How to Determine if a Point Lies to the Left or Right of a Line?

Determine the position of the point relative to the line

In order to separate a set of points into two different sets based on their position relative to the line, you need to determine whether a point is to the left or right of the line. Here's one way to accomplish this:

The cross product of two vectors provides the orientation of a point relative to a line. Given a straight line defined by two points a and b and a point c, the cross product formula is:

<code>(b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x)</code>

If the result is positive, it means that point c is to the left of line a-b. Conversely, if the result is negative, then c is to the right of the line. If the result is 0, then c is collinear with the line (that is, it lies on the line).

Example implementation

This is a Python code implementation using the cross product method:

<code class="language-python">def isLeft(a, b, c):
  return (b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x) > 0</code>

Where a, b and c represent three points.

Other notes

If the line is horizontal, you can adjust the cross product formula to determine whether the point is above or below the line:

<code>(b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x) > 0  (上方)</code>

This method provides a simple and efficient way to determine the position of a point relative to a line, allowing points to be divided into two sets based on their position on either side of the line.

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