我有一个三角形和一个平面(在3维空间中),如何计算两个相交的线段,如果没有相交,则需要检测这种情况。
我正在寻找的最终结果是两个3维向量,它们定义了线段的起点和终点。
为了帮助您一点,我已经计算了脸部平面和平面之间的相交光线,我只需要找到端点以将该光线裁剪为线段即可。
对于那些喜欢阅读代码的人:
Face face; //a face, defined by 3 points
Plane plane; //a plane, defined by a normal vector and a distance
Ray intersection; //a ray, defined by a point and a direction, initialised to the intersection of the face plane and the face
Segment s = CalculateSegment(face, plane, intersection); //this method needs defining
最佳答案
这是一些建议的伪代码。首先是简单版本,后来是更强大的版本(只是为了帮助将原理与新特性区分开)。
简单版本:
// Assume the plane is given as the equation dot(N,X) + d = 0, where N is a (not
// neccessarily normalized) plane normal, and d is a scalar. Any way the plane is given -
// DistFromPlane should just let the input vector into the plane equation.
vector3d planeN;
float planeD;
float DistFromPlane( vector3d P)
{
// if N is not normalized this is *not* really the distance,
// but the computations work just the same.
return dot(planeN,P) + planeD;
}
bool GetSegmentPlaneIntersection( vector3d P1, vector3d P2, vector3d& outP)
{
float d1 = DistFromPlane(P1),
d2 = DistFromPlane(P2);
if (d1*d2 > 0) // points on the same side of plane
return false;
float t = d1 / (d1 - d2); // 'time' of intersection point on the segment
outP = P1 + t * (P2 - P1);
return true;
}
void TrianglePlaneIntersection(vector3d triA, vector3d triB, vector3d triC,
vector3dArray& outSegTips)
{
vector3d IntersectionPoint;
if( GetSegmentPlaneIntersection( triA, triB, IntersectionPoint))
outSegTips.Add(IntersectionPoint);
if( GetSegmentPlaneIntersection( triB, triC, IntersectionPoint))
outSegTips.Add(IntersectionPoint);
if( GetSegmentPlaneIntersection( triC, triA, IntersectionPoint))
outSegTips.Add(IntersectionPoint);
}
现在增加一些健壮性:
[编辑:增加了平面上单个顶点情况的明确考虑]
vector3d planeN;
float planeD;
float DistFromPlane( vector3d P)
{
return dot(planeN,P) + planeD;
}
void GetSegmentPlaneIntersection( vector3d P1, vector3d P2, vector3dArray& outSegTips)
{
float d1 = DistFromPlane(P1),
d2 = DistFromPlane(P2);
bool bP1OnPlane = (abs(d1) < eps),
bP2OnPlane = (abs(d2) < eps);
if (bP1OnPlane)
outSegTips.Add(P1);
if (bP2OnPlane)
outSegTips.Add(P2);
if (bP1OnPlane && bP2OnPlane)
return;
if (d1*d2 > eps) // points on the same side of plane
return;
float t = d1 / (d1 - d2); // 'time' of intersection point on the segment
outSegTips.Add( P1 + t * (P2 - P1) );
}
void TrianglePlaneIntersection(vector3d triA, vector3d triB, vector3d triC,
vector3dArray& outSegTips)
{
GetSegmentPlaneIntersection( triA, triB, outSegTips));
GetSegmentPlaneIntersection( triB, triC, outSegTips));
GetSegmentPlaneIntersection( triC, triA, outSegTips));
RemoveDuplicates(outSegTips); // not listed here - obvious functionality
}
希望能给出一个主意,但仍有许多潜在的优化方法。例如,如果要为大型网格中的每个三角形计算这些相交,则可以为每个顶点计算并缓存一次DistanceFromPlane,然后针对该顶点参与的每个边检索它。还可以进行更高级的缓存,取决于您的方案和数据表示形式。
关于language-agnostic - 确定三角形与平面的交点,我们在Stack Overflow上找到一个类似的问题:https://stackoverflow.com/questions/3142469/