You need two direction vectors. One is along the line AB given by
Vector3D e = Normalize(B-A)
and one to descibe the "up" direction for the cross section. This can be given, or it can be calculated with the following algorithm (with preference towards +y)
if( e.X != 0 || e.Z != 0 )
{
// choose direction perpendicular to line closest to +y direction
Vector3D n = [-e.X*e.Y, e.X*e.X+e.Z*e.Z, -e.Z*e.Y];
} else {
// if line along +y already then choose +z for up vector
Vector3D n = [ 0, 0, 1];
}
Now you can calculate the 3rd direction to form a coordinate system
Vector3D k = Normalize( Cross(e,n) )
And you assemble the 3×3 rotation matrix that transforms local coordinates to world coordinates with
| k.X n.X e.X |
R = | k.Y n.Y e.Y |
| k.Z n.Z e.Z |
The local coordinate have the direction along the line as +z such that
Point3D a = A + R*[w,h,0]
Point3D b = A + R*[-w,h,0]
Point3D c = A + R*[w,-h,0]
Point3D d = A + R*[-w,-h,0]
Point3D e = B + R*[w,h,0]
Point3D f = B + R*[-w,h,0]
Point3D g = B + R*[w,-h,0]
Point3D h = B + R*[-w,-h,0]
where R*[x,y,z]
designates a matrix-vector multiplication, w
and h
are the width and height of the rectangular cross section, and A
, B
are the point A and B position vectors. Addition here between vectors is element by element.
I have checked the code in my own code and it works. vec3
is alias for a 3D vector, mat3
is alias for 3×3 matrix.
vec3 u=(B-A).Normalized();
vec3 n = vec3.O;
if(Math.Abs(u.X)<=Math.Abs(u.Y)&&Math.Abs(u.X)<=Math.Abs(u.Z))
{
n=new vec3(u.Y*u.Y+u.Z*u.Z, -u.Y*u.X, -u.Z*u.X);
}
else if(Math.Abs(u.Y)<=Math.Abs(u.X)&&Math.Abs(u.Y)<=Math.Abs(u.Z))
{
n=new vec3(-u.X*u.Y, u.X*u.X+u.Z*u.Z, -u.Z*u.Y);
}
else if(Math.Abs(u.Z)<=Math.Abs(u.X)&&Math.Abs(u.Z)<=Math.Abs(u.Y))
{
n=new vec3(-u.X*u.Z, -u.Y*u.Z, u.X*u.X+u.Y*u.Y);
}
vec3 v=n.Cross(u);
mat3 R=mat3.Combine(v, n, u);
var a=A+R*new vec3(wt, ht, 0);
var b=A+R*new vec3(-wt, ht, 0);
var c=A+R*new vec3(wt, -ht, 0);
var d=A+R*new vec3(-wt, -ht, 0);
var e=B+R*new vec3(wt, ht, 0);
var f=B+R*new vec3(-wt, ht, 0);
var g=B+R*new vec3(wt, -ht, 0);
var h=B+R*new vec3(-wt, -ht, 0);