OpenCASCADE Linear Extrusion Surface
Abstract. OpenCASCADE linear extrusion surface is a generalized cylinder. Such a surface is obtained by sweeping a curve (called the “extruded curve” or “basis”) in a given direction (referred to as the direction of extrusion and defined by a unit vector). The u parameter is along the extruded curve. The v parameter is along the direction of extrusion. The form of a surface of linear extrusion is generally a ruled surface. It can be a cylindrical surface, or a planar surface.
Key Words. OpenCASCADE, Extrusion Surface, Sweeping
1. Introduction
一般柱面（The General Cylinder）可以由一段或整個(gè)圓弧沿一個(gè)方向偏移一定的距離得到。如下圖所示：
Figure 1.1 Extrusion Shapes
當(dāng)將頂點(diǎn)拉伸時(shí)，會(huì)生成一條邊；當(dāng)將邊拉伸時(shí)，會(huì)生成面；當(dāng)將Wire拉伸時(shí)，會(huì)生成Shell，當(dāng)將面拉伸時(shí)，會(huì)生成體。當(dāng)將曲線沿一個(gè)方向拉伸時(shí)，會(huì)形成一個(gè)曲面，如果此方向?yàn)橹本€，則會(huì)生成一般柱面。如果此方向是曲線時(shí)，會(huì)生成如下圖所示曲面：
Figure 1.2 Swept surface/ loft surface
本文主要介紹將曲線沿直線方向拉伸的算法，即一般柱面生成算法。并將生成的曲面在OpenSceneGraph中進(jìn)行顯示。
2. Cylinder Surface Definition
設(shè) W是一個(gè)單位向量，C(u)是定義在節(jié)點(diǎn)矢量U上，權(quán)值為wi的p次NURBS曲線。我們要得到一般柱面S(u,v)的表達(dá)式，S(u,v)是通過將 C(u)沿方向W平行掃描（sweep）距離d得到的。記掃描方向的參數(shù)為v, 0<v<1，顯然，S(u,v)必須滿足以下兩個(gè)條件：
v 對(duì)于固定的u0, S(u0, v)為由C(u0)到C(u0)+dW的直線段；
v 對(duì)于固定的v0：
所要求的柱面的表達(dá)式為：
S(u,v)定義在節(jié)點(diǎn)矢量U和V上，這里V＝{0，0，1，1}，U為C(u)的節(jié)點(diǎn)矢量。控制頂點(diǎn)由Pi,0=Pi和Pi,1=Pi+dW給出，權(quán)值wi,0=wi,1=wi。如下圖所示為一般柱面：
Figure 2.1 A general cylinder obtained by translating C(u) a distance d along W.
其中OpenCASCADE中一般柱面的表達(dá)式如下所示：
其取值范圍的代碼如下所示：
//
=======================================================================
//
function : Bounds
//
purpose :
//
=======================================================================
void
Geom_SurfaceOfLinearExtrusion::Bounds ( Standard_Real&
U1,
Standard_Real
&
U2,
Standard_Real
&
V1,
Standard_Real
& V2 )
const
{
V1
= -Precision::Infinite(); V2 =
Precision::Infinite();
U1
= basisCurve->FirstParameter(); U2 = basisCurve->
LastParameter();
}
由上代碼可知，參數(shù)在v方向上是趨于無窮的；在u方向上參數(shù)的范圍為曲線的范圍。計(jì)算柱面上點(diǎn)的方法代碼如下所示：
//
=======================================================================
//
function : D0
//
purpose :
//
=======================================================================
void
Geom_SurfaceOfLinearExtrusion::D0 (
const
Standard_Real U,
const
Standard_Real V,
Pnt
& P)
const
{
XYZ Pxyz
=
direction.XYZ();
Pxyz.Multiply (V);
Pxyz.Add (basisCurve
->
Value (U).XYZ());
P.SetXYZ(Pxyz);
}
即將柱面上點(diǎn)先按V方向來計(jì)算，再按U方向來計(jì)算，最后將兩個(gè)方向的值相加即得到柱面上的點(diǎn)。
由上述代碼可知，OpenCASCADE中一般柱面沒有使用NURBS曲面來表示。根據(jù)這個(gè)方法，可以將任意曲線沿給定的方向來得到一個(gè)柱面，這個(gè)曲線可以是直線、圓弧、圓、橢圓等。關(guān)于柱面上更多算法，如求微分等，可以參考源程序。
3. Display the Surface
還是在OpenSceneGraph中來對(duì)一般柱面進(jìn)行可視化，來驗(yàn)證結(jié)果。因?yàn)镺penSceneGraph的簡(jiǎn)單易用，顯示曲面的程序代碼如下所示：
/*
* Copyright (c) 2013 to current year. All Rights Reserved.
*
* File : Main.cpp
* Author : eryar@163.com
* Date : 2014-11-23 10:18
* Version : OpenCASCADE6.8.0
*
* Description : Test the Linear Extrusion Surface of OpenCASCADE.
*
* Key Words : OpenCascade, Linear Extrusion Surface, General Cylinder
*
*/
//
OpenCASCADE.
#define
WNT
#include
<Precision.hxx>
#include
<gp_Circ.hxx>
#include
<Geom_SurfaceOfLinearExtrusion.hxx>
#include
<GC_MakeCircle.hxx>
#include
<GC_MakeSegment.hxx>
#include
<GC_MakeArcOfCircle.hxx>
#pragma
comment(lib, "TKernel.lib")
#pragma
comment(lib, "TKMath.lib")
#pragma
comment(lib, "TKG3d.lib")
#pragma
comment(lib, "TKGeomBase.lib")
//
OpenSceneGraph.
#include <osgViewer/Viewer>
#include
<osgViewer/ViewerEventHandlers>
#include
<osgGA/StateSetManipulator>
#pragma
comment(lib, "osgd.lib")
#pragma
comment(lib, "osgGAd.lib")
#pragma
comment(lib, "osgViewerd.lib")
const
double
TOLERANCE_EDGE = 1e-
6
;
const
double
APPROXIMATION_DELTA =
0.05
;
/*
*
* @brief Render 3D geometry surface.
*/
osg::Node
* BuildSurface(
const
Handle_Geom_Surface&
theSurface)
{
osg::ref_ptr
<osg::Geode> aGeode =
new
osg::Geode();
Standard_Real aU1
=
0.0
;
Standard_Real aV1
=
0.0
;
Standard_Real aU2
=
0.0
;
Standard_Real aV2
=
0.0
;
Standard_Real aDeltaU
=
0.0
;
Standard_Real aDeltaV
=
0.0
;
theSurface
->
Bounds(aU1, aU2, aV1, aV2);
//
trim the parametrical space to avoid infinite space.
Precision::IsNegativeInfinite(aU1) ? aU1 = -
1.0
: aU1;
Precision::IsInfinite(aU2)
? aU2 =
1.0
: aU2;
Precision::IsNegativeInfinite(aV1)
? aV1 = -
1.0
: aV1;
Precision::IsInfinite(aV2)
? aV2 =
1.0
: aV2;
//
Approximation in v direction.
aDeltaU = (aU2 - aU1) *
APPROXIMATION_DELTA;
aDeltaV
= (aV2 - aV1) *
APPROXIMATION_DELTA;
for
(Standard_Real u = aU1; (u - aU2) <= TOLERANCE_EDGE; u +=
aDeltaU)
{
osg::ref_ptr
<osg::Geometry> aLine =
new
osg::Geometry();
osg::ref_ptr
<osg::Vec3Array> aPoints =
new
osg::Vec3Array();
for
(Standard_Real v = aV1; (v - aV2) <= TOLERANCE_EDGE; v +=
aDeltaV)
{
gp_Pnt aPoint
= theSurface->
Value(u, v);
aPoints
->
push_back(osg::Vec3(aPoint.X(), aPoint.Y(), aPoint.Z()));
}
//
Set vertex array.
aLine->
setVertexArray(aPoints);
aLine
->addPrimitiveSet(
new
osg::DrawArrays(osg::PrimitiveSet::LINE_STRIP,
0
, aPoints->
size()));
aGeode
->addDrawable(aLine.
get
());
}
//
Approximation in u direction.
for
(Standard_Real v = aV1; (v - aV2) <= TOLERANCE_EDGE; v +=
aDeltaV)
{
osg::ref_ptr
<osg::Geometry> aLine =
new
osg::Geometry();
osg::ref_ptr
<osg::Vec3Array> aPoints =
new
osg::Vec3Array();
for
(Standard_Real u = aU1; (u - aU2) <= TOLERANCE_EDGE; u +=
aDeltaU)
{
gp_Pnt aPoint
= theSurface->
Value(u, v);
aPoints
->
push_back(osg::Vec3(aPoint.X(), aPoint.Y(), aPoint.Z()));
}
//
Set vertex array.
aLine->
setVertexArray(aPoints);
aLine
->addPrimitiveSet(
new
osg::DrawArrays(osg::PrimitiveSet::LINE_STRIP,
0
, aPoints->
size()));
aGeode
->addDrawable(aLine.
get
());
}
return
aGeode.release();
}
/*
*
* @brief Build the test scene.
*/
osg::Node
* BuildScene(
void
)
{
osg::ref_ptr
<osg::Group> aRoot =
new
osg::Group();
//
test the linear extrusion surface.
//
test linear extrusion surface of a line.
Handle_Geom_Curve aSegment = GC_MakeSegment(gp_Pnt(
3.0
,
0.0
,
0.0
), gp_Pnt(
6.0
,
0.0
,
0.0
));
Handle_Geom_Surface aPlane
=
new
Geom_SurfaceOfLinearExtrusion(aSegment, gp::DZ());
aRoot
->
addChild(BuildSurface(aPlane));
//
test linear extrusion surface of a arc.
Handle_Geom_Curve aArc = GC_MakeArcOfCircle(gp_Circ(gp::ZOX(),
2.0
),
0.0
, M_PI,
true
);
Handle_Geom_Surface aSurface
=
new
Geom_SurfaceOfLinearExtrusion(aArc, gp::DY());
aRoot
->
addChild(BuildSurface(aSurface));
//
test linear extrusion surface of a circle.
Handle_Geom_Curve aCircle = GC_MakeCircle(gp::XOY(),
1.0
);
Handle_Geom_Surface aCylinder
=
new
Geom_SurfaceOfLinearExtrusion(aCircle, gp::DZ());
aRoot
->
addChild(BuildSurface(aCylinder));
return
aRoot.release();
}
int
main(
int
argc,
char
*
argv[])
{
osgViewer::Viewer aViewer;
aViewer.setSceneData(BuildScene());
aViewer.addEventHandler(
new
osgGA::StateSetManipulator(
aViewer.getCamera()
->
getOrCreateStateSet()));
aViewer.addEventHandler(
new
osgViewer::StatsHandler);
aViewer.addEventHandler(
new
osgViewer::WindowSizeHandler);
return
aViewer.run();
return
0
;
}
上述顯示方法只是顯示線框的最簡(jiǎn)單的算法，只為驗(yàn)證一般柱面結(jié)果，不是高效算法。顯示結(jié)果如下圖所示：
Figure 3.1 General Cylinder for: Circle, Arc, Line
如上圖所示分別為對(duì)圓、圓弧和直線進(jìn)行拉伸得到的一般柱面。根據(jù)這個(gè)原理可以將任意曲線沿給定方向進(jìn)行拉伸得到一個(gè)柱面。
4. Conclusion
通 過對(duì)OpenCASCADE中一般柱面的類中代碼進(jìn)行分析可知，OpenCASCADE的這個(gè)線性拉伸柱面 Geom_SurfaceOfLinearExtrusion是根據(jù)一般柱面的定義實(shí)現(xiàn)的，并不是使用NURBS曲面來表示的。當(dāng)需要用NURBS曲面來 表示一般柱面時(shí)，需要注意控制頂點(diǎn)及權(quán)值的計(jì)算取值。
5. References
1. 趙罡，穆國旺，王拉柱譯. 非均勻有理B樣條. 清華大學(xué)出版社. 2010
2. Les Piegl, Wayne Tiller. The NURBS Book. Springer-Verlag. 1997
3. OpenCASCADE Team, OpenCASCADE BRep Format. 2014
4. Donald Hearn, M. Pauline Baker. Computer Graphics with OpenGL. Prentice Hall. 2009
5. 莫蓉，常智勇. 計(jì)算機(jī)輔助幾何造型技術(shù). 科學(xué)出版社. 2009
PDF Version and Source Code: OpenCASCADE Linear Extrusion Surface
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