references -->
<element name="Types"/>
<element name="Classes"/>
<element name="SysUtils"/>
<element name="Math"/>
<element name="GraphType"/>
<element name="LazUtilities"/>

<element name="TFloatPoint">
<short>
<var>TFloatPoint</var> is an extended precision record specifying the X and
Y coordinates of a point in a graphic environment.
</short>
<descr/>
<seealso/>
</element>
<element name="TFloatPoint.X">
<short>
Horizontal position for the coordinate.
</short>
</element>
<element name="TFloatPoint.Y">
<short>
Vertical position for the coordinate.
</short>
</element>

<element name="TBezier">
<short>
Array type used to store the coordinates for Bezier control points as 
floating point values.
</short>
<descr>
<p>
<var>TBezier</var> allows up to 4 coordinates to be specified which represent 
the control points for the parametric curve. Each coordinate is implemented 
using the TFloatPoint type, and stored as elements in the array.
</p>
<p>
TBezier is the type returned by the Bezier function. The type is passed as an 
argument to routines like: Arc2Bezier, Bezier2Polyline, BezierMidPoint, and 
SplitBezier.
</p>
</descr>
<seealso>
<link id="Bezier"/>
<link id="Arc2Bezier"/>
<link id="Bezier2Polyline"/>
<link id="BezierMidPoint"/>
<link id="SplitBezier"/>
</seealso>
</element>

<element name="PPoint">
<short>
Pointer to the TPoint type.
</short>
<descr/>
<seealso>
<link id="#rtl.classes.TPoint">TPoint</link>
</seealso>
</element>

<element name="Angles2Coords">
<short>
Converts an Eccentric Angle and an Angle-Length, into the coordinates for
the Start and End radial Points.
</short>
<descr>
<p>
Use <var>Angles2Coords</var> to convert an Eccentric (Radial) angle and an
angle-length, such as are used in X-Windows and GTK, into the coordinates
for the Start and End radial Points. Like those used in the Arc, Pie, and
Chord routines from the Windows API.
</p>
<p>
The angles are specified in 1/16th of a degree. For example, a full circle
equals 5760 (16*360).
</p>
<p>
Positive values in Angle and AngleLength mean counter-clockwise, while
negative values mean clockwise direction. Zero degrees is at the 3
o&apos;clock position.
</p>
</descr>
<seealso/>
</element>
<element name="Angles2Coords.X">
<short/>
</element>
<element name="Angles2Coords.Y">
<short/>
</element>
<element name="Angles2Coords.Width">
<short/>
</element>
<element name="Angles2Coords.Height">
<short/>
</element>
<element name="Angles2Coords.Angle1">
<short/>
</element>
<element name="Angles2Coords.Angle2">
<short/>
</element>
<element name="Angles2Coords.SX">
<short/>
</element>
<element name="Angles2Coords.SY">
<short/>
</element>
<element name="Angles2Coords.EX">
<short/>
</element>
<element name="Angles2Coords.EY">
<short/>
</element>

<element name="Arc2Bezier">
<short>
Converts an Arc and ArcLength into a Bezier Approximation of the Arc.
</short>
<descr>
<p>
Use <var>Arc2Bezier</var> to convert an elliptical arc into a Bezier 
approximation of the arc.
</p>
<p>
The arc is defined by a rectangle which bounds the full ellipse. The rectangle 
has the specified Width and Height, its top/left corner is at the point 
<var>X</var>, <var>Y</var>. The start of the arc is given by the eccentric 
<var>Angle1</var>, and its length is given by eccentric <var>Angle2</var>. 
Both angles are expressed in 1/16th of a degree, a full circle, for example, 
equals to 5760 (16*360). A positive value of Angle2 means counter-clockwise 
while a negative value means clockwise direction. Zero degrees is at the 3 
o&apos;clock position.
</p>
<p>
The <var>Rotation</var> parameter accepts an angle for a rotated ellipse - for 
a non-rotated ellipse this value would be 0, or 360*16.
</p>
<p>
If the angle length is greater than 90 degrees or is equal to 0, the procedure 
automatically exits, as Bezier cannot accurately approximate any angle greater 
then 90 degrees, and in fact for best results no angle greater than 45 should 
be converted, instead an array of Beziers should be created, each Bezier 
describing a portion of the total arc no greater than 45 degrees.
</p>
</descr>
<seealso/>
</element>
<element name="Arc2Bezier.X">
<short>
Left coordinate for the upper/left corner of the rectangle where the arc is 
located.
</short>
</element>
<element name="Arc2Bezier.Y">
<short>
Top coordinate for the upper/left corner of the rectangle where the arc is 
located.
</short>
</element>
<element name="Arc2Bezier.Width">
<short>
Width for the rectangle where the arc is located.
</short>
</element>
<element name="Arc2Bezier.Height">
<short>
Height for the rectangle where the arc is located.
</short>
</element>
<element name="Arc2Bezier.Angle1">
<short>
Eccentric angle for the start of the arc.
</short>
</element>
<element name="Arc2Bezier.Angle2">
<short>
Eccentric angle which defines the length of the arc.
</short>
</element>
<element name="Arc2Bezier.Rotation">
<short>
Rotation angle for the arc specified in 1/16ths of a degree.
</short>
</element>
<element name="Arc2Bezier.Points">
<short>
TBezier instance where the array of floating point values for the bezier 
approximation is stored.
</short>
</element>

<element name="Bezier">
<short>
Gets a TBezier instance representing the specified Bezier control points.
</short>
<descr/>
<seealso/>
</element>
<element name="Bezier.Result">
<short>
TBezier instance with the values in C1, C2, C3, and C4.
</short>
</element>
<element name="Bezier.C1">
<short>
Control point on the parametric curve. C1 is an endpoint.
</short>
</element>
<element name="Bezier.C2">
<short>
Control point on the parametric curve. C2 is a directional control point for 
a Quadratic or Cubic Bezier.
</short>
</element>
<element name="Bezier.C3">
<short>
Control point on the parametric curve. C3 is a directional control point for 
a Cubic Bezier.
</short>
</element>
<element name="Bezier.C4">
<short>
Control point on the parametric curve. C4 is an endpoint.
</short>
</element>

<element name="Bezier2Polyline">
<short>
<var>Bezier2Polyline</var> - convert a 4-Point Bezier into a Pointer Array
of TPoint and a Count variable.
</short>
<descr>
<p>
Use BezierToPolyline to convert a 4-Point Bezier into a Pointer Array of
TPoint and a Count variable which can then be used within either a
Polyline, or Polygon routine. It is primarily for use within
PolyBezier2Polyline.
</p>
<p>
If Points is not initialized or Count is less then 0, it is set to nil and
the array starts at 0, otherwise it tries to append points to the array
starting at Count. Points should ALWAYS be Freed when done by calling to
ReallocMem(Points, 0) or FreeMem.
</p>
</descr>
<seealso/>
</element>
<element name="Bezier2Polyline.Bezier">
<short/>
</element>
<element name="Bezier2Polyline.Points">
<short/>
</element>
<element name="Bezier2Polyline.Count">
<short/>
</element>

<element name="BezierArcPoints">
<short>
Converts an Arc and ArcLength into a Pointer to an Array of TPoints for use 
with Polyline or Polygon.
</short>
<descr>
<p>
Use <var>BezierArcPoints</var> to convert an elliptical arc to a pointer to an 
array of TPoints for use with Polyline or Polygon.
</p>
<p>
The arc is defined by a rectangle which bounds the full ellipse. The
rectangle has the specified Width and Height, its top/left corner is at the
point <var>X</var>, <var>Y</var>. The start of the arc is given by the 
eccentric <var>Angle1</var>, and its length is given by eccentric 
<var>Angle2</var>. Both angles are expressed in 1/16th of a degree. 
A full circle, for example, equals to 5760 (16*360). A positive value in  
Angle2 means counter-clockwise while a negative value means clockwise 
direction. Zero degrees is at the 3 o&apos;clock position.
</p>
<p>
The <var>Rotation</var> parameter accepts a rotation angle for a rotated 
ellipse - for a non-rotated ellipse this value would be 0, or 360*16.
</p>
<p>
The result is an approximation based on one or more Beziers. If the angle 
length is greater than 90 degrees, the procedure calls PolyBezierArcPoints, 
otherwise it converts the angles into a Bezier by calling Arc2Bezier which 
then is converted to an array of points by calling Bezier2PolyLine.
</p>
<p>
If <var>Points</var> is not initialized or Count is less than 0, it is set to 
<b>Nil</b> and the array starts at 0, otherwise it tries to append points to 
the array starting at Count. Points should <b>ALWAYS</b> be free'd when done 
by calling ReallocMem(Points, 0) or FreeMem.
</p>
</descr>
<seealso/>
</element>
<element name="BezierArcPoints.X">
<short>
Left coordinate for the rectangle where the ellipse is positioned.
</short>
</element>
<element name="BezierArcPoints.Y">
<short>
Top coordinate for the rectangle where the ellipse is positioned.
</short>
</element>
<element name="BezierArcPoints.Width">
<short>
Width for the rectangle where the ellipse is positioned.
</short>
</element>
<element name="BezierArcPoints.Height">
<short>
Height for the rectangle where the ellipse is positioned.
</short>
</element>
<element name="BezierArcPoints.Angle1">
<short>
Eccentric angle which defines the start of the arc.
</short>
</element>
<element name="BezierArcPoints.Angle2">
<short>
Eccentric angle which defines the length of the arc.
</short>
</element>
<element name="BezierArcPoints.Rotation">
<short>
Rotation angle for the ellipse specified in 1/16ths of a degree.
</short>
</element>
<element name="BezierArcPoints.Points">
<short>
Returns a pointer to the array of points with the approximation based on one 
or more Beziers, or Nil when Count is less than 0.
</short>
</element>
<element name="BezierArcPoints.Count">
<short>
Returns the number of points stored in the Points argument.
</short>
</element>

<element name="BezierMidPoint">
<short>
<var>BezierMidPoint</var> - get the Mid-Point of any 4-Point Bezier. It is
primarily for use in SplitBezier.
</short>
<descr/>
<seealso/>
</element>
<element name="BezierMidPoint.Result">
<short/>
</element>
<element name="BezierMidPoint.Bezier">
<short/>
</element>

<element name="CenterPoint">
<short>
<var>CenterPoint</var> - get the Center-Point of any rectangle. It is
primarily for use with, and in, other routines such as Quadrant, and
RadialPoint.
</short>
<descr/>
<seealso/>
</element>
<element name="CenterPoint.Result">
<short/>
</element>
<element name="CenterPoint.Rect">
<short/>
</element>


<element name="CalculateLeftTopWidthHeight">
<short>
Calculates the values for the output variables in Left, Top, Width, and Height.
</short>
<descr>
<p>
<var>CalculateLeftTopWidthHeight</var> checks values in the X1, X2, Y1, and Y2 
arguments and sets the values for the Left, Top, Width, and Height output 
parameters accordingly.
</p>
<dl>
<dt>Left</dt>
<dd>
Set to the smaller of the two values in X1 and X2.
</dd>
<dt>Width</dt>
<dd>
Set to the difference between the X1 and X2 values. The value will be a 
positive integer value or zero (0).
</dd>
<dt>Top</dt>
<dd>
Set to the smaller of the two values in Y1 and Y2.
</dd>
<dt>Height</dt>
<dd>
Set to the difference between Y1 and Y2. The value will be a positive integer 
value or zero (0).
</dd>
</dl>
<p>
Used in the implementation of the Rectangle and Ellipse routines for the GTK 
widgetset.
</p>
</descr>
<version>
Added in LazUtils version 3.0.
</version>
<seealso/>
</element>
<element name="CalculateLeftTopWidthHeight.X1">
<short>
Horizontal coordinate examined in the routine.
</short>
</element>
<element name="CalculateLeftTopWidthHeight.Y1">
<short>
Vertical coordinate examined in the routine.
</short>
</element>
<element name="CalculateLeftTopWidthHeight.X2">
<short>
Horizontal coordinate examined in the routine.
</short>
</element>
<element name="CalculateLeftTopWidthHeight.Y2">
<short>
Vertical coordinate examined in the routine.
</short>
</element>
<element name="CalculateLeftTopWidthHeight.Left">
<short>
Returns the left coordinate for the specified values.
</short>
</element>
<element name="CalculateLeftTopWidthHeight.Top">
<short>
Returns the top coordinate for the specified values.
</short>
</element>
<element name="CalculateLeftTopWidthHeight.Width">
<short>
Returns the width for the specified horizontal coordinates.
</short>
</element>
<element name="CalculateLeftTopWidthHeight.Height">
<short>
Returns the height for the specified vertical coordinates.
</short>
</element>

<element name="Coords2Angles">
<short>
<var>Coords2Angles</var> - convert the coords for Start and End Radial-
Points into an Eccentric counter clockwise Angle and an Angle-Length.
</short>
<descr>
<p>
Use Coords2Angles to convert the coords for Start and End Radial-Points,
such as are used in the Windows API Arc Pie and Chord routines, into an
Eccentric (aka Radial) counter clockwise Angle and an Angle-Length, such as
are used in X-Windows and GTK.
</p>
<p>
The angles angle1 and angle2 are returned in 1/16th of a degree. For
example, a full circle equals 5760 (16*360).
</p>
<p>
Zero degrees is at the 3 o&apos;clock position.
</p>
</descr>
<seealso/>
</element>
<element name="Coords2Angles.X">
<short/>
</element>
<element name="Coords2Angles.Y">
<short/>
</element>
<element name="Coords2Angles.Width">
<short/>
</element>
<element name="Coords2Angles.Height">
<short/>
</element>
<element name="Coords2Angles.SX">
<short/>
</element>
<element name="Coords2Angles.SY">
<short/>
</element>
<element name="Coords2Angles.EX">
<short/>
</element>
<element name="Coords2Angles.EY">
<short/>
</element>
<element name="Coords2Angles.Angle1">
<short/>
</element>
<element name="Coords2Angles.Angle2">
<short/>
</element>

<element name="Distance">
<short>
Gets the distance between two points, or the distance of a point from a 
specified line.
</short>
<descr>
<p>  
<var>Distance</var> is an overloaded function with variants that operate on 
either two point coordinates, or on a point and a line defined by two 
additional points values. The Distance() function is used primarily for 
internal purposes (such as in Bezier2PolyLine and EccentricAngle) 
but can be used for any purpose.
</p>
<p>
The return value is an Extended type with the calculated distance between the 
argument values. The return value is always a positive value.
</p>
<p>
The variants using two point arguments (TPoint or TFloatPoint) calculates the 
length of a straight line between the coordinates in PT1 and PT2 using the 
Pythagorean theorem. The distance between identical points is always zero (0). 
</p>
<p>
The variant with three TFloatPoint arguments calculates the distance between 
the point Pt and the line represented by the points in SP and EP using 
Euclidean geometry. The distance is derived by finding the length of an 
imaginary line between Pt and the closest point that intersects the slope of 
the line in SP and EP. The distance for a point which lies on the defined line 
is always zero (0).
</p>
</descr>
<seealso/>
</element>
<element name="Distance.Result">
<short>
Straight-line distance between the specified coordinates or objects.
</short>
</element>
<element name="Distance.PT1">
<short>
Starting point for the calculated distance.
</short>
</element>
<element name="Distance.PT2">
<short>
Ending point for the calculated distance.
</short>
</element>
<element name="Distance.Pt">
<short>
Fixed point for the calculated distance.
</short>
</element>
<element name="Distance.SP">
<short>
Starting point for the line used in the distance calculation.
</short>
</element>
<element name="Distance.WP">
<short>
Ending point for the line used in the distance calculation.
</short>
</element>

<element name="EccentricAngle">
<short>
<var>EccentricAngle</var> - get the Eccentric Angle of a given point on any
non-rotated ellipse.
</short>
<descr>
<p>
Use EccentricAngle to get the Eccentric( aka Radial ) Angle of a given
point on any non-rotated ellipse. It is primarily for use in Coords2Angles.
The result is in 1/16th of a degree. For example, a full circle equals 5760
(16*360). Zero degrees is at the 3 o&apos;clock position.
</p>
</descr>
<seealso/>
</element>
<element name="EccentricAngle.Result">
<short/>
</element>
<element name="EccentricAngle.PT">
<short/>
</element>
<element name="EccentricAngle.Rect">
<short/>
</element>

<element name="EllipseRadialLength">
<short>
<var>EllipseRadialLength</var> - Radial-Length of non-rotated ellipse at
any given Eccentric Angle.
</short>
<descr>
<p>
Use EllipseRadialLength to get the Radial-Length of non-rotated ellipse at
any given Eccentric( aka Radial ) Angle. It is primarily for use in other
routines such as RadialPoint. The Eccentric angle is in 1/16th of a degree.
For example, a full circle equals 5760 (16*360). Zero degrees is at the 3
o&apos;clock position.
</p>
</descr>
<seealso/>
</element>
<element name="EllipseRadialLength.Result">
<short/>
</element>
<element name="EllipseRadialLength.Rect">
<short/>
</element>
<element name="EllipseRadialLength.EccentricAngle">
<short/>
</element>

<element name="EllipsePolygon">
<short>
Gets an array of points for a polygon which approximates an ellipse in the 
specified rectangle.
</short>
<descr>
<p>
<var>EllipsePolygon</var> is a <var>TPointArray</var> function used to 
calculate the array of points which approximate an ellipse bounded by the 
rectangle specified in the ARect argument. In EllipsePolygon, the ellipse is 
aligned to X and Y axes in the rectangle. It calculates the center point, 
radii, and diameter for the ellipse using ARect as well as the minimum number 
of points needed such that the distance between the edges of the rectangle and 
the mathematical circle is less than the rounding error (0.5) and a smoother 
stepping value of 0.4. The points for the polygon are calculated and stored in 
the return value.
</p>
</descr>
<version>
Added in LCL version 4.0.  
</version>
<seealso/>
</element>
<element name="EllipsePolygon.Result">
<short>
Array with the points for the polygon.
</short>
</element>
<element name="EllipsePolygon.ARect">
<short>
Rectangle instance with the bounds for an X- and Y-aligned ellipse.
</short>
</element>

<element name="FloatPoint">
<short>
<var>FloatPoint</var> - it is essentially like Classes.Point in use, except
that it accepts Extended Parameters. It is Primarily for use with and in
Bezier routines.
</short>
<descr/>
<seealso/>
</element>
<element name="FloatPoint.Result">
<short/>
</element>
<element name="FloatPoint.AX">
<short/>
</element>
<element name="FloatPoint.AY">
<short/>
</element>

<element name="LineEndPoint">
<short>
<var>LineEndPoint</var> - get the End-Point of a line of any given Length
at any given angle with any given Start-Point.
</short>
<descr>
<p>
Use LineEndPoint to get the End-Point of a line of any given Length at any
given angle with any given Start-Point. It is primarily for use in other
routines such as RadialPoint. The angle is in 1/16th of a degree. For
example, a full circle equals 5760 (16*360). Zero degrees is at the 3
o&apos;clock position.
</p>
</descr>
<seealso/>
</element>
<element name="LineEndPoint.Result">
<short/>
</element>
<element name="LineEndPoint.StartPoint">
<short/>
</element>
<element name="LineEndPoint.Angle">
<short/>
</element>
<element name="LineEndPoint.Length">
<short/>
</element>

<element name="MakeMinMax">
<short>
Ensures that the i1 argument is the smaller of the two Integer values.
</short>
<descr>
<p>
<var>MakeMinMax</var> swaps the values in i1 and i2 variable parameters when 
i1 is larger than i2. No actions are performed in the routine if i1 is smaller 
than i2, or the arguments have the same value.
</p>
</descr>
<version>
Added in LazUtils version 3.0.
</version>
<seealso/>
</element>
<element name="MakeMinMax.i1">
<short>
Integer value compared (and potentially updated) in the routine.
</short>
</element>
<element name="MakeMinMax.i2">
<short>
Integer value compared (and potentially updated) in the routine.
</short></element>

<element name="MoveRect">
<short>
Moves the specified rectangle to the origin in the x and y arguments.
</short>
<descr>
<p>
<var>MoveRect</var> sets the Left and Top members in ARect to the values 
specified in x and y (respectively). The Right and Bottom members in ARect are 
updated to reflect the relative distance from the original Top and Left on 
entry.
</p>
</descr>
<version>
Added in LazUtils version 3.0.
</version>
<seealso/>
</element>
<element name="MoveRect.ARect">
<short>
TRect instance updated in the routine.
</short>
</element>
<element name="MoveRect.x">
<short>
New position for the Top coordinate in the rectangle.
</short>
</element>
<element name="MoveRect.y">
<short>
New position for the Left coordinate in the rectangle.
</short>
</element>

<element name="MoveRectToFit">
<short>
Moves and potentially resizes a rectangle to fit within the specified target 
rectangle.
</short>
<descr>
<p>
<var>ARect</var> is the TRect instance repositioned in the routine. Values in 
the Left, Right, Top, and Bottom members may be updated in the routine if the 
rectangle is not located within the bounds for the target rectangle.
</p>
<p>
<var>MaxRect</var> is the TRectangle instance where ARect is repositioned. It 
also establishes the maximum size for ARect after it has been repositioned. If 
aRect is larger than MaxRect, ARect is resized to fit with in the constraints 
in MaxRect.
</p>
<p>
MoveRectToFit is used in the implementation for the DoDock method in TControl.
</p>
</descr>
<version>
Added in LazUtils version 3.0.
</version>
<seealso/>
</element>
<element name="MoveRectToFit.ARect">
<short>
TRect instance moved and optionally resized in the routine.
</short>
</element>
<element name="MoveRectToFit.MaxRect">
<short>
TRect instance where the rectangle is relocated. It also specifies the maximum 
size for the relocated rectangle.
</short>
</element>

<element name="SameRect">
<short>
Indicates whether member in the specified rectangles have the same values.
</short>
<descr>
<p>
<var>SameRect</var> is <var>Boolean</var> function used to determine whether 
the <var>TRect</var> instances R1 and R2 represent the same rectangular areas. 
It compares the values for the Left, Right, Top, and Bottom members in the 
TRect instances. The return value is <b>True</b> when members in R1 have the 
same values as the corresponding members in R2.
</p>
<p>
SameRect replaces the deprecated CompareRect routine in the 
<file>lclproc.pas</file> unit in the LCL.
</p>
</descr>
<version>
Added in LazUtils version 3.99.
</version>
<seealso>
<link id="#lcl.lclproc.CompareRect">CompareRect</link>
<link id="#rtl.types.PRect">PRect</link>
<link id="#rtl.types.TRect">TRect</link>
</seealso>
</element>
<element name="SameRect.Result">
<short>
True when the specified rectangles have the same values in the members.
</short>
</element>
<element name="SameRect.R1">
<short>
Pointer to a TRect instance examined in the routine.
</short>
</element>
<element name="SameRect.R2">
<short>
Pointer to a TRect instance examined in the routine.
</short>
</element>

<element name="PolyBezier2Polyline">
<short>
<var>PolyBezier2Polyline</var> - convert an array of 4-Point Bezier into a
Pointer Array of TPoint and a Count variable.
</short>
<descr>
<p>
Use BezierToPolyline to convert an array of 4-Point Bezier into a Pointer
Array of TPoint and a Count variable which can then be used within either a
Polyline, or Polygon routine. Points is automatically initialized, so any
existing information is lost, and the array starts at 0. Points should
ALWAYS be Freed when done by calling to ReallocMem(Points, 0).
</p>
</descr>
<seealso/>
</element>
<element name="PolyBezier2Polyline.Beziers">
<short/>
</element>
<element name="PolyBezier2Polyline.Points">
<short/>
</element>
<element name="PolyBezier2Polyline.Count">
<short/>
</element>

<element name="PolyBezierArcPoints">
<short>
<var>PolyBezierArcPoints</var> - convert an Arc and ArcLength into a
Pointer Array of TPoints for use with Polyline or Polygon.
</short>
<descr>
<p>
Use PolyBezierArcPoints to convert an arc between two angles Angle1 and Angle2
into a pointer array of TPoints for use with Polyline or Polygon.
The Rotation parameter accepts a rotation angle for a rotated ellipse - for
a non-rotated ellipse this value would be 0, or 360*16.
</p>
<p>
The result is an approximation based on 1 or more Beziers. If the angle length
is greater than 45*16 degrees, it recursively breaks the arc into arcs of
45*16 degrees or less, and converts them into beziers with BezierArcPoints.
The angles are 1/16th of a degree. For example, a full circle equals
5760 (16*360).
</p>
<p>
Positive values in Angle1 and Angle2 mean counter-clockwise while negative
values mean clockwise direction. Zero degrees is at the 3'o clock position.
Points is automatically initialized, so any existing information is lost,
and the array starts at 0. Points should ALWAYS be freed when done by calling
to ReallocMem(Points, 0).
</p>
</descr>
<seealso/>
</element>
<element name="PolyBezierArcPoints.X">
<short/>
</element>
<element name="PolyBezierArcPoints.Y">
<short/>
</element>
<element name="PolyBezierArcPoints.Width">
<short/>
</element>
<element name="PolyBezierArcPoints.Height">
<short/>
</element>
<element name="PolyBezierArcPoints.Angle1">
<short/>
</element>
<element name="PolyBezierArcPoints.Angle2">
<short/>
</element>
<element name="PolyBezierArcPoints.Rotation">
<short/>
</element>
<element name="PolyBezierArcPoints.Points">
<short/>
</element>
<element name="PolyBezierArcPoints.Count">
<short/>
</element>

<element name="Quadrant">
<short>
Determine the <var>Quadrant</var> of any point, given the Center.
</short>
<descr>
<p>
Use Quadrant to determine the Quadrant of any point, given the Center. It
is primarily for use in other routines such as EccentricAngle. A result of
1-4 represents the primary 4 quadrants. A result of 5-8 means the point
lies on one of the Axis, 5 = -Y Axis, 6 = +X Axis, 7 = +Y Axis, and 8 = -X
Axis. A result of -1 means that it does not fall in any quadrant, that is,
it is the Center.
</p>
</descr>
<seealso/>
</element>
<element name="Quadrant.Result">
<short/>
</element>
<element name="Quadrant.PT">
<short/>
</element>
<element name="Quadrant.Center">
<short/>
</element>

<element name="RadialPoint">
<short>
Get the <var>RadialPoint</var> at any given Eccentric angle on any non-
rotated ellipse.
</short>
<descr>
<p>
Use RadialPoint to get the Radial-Point at any given Eccentric(aka Radial)
angle on any non-rotated ellipse. It is primarily for use in Angles2Coords.
The EccentricAngle is in 1/16th of a degree. For example, a full circle
equals 5760 (16*360). Zero degrees is at the 3 o&apos;clock position.
</p>
</descr>
<seealso/>
</element>
<element name="RadialPoint.Result">
<short/>
</element>
<element name="RadialPoint.EccentricAngle">
<short/>
</element>
<element name="RadialPoint.Rect">
<short/>
</element>

<element name="RotatePoint">
<short>
Rotates a point around the origin by the specified angle (in radians).
</short>
<descr>
<p>
Rotates a point around the origin (0,0) by the angle in AAngle. The angle is
in radians and positive for counter-clockwise rotation.
Note that y points downwards.
</p>
</descr>
<seealso/>
</element>
<element name="RotatePoint.Result">
<short>TPoint with the coordinates after rotation.</short>
</element>
<element name="RotatePoint.APoint">
<short>TPoint with coordinates rotated in the routine.</short>
</element>
<element name="RotatePoint.AAngle">
<short>Rotation angle in radians.</short>
</element>

<element name="RotateRect">
<short>
Rotates a rectangle with the specified dimensions by the specified angle
(in radians).
</short>
<descr>
<p>
Rotates the rectangle (0, 0, AWidth, AHeight) around its top-left corner
(0,0) by the angle in AAngle (in radians).
Note that y points downwards.
</p>
</descr>
<seealso/>
</element>
<element name="RotateRect.Result">
<short>The smallest TRect which contains the rotated rectangle.</short>
</element>
<element name="RotateRect.AWidth">
<short>Width for the rectangle.</short>
</element>
<element name="RotateRect.AHeight">
<short>Height for the rectangle.</short>
</element>
<element name="RotateRect.AAngle">
<short>Rotation angle in radians.</short>
</element>

<element name="SplitBezier">
<short>
<var>SplitBezier</var> - split any 4-Point Bezier into two 4-Point Beziers:
a 'Left' and a 'Right'
</short>
<descr>
<p>
Use SplitBezier to split any 4-Point Bezier into two 4-Point Beziers: a
'Left' and a 'Right'. It is primarily for use in Bezier2Polyline.
</p>
</descr>
<seealso/>
</element>
<element name="SplitBezier.Bezier">
<short/>
</element>
<element name="SplitBezier.Left">
<short/>
</element>
<element name="SplitBezier.Right">
<short/>
</element>

<element name="operator +(TFloatPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Add operator (+) for values using the TFloatPoint type.
</short>
<descr>
<p>
Values for the X and Y members in the addends are summed and stored in the 
corresponding members in the TFloatPoint result.
</p>
</descr>
</element>

<element name="operator +(TFloatPoint,Extended):TFloatPoint">
<short>
Implements the Add operator (+) for values using the TFloatPoint and Extended 
types.
</short>
<descr>
<p>
The Extended value is added to both the X and Y members in the TFloatPoint 
type and used in the result for the operator.
</p>
</descr>
<seealso/>
</element>

<element name="operator +(Extended,TFloatPoint):TFloatPoint">
<short>
Implements the Add operator (+) for values using the Extended and TFloatPoint 
types.
</short>
<descr>
<p>
The Extended value is added to both the X and Y members in the TFloatPoint 
type and used in the result for the operator.
</p>
</descr>
<seealso/>
</element>

<element name="operator +(TFloatPoint,TPoint):TFloatPoint">
<short>
Implements the Add operator (+) for values using the TFloatPoint and TPoint 
types.
</short>
<descr/>
<seealso/>
</element>

<element name="operator +(TPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Add operator (+) for values using the TPoint and TFloatPoint 
types.
</short>
<descr/>
<seealso/>
</element>

<element name="operator -(TFloatPoint,Extended):TFloatPoint">
<short>
Implements the Subtract operator (-) for values using the TFloatPoint and 
Extended types.
</short>
<descr>
<p>
The value in the Extended type is subtracted from both the X and Y members in 
the TFloatPoint type and used in the result for the operator.
</p>
</descr>
</element>

<element name="operator -(TFloatPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Subtract operator (-) for values using the TFloatPoint type.
</short>
<descr/>
</element>

<element name="operator -(TFloatPoint,TPoint):TFloatPoint">
<short>
Implements the Subtract operator (-) for values using the TFloatPoint and 
TPoint types.
</short>
<descr/>
</element>

<element name="operator -(TPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Subtract operator (-) for values using the TPoint and 
TFloatPoint types.
</short>
<descr/>
</element>

<element name="operator *(TFloatPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Multiply operator (*) for values using the TFloatPoint type.
</short>
<descr/>
</element>

<element name="operator *(TFloatPoint,Extended):TFloatPoint">
<short>
Implements the Multiply operator (*) for values using the TFloatPoint and 
Extended types.
</short>
<descr>
<p>
The value in the Extended type is applied to both the X and Y members in the TFloatPoint type and used in the result for the operator.
</p>  
</descr>
</element>

<element name="operator *(Extended,TFloatPoint):TFloatPoint">
<short>
Implements the Multiply operator (*) for values using the Extended and 
TFloatPoint types.
</short>
<descr>
<p>
The value in the Extended type is applied to both the X and Y members in the TFloatPoint type and used in the result for the operator.
</p>  
</descr>
</element>

<element name="operator *(TFloatPoint,TPoint):TFloatPoint">
<short>
Implements the Multiply operator (*) for values using the TFloatPoint and 
TPoint types.
</short>
<descr/>
</element>

<element name="operator *(TPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Multiply operator (*) for values using the TPoint and 
TFloatPoint types.
</short>
<descr/>
</element>

<element name="operator /(TFloatPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Divide operator (/) for values using the TFloatPoint type.
</short>
<descr/>
</element>

<element name="operator /(TFloatPoint,Extended):TFloatPoint">
<short>
Implements the Divide operator (/) for values using the TFloatPoint and 
Extended types.
</short>
<descr>
<p>
The value in the Extended type is applied to both the X and Y members in the 
TFloatPoint type and used in the result for the operator. No validation is 
performed for the divisor in the operator.
</p>
</descr>
</element>

<element name="operator /(TFloatPoint,TPoint):TFloatPoint">
<short>
Implements the Divide operator (/) for values using the TFloatPoint and 
TPoint types.
</short>
<descr/>
</element>

<element name="operator /(TPoint,TFloatPoint):TFloatPoint">
<short>
Implements the Divide operator (/) for values using the TPoint and 
TFloatPoint types.
</short>
<descr/>
</element>

<element name="operator =(TPoint,TPoint):Boolean">
<short>
Implements the Equal operator (=) to compare values using the TPoint type.
</short>
<descr>
<p>
The result is <b>True</b> when the X and Y members in the compared values are 
the same.
</p>
</descr>
</element>

<element name="operator =(TFloatPoint,TFloatPoint):Boolean">
<short>
Implements the Equal operator (=) to compare values using the TFloatPoint type.
</short>
<descr>
<p>
The result is <b>True</b> when the X and Y members in the compared values are 
the same.
</p>
</descr>
</element>

<element name="operator =(TRect,TRect):Boolean">
<short>
Implements the Equal operator (=) to compare values using the TRect type.
</short>
<descr>
<p>
The result is <b>True</b> when the Left, Top, Right, and Bottom members in the 
compared values are the same.
</p>
</descr>
</element>

<element name="operator :=(TFloatPoint):TPoint">
<short>
Implements the Assign operator (=) to store a value using the TFloatPoint type 
to a TPoint instance.
</short>
<descr>
<p>
<var>SimpleRoundTo</var> in the RTL <file>math.pp</file> unit is called to 
round both the X and Y members to a Double value with 0 decimals in the 
precision. The Double values are truncated and stored to the Longint types 
used for the X and Y members in the result for the operator.
</p>
</descr>
</element>

<element name="operator :=(TPoint):TFloatPoint">
<short>
Implements the Assign operator (=) to store a value using the TPoint type to a 
TFloatPoint instance.
</short>
<descr/>
</element>

<topic name="GraphMathOperators">
<short>
<b>GraphMath Operators</b>.
</short>
<descr>
<p>
This Unit contains a number of routines for calculating and converting
series of graphic points from one coordinate system to another.
</p>
<p>
A fundamental type is introduced, called TFloatPoint. It is an extended
precision record containing an X and a Y coordinate for a graphic point.
Its structure is as follows:
</p>
<code>
type
  TFloatPoint = record
    X, Y: Extended;
  end;
</code>
<p>
The Unit contains definitions for mathematical operators which extend the
normal definitions of addition, subtraction, multiplication, division and
comparison to cover manipulations with TFloatPoints, allowing, for example,
addition or multiplication of two TFloatPoints, a TFloatPoint and a TPoint,
or a TFloatPoint and an Extended precision number.
</p>
</descr>
</topic>
</module>
<!-- GraphMath -->
</package>
</fpdoc-descriptions>
