OpenCASCADE Make Primitives-Sphere
eryar@163.com
Abstract. The sphere is the simplest topology shape of the BRep structure. But there are several import concept of the sphere edges, such as degenerated edge and seam edge. So construct a sphere by code, you will learn these.
Key Words. OpenCASCADE, Sphere, BRep
1. Introduction
球體（sphere）是邊界表示法（BRep）中最簡(jiǎn)單的一個(gè)拓樸形狀了，因?yàn)樗苯佑汕蛎鎭?lái)構(gòu)造。但是其中包含了一些重要的概念，如退化邊（degenerated edge）、銜接邊（seam edge）。由代碼手工來(lái)構(gòu)造一個(gè)球體，可以學(xué)習(xí)這些概念。首先要知道OpenCASCADE中球面的參數(shù)方程：
在Draw Test Harness中顯示如下圖所示：
Figure 1.1 Sphere in Draw Test Harness
由球面的參數(shù)方程可知，當(dāng)參數(shù)u=0或2PI時(shí)，對(duì)應(yīng)球面上的點(diǎn)就是上圖所示的綠線，實(shí)際上是由兩個(gè)線重合在一起了。
當(dāng)參數(shù)v=-PI/2或PI/2時(shí)，對(duì)應(yīng)球面上兩個(gè)極點(diǎn)，因?yàn)榍蛎娴膬蓚€(gè)極點(diǎn)處法向?yàn)榱悖蛎嬖趦蓚€(gè)極點(diǎn)處的法向是存在的，所以這樣的點(diǎn)即為邊退化而成，稱(chēng)為退化邊。
三維曲線圓的參數(shù)方程如下所示：
通過(guò)代碼從點(diǎn)開(kāi)始來(lái)構(gòu)造一個(gè)球體，從而來(lái)加深理解OpenCASCADE的BRep表示法。
2.Make the Sphere
2.1 Make Vertex
從頂點(diǎn)開(kāi)始來(lái)創(chuàng)建球體。因?yàn)榍蝮w就是一個(gè)球面，為了得到Face的Wire，需要構(gòu)造一個(gè)閉合的區(qū)域。這里選擇兩個(gè)極點(diǎn)作為球體的頂點(diǎn)。創(chuàng)建球體的兩個(gè)極點(diǎn)，程序代碼如下所示：
// make the north and south poles.
aBuilder.MakeVertex(aNorthPole, aPoints[0], Precision::Confusion());
aBuilder.MakeVertex(aSouthPole, aPoints[1], Precision::Confusion());
2.2 Make Edge
為了得到閉合的Wire，需要四條邊，其中在球面兩個(gè)極點(diǎn)處的兩條退化邊，還有連接兩個(gè)極點(diǎn)的重合的銜接邊。創(chuàng)建邊的代碼如下所示：
// make the seam edge with the 3D geometry curve.
aBuilder.MakeEdge(aSeamEdge, new Geom_Circle(aCircle), Precision::Confusion());
// there is no 3D geometry curve in the degenerated edge.
aBuilder.MakeEdge(aNorthEdge);
aBuilder.Degenerated(aNorthEdge, Standard_True);
// there is no 3D geometry curve in the degenerated edge.
aBuilder.MakeEdge(aSouthEdge);
aBuilder.Degenerated(aSouthEdge, Standard_True);
// set the vertex info of the seam edges.
{
TopoDS_Vertex V1 = aNorthPole;
TopoDS_Vertex V2 = aSouthPole;
V1.Reverse();
aBuilder.Add(aSeamEdge, V1);
aBuilder.Add(aSeamEdge, V2);
aBuilder.UpdateVertex(V1, ElCLib::Parameter(aCircle, aPoints[0]), aSeamEdge, Precision::Confusion());
aBuilder.UpdateVertex(V2, ElCLib::Parameter(aCircle, aPoints[1]), aSeamEdge, Precision::Confusion());
BRepTools::Update(aSeamEdge);
}
// set the vertex info of the north degenerated edge.
{
TopoDS_Vertex V1 = aNorthPole;
TopoDS_Vertex V2 = aNorthPole;
V2.Reverse();
aBuilder.Add(aNorthEdge, V1);
aBuilder.Add(aNorthEdge, V2);
BRepTools::Update(aNorthEdge);
}
// set the vertex info of the south degenerated edge.
{
TopoDS_Vertex V1 = aSouthPole;
TopoDS_Vertex V2 = aSouthPole;
V2.Reverse();
aBuilder.Add(aSouthEdge, V1);
aBuilder.Add(aSouthEdge, V2);
BRepTools::Update(aSouthEdge);
}
由上述代碼可知，銜接邊中包含了幾何信息：三維曲線圓；退化邊中未包含幾何信息，但將其退化邊屬性設(shè)置為true。之后將邊上頂點(diǎn)在曲線上對(duì)應(yīng)的參數(shù)值設(shè)置到邊中，退化邊不需要設(shè)置。
2.3 Make Wire
創(chuàng)建Wire需要確保組成Wire的邊要閉合。程序代碼如下所示：
// make wire.
aBuilder.MakeWire(aWire);
// add edges to the wire.
{
TopoDS_Edge E1 = aNorthEdge;
TopoDS_Edge E2 = aSeamEdge;
TopoDS_Edge E3 = aSouthEdge;
TopoDS_Edge E4 = aSeamEdge;
E1.Reverse();
E4.Reverse();
aBuilder.Add(aWire, E1);
aBuilder.Add(aWire, E2);
aBuilder.Add(aWire, E3);
aBuilder.Add(aWire, E4);
BRepTools::Update(aWire);
}
2.4 Make Face
創(chuàng)建面后，將邊與面關(guān)聯(lián)起來(lái)至關(guān)重要，即PCurve的設(shè)置。程序代碼如下所示：
// make face.
aBuilder.MakeFace(aFace, new Geom_SphericalSurface(aSphere), Precision::Confusion());
// set the pcurve info between edge and face.
{
aBuilder.Range(aNorthEdge, 0.0, 2 * M_PI);
aBuilder.UpdateEdge(aNorthEdge, new Geom2d_Line(aLines[0]), aFace, Precision::Confusion());
aBuilder.Range(aSeamEdge, 1.5 * M_PI, 2.5 * M_PI);
aBuilder.UpdateEdge(aSeamEdge, new Geom2d_Line(aLines[1]), new Geom2d_Line(aLines[2]), aFace, Precision::Confusion());
aBuilder.Continuity(aSeamEdge, aFace, aFace, GeomAbs_CN);
aBuilder.Range(aSouthEdge, 0.0, 2 * M_PI);
aBuilder.UpdateEdge(aSouthEdge, new Geom2d_Line(aLines[3]), aFace, Precision::Confusion());
BRepTools::Update(aFace);
}
由上述代碼可知，球面中包含了一個(gè)幾何的曲面。創(chuàng)建球面后，將相關(guān)的邊與面關(guān)聯(lián)起來(lái)。參數(shù)曲線PCurve的范圍Range在球面的參數(shù)空間中應(yīng)該閉合。其中兩個(gè)退化邊的范圍都是從0到2PI，而銜接邊的范圍設(shè)置不當(dāng)，會(huì)產(chǎn)生不正確的結(jié)果，如下圖所示：
Figure 2.4.1 Seam Edge Range[-PI/2, PI/2]
線框模式顯示正常，但是不能切換到渲染模式，即不能顯示出面。結(jié)合其PCurve的范圍可以發(fā)現(xiàn)組成Wire的邊的PCurve不能閉合。
當(dāng)Seam邊的三維曲線方向不當(dāng)時(shí)，會(huì)不與球面的Seam重合，如下圖所示：
Figure 2.4.2 Circle in Seam Edge Range [-PI/2, PI/2]
Figure 2.4.3 Wrong Seam Edge Geometry Curve
Figure 2.4.4 Wrong Seam Edge Geometry Curve
3. Test the Sphere
正確生成球體后導(dǎo)出為brep文件即可以在Draw Test Harness中來(lái)顯示及進(jìn)行一些操作來(lái)驗(yàn)證結(jié)果的正確性。在Draw Test Harness中打開(kāi)brep文件并顯示球體如下圖所示：
Figure 3.1 Show the Sphere from file in Draw Test Harness
將其與一個(gè)長(zhǎng)方體進(jìn)行布爾運(yùn)算，效果如下圖所示：
Figure 3.2 Spher and a Box
Figure 3.3 Sphere cut a Box
由上圖可知，球體與長(zhǎng)方體布爾運(yùn)算結(jié)果正確。
4. Conclusion
通過(guò)生成一個(gè)球體，示例了特殊邊的構(gòu)造，如退化邊和銜接邊。需要注意的事項(xiàng)還是組成Wire的所有邊中的PCurve必須在面的參數(shù)空間中閉合。由PCurve可知，球面對(duì)應(yīng)的參數(shù)空間不是幾何曲面的范圍，而是在v方向上偏移了2PI。
5. References
1. OpenCascade Primitives BRep - Sphere,
http://www.shnenglu.com/eryar/archive/2014/03/22/206279.html
2. PCurve - Curve on Surface,
http://www.shnenglu.com/eryar/archive/2014/03/15/206180.html
3. Topology and Geometry in OpenCascade-Face,
http://www.shnenglu.com/eryar/archive/2013/09/12/203199.html