Is it possible to calculate the length of a vector by rotating it to, and along an axis, and then use that axis to measure the length of the vector? If so, is it less expensive than calculating length by Pythagoras/square-root? i am working in unity (C#)
Example:
Vector3 myVector(x, y, z);
Vector3 myVectorRealigned = Quaternion.FromToRotation(myVector, Vector3.up) * myVector;
float myVectorLength1 = sqrt(myVector.x^2 + myVector.y^2 + myVector.z^2);
float myVectorLength2 = myVectorRealigned.y;
when i tried this it seemed to work! however which of these methods is the best to use/is the least expensive?
I am no mathematician, so please correct me if I am wrong.
As you have tested, both approaches should work, but I guess that the Quaternion approach is more costly.
The norm approach requires 3 multiplications, 2 additions and 1 sqrt.
In contrast, the first step in the quaternion approach (Quaternion.FromToRotation) alone requires is more costly than calculating the norm. A way of calculating the quaternion describing the rotation from one vector to another is this:
Quaternion q;
vector a = crossproduct(v1, v2)
q.xyz = a;
q.w = sqrt((v1.Length ^ 2) * (v2.Length ^ 2)) + dotproduct(v1, v2)
As you see, this alone is more costly than norm = sqrt(myVector.x^2 + myVector.y^2 + myVector.z^2).
Additionally you use that quaternion to rotate your vector again involves dot products, cross products and several multiplications and additions.
E.g.: rotatedVec = 2 * dot(q.xyz, v) * q.xyz + (q.w^2 - dot(q.xyz, q.xyz)) * v + 2 * q.w * cross(q.xyz, v).
References: from-to-quaternion, rotate vector by quaternion
Side Note: If you are concerned with performance and you don't need the exact length of the vector (e.g. for determining the closest object to position x), you can use Vector3.sqrMagnitude (squared norm) as well.