Basic task - to find the intersection point between a line segment with corner points S1 and S2 and a triangle with vertices A,B,C.

To do this I wrote a function that calculates partial coefficients or "weights" of each triangle' vertices at the intersection point (Pint), so it can be found as Pint=Wa*A+Wb*B+Wc*C.

I intuitively did it through volumes, though I haven't seen such algorithm anywhere, so I would like a confirmation if this is correct (it seems so...). Here is a working code of this function:

Code :

//Macros for calculating determinant of 3x3 matrix composed from 3 vectors
#define Determinant3xV(a,b,c) (a[0]*b[1]*c[2]+a[1]*b[2]*c[0]+a[2]*b[0]*c[1]-a[2]*b[1]*c[0]-a[1]*b[0]*c[2]-a[0]*b[2]*c[1])
//------------------------------------------------------------------------------
//Function checks if specified line' segment S1-S2 crosses triangle with given
//vertexes (A,B,C), and writes each triangle' vertex' attribute-part-value (at
//the intersection point) to PartsABC[3] variable
bool PolarizeTriangleWithSegment(const float S1[3], const float S2[3],
const float A[3], const float B[3], const float C[3],
float *PartsABC){
//Calculate vectors
float R[3] = {S2[0]-S1[0], S2[1]-S1[1], S2[2]-S1[2]},
S1A[3] = {A[0]-S1[0], A[1]-S1[1], A[2]-S1[2]},
S1B[3] = {B[0]-S1[0], B[1]-S1[1], B[2]-S1[2]},
S1C[3] = {C[0]-S1[0], C[1]-S1[1], C[2]-S1[2]},
S2A[3] = {A[0]-S2[0], A[1]-S2[1], A[2]-S2[2]},
S2B[3] = {B[0]-S2[0], B[1]-S2[1], B[2]-S2[2]},
S2C[3] = {C[0]-S2[0], C[1]-S2[1], C[2]-S2[2]};
//Calculate 6X volumes
float S1ABC = Determinant3xV(S1A,S1B,S1C);
float S2BAC = Determinant3xV(S2B,S2A,S2C);
if(S1ABC*S2BAC<0.f)return false; //No intersection with segment
float S1ABCS2 = S1ABC + S2BAC;
if(S1ABCS2==0.f)return false; //Triangle is degenerate or S1==S2
//Calculate volumes' parts
PartsABC[0] = Determinant3xV(S1B,S1C,R)/S1ABCS2;
if(PartsABC[0]<0.f)return false; //No intersection with segment
PartsABC[1] = Determinant3xV(S1C,S1A,R)/S1ABCS2;
if(PartsABC[1]<0.f)return false; //No intersection with segment
PartsABC[2] = Determinant3xV(S1A,S1B,R)/S1ABCS2;
if(PartsABC[2]<0.f)return false; //No intersection with segment
//Weights are calculated; Pint = PartsABC[0]*A + PartsABC[1]*B + PartsABC[2]*C
return true;
}

Is this correct? ]]>

Lets say we have the two vectors a and b.

if their dot product a.b > 0 , it implies that the angle between them is acute and pointing mostly in the same direction.

if a.b == 0. The vectors are perpendicular to each other..

If a.b < 0, the vectors are pointing mostly in the opposite direction.

What if we have the three vector x, y and z. And we have the following condition

Code :

if(std::abs(x.dot(y)) < std::abs(x.dot(z)))
{
}

I am trying to visualize the above snippet. Some help would be helpful.

Lets get even more specific . If both x.dot(y) > 0 and x.dot(z) > 0 , but x.dot(y) < x.dot(z), then what can we imply ?

Thanks ]]>