Determining the Angle Between a Line and the Horizontal Axis

To calculate the angle between a line and the horizontal axis, there are several steps involved. Firstly, it's essential to find the difference between the start and endpoints, as it represents a directed line segment, not an infinite line. The difference can be computed as:

deltaY = P2_y - P1_y
deltaX = P2_x - P1_x

Subsequently, the angle is calculated from the positive X axis at P1 to the positive Y axis at P1. Here's a common approach:

angleInDegrees = arctan(deltaY / deltaX) * 180 / PI

However, arctan may not always be ideal. To account for the proper quadrant and preserve the distinction between quadrants, an alternative solution utilizing atan2 is recommended:

angleInDegrees = atan2(deltaY, deltaX) * 180 / PI

For a more elegant approach, particularly when only the angle's cosine and sine are required, consider the following technique:

1. Create a vector representation of (deltaX, deltaY).
2. Normalize the vector to obtain a unit vector.
3. The normalized deltaX will represent the cosine of the angle, while deltaY will represent the sine.

It's worth noting that the signs of deltaX and deltaY provide valuable information regarding the quadrant in which the angle resides. For example:

• When both deltaX and deltaY are positive, the angle falls within the first quadrant (0 to 90 degrees).
• When deltaX is negative and deltaY is positive, the angle lies in the second quadrant (90 to 180 degrees).
• When both deltaX and deltaY are negative, the angle is located in the third quadrant (180 to 270 degrees).
• When deltaX is positive and deltaY is negative, the angle resides in the fourth quadrant (270 to 360 degrees).