如何通过旋转矩阵第二行与向量点积判断点的平面侧别？
Hey there, let’s unpack this trick—this is a classic optimization in 3D geometry that’s easy to overlook if you don’t connect rotation matrix rows to coordinate system transformations.
First, let’s recap what that composite rotation matrix (X-axis then Y-axis) actually represents:
- When you build a rotation matrix by combining X and Y rotations, each row of the matrix is a unit vector representing one axis of the new, rotated coordinate system in the original coordinate system’s terms.
- Row 1 = Unit vector for the rotated X'-axis (in original X/Y/Z)
- Row 2 = Unit vector for the rotated Y'-axis (in original X/Y/Z)
- Row 3 = Unit vector for the rotated Z'-axis (in original X/Y/Z)
Now, the key part: when you take the dot product of the matrix’s second row with your original point vector, you’re calculating something very specific:
That dot product equals the Y'-coordinate of your point in the rotated coordinate system.
Here’s why that matters for plane side checks:
- Imagine in the rotated coordinate system, we have a plane that splits space at
Y' = 0(a plane perpendicular to the Y'-axis, passing through the origin). To figure out which side of this plane a point is on, you just look at the sign of its Y' coordinate:- Positive Y' → point is on one side of the plane
- Negative Y' → point is on the opposite side
- Zero → point lies exactly on the plane
Instead of doing the full rotation (multiplying the entire matrix by your vector to get all three new coordinates), the code is cutting corners (smartly!)—it only calculates the Y' component via the dot product with the second row. This saves computation time (3 multiply-add operations instead of 9), which was a big deal in older codebases where processing power was limited.
To tie this back to your existing logic: if you were to run the full rotation on your point and then check the sign of the resulting Y-coordinate, you’d get exactly the same result as this dot product check. The code is just skipping the unnecessary calculations for X' and Z' since it only cares about the plane side.
内容的提问来源于stack exchange，提问作者PaulB
