How to identify unit vectors with angle threshold quickly? - geometry

Question

I am writing a program about computing geometry.
In this program, I need to identify unit vectors. (The word identify maybe not accurate)
i.e., writing a program to check whether a unit vector already exists.
This procedure is used when checking whether two polygons are on one plane. The first step is to check whether normal of two polygons are very close (angle < 1.0 degree).
So, we can assume that
all vectors are unit vectors
vectors are random
For example, set the angle threshold to 1.0 degree. And we have 6 vectors.
(1,0,0)
(0,1,0)
(1,0,1e-8) // in program, this will be normalized
(1,0,0)
(sin(45), cos(45),0)
(sin(44.9), cos(44.9),0)
then, the index of each vector is
0 1 0 0 2 2
i.e., the 1st / 3rd / 4th vectors are the same one because their angle is within 1.0 degree or just the same direction. angle between the 5th/6th vector is smaller than 1.0 degree.
Now, the problem comes, I have hundreds of thousands unit vectors to identify in different stages. This procedure costs about half of total time.
example code
std::vector<Vector3d> unitVecs; // all unit vectors
// more than 100,000 unit vectors in real case
int getVectorID(const Vector3d& vec)
{
for(int i=0; i<unitVecs.size(); ++i) {
if(calcAngle(unitVecs[i], vec) <1.0) // 1.0 is angle degree threshold
return i;
/// alternatively, check with cos value
if(unitVecs[i].dot(vec)>cos(1.0*RADIAN))
return i;
}
return -1;
}
int insertVector(const Vector3d& vec)
{
int idx = getVectorID(vec);
if(idx!=-1) return idx;
unitVecs.push_back(vec);
return unitVecs.size()-1;
}
Does anyone have good ideas to accelerate this process ?

Answer 1

If you are able to accept vectors which are merely "very close to being unit vectors", as opposed to vectors which are strictly less than or equal to 1 degree from being a unit vector, you can simply check that for a given vector 3 values are very close to 0, and one value is very close to 1:
int valueCloseTo(float value, float trg, float epsilon=0.0001) {
return abs(value) - trg <= epsilon;
}
int isRoughlyUnitVector(float x, float y, float z, float epsilon=0.0001) {
// We can quickly return false if units don't add near 1
// Could also consider multiplying `epsilon` x 3 here to account for accumulated error
if (!valueCloseTo(x + y + z, 1, epsilon)) return false;
// Now ensure that of x, y, and z, two are ~0 and one is ~1
int numZero = 0;
int numOne = 0;
std::vector<float> vec{ x, y, z };
for (float v : vec) {
// Count another ~0 value
if (valueCloseTo(v, 0, epsilon)) numZero++;
// Count another ~1 value
else if (valueCloseTo(v, 1, epsilon)) numOne++;
// If any value isn't close to 0 or 1, (x,y,z) is not a unit vector
else return false;
// False if we exceed 2 values near 0, and one value near 1
if (numZero > 2 || numOne > 1) return false;
}
return true;
}
Note that this method does not give any way to define a "maximum offset angle" (like 1deg in your question) - instead it lets us work with an epsilon value, which isn't an angle but rather a simple linear value. As epsilon increases vectors that are further from being unit vectors get accepted, but epsilon doesn't have an "angular" nature to it.