已知角平分线垂线时求角平分线夹角的MATLAB计算疑问咨询

Hey there! Let's work through your question step by step to confirm if your math is on track, especially considering MATLAB's flipped image coordinate system.

First, let's recap your setup:

• You're calculating vertebra angles from radiographs, then finding the angle B between two vertebral bisectors using the method shown in your reference image.
• You currently think ( B = \alpha_2 - \alpha_1 ), but you're unsure because MATLAB's image coordinate system has a reversed y-axis (positive y points downward, unlike the standard Cartesian system where positive y points upward).

Key Analysis of Coordinate System Impact #

The good news is that as long as you calculate both ( \alpha_1 ) and ( \alpha_2 ) using the same coordinate system rules (i.e., consistently using MATLAB's image axes), the relative angle between them will remain valid. Here's why:

• In MATLAB, when you compute an angle using atan2(dy, dx) for a line segment, dy is the change in the image's y-coordinate (downward = positive). This gives you an angle measured clockwise from the positive x-axis, whereas standard Cartesian atan2 measures counterclockwise.
• However, this flip affects both ( \alpha_1 ) and ( \alpha_2 ) equally. When you take the difference ( \alpha_2 - \alpha_1 ), the directional flip cancels out, leaving you with the correct relative angle between the two bisectors.

Quick Test to Verify #

Let's use a simple example to confirm:

1. Suppose ( \alpha_1 ) is a horizontal line pointing right: in MATLAB, dy=0, dx>0, so ( \alpha_1 = atan2(0, dx) = 0 ).
2. Suppose ( \alpha_2 ) is a line sloping 45° downward (right and down): dy=1, dx=1, so ( \alpha_2 = atan2(1,1) = \pi/4 ) (45° clockwise from x-axis).
3. The difference ( \alpha_2 - \alpha_1 = \pi/4 ), which is exactly the angle B between the two lines—this matches what you'd expect visually.

If you get a negative difference (e.g., ( \alpha_2 ) is sloping upward relative to ( \alpha_1 )), just take the absolute value to get the positive angle between the lines (since angles between two lines are typically reported as values between 0 and ( \pi ) radians, or 0° and 180°).

Final Takeaway #

Your initial conclusion that ( B = \alpha_2 - \alpha_1 ) (or its absolute value, depending on direction) is correct! The flipped y-axis in MATLAB doesn't invalidate this calculation because both angles are computed using the same coordinate system rules, so their relative difference remains accurate.

备注：内容来源于stack exchange，提问作者Nathan S Hogaboom