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

Determine which side of the line the point is on

To determine the position of a point relative to a line, a frame must be established. This framework involves selecting two points (A and B) on a line and forming an imaginary line between them. The goal is to divide the given set of points into two distinct sets: points that lie to the left of the line and points that lie to the right of the line.

Initially, try to determine the location of point Z using the angle between vectors A-Z-B. Assume that angles less than 180 degrees are to the right of a line, while angles greater than 180 degrees are to the left. However, due to mathematical limitations, this approach leads to inconsistent results, as the calculated angle is always less than 180 degrees.

To overcome this limitation, a more reliable method is to use cross product. Given a line A--B and point C, you can use the following formula to determine the location of point C:

<code class="language-javascript">isLeft(a, b, c) {
  return (b.x - a.x)*(c.y - a.y) - (b.y - a.y)*(c.x - a.x) > 0;
}</code>

If the point is on the right side of the straight line, the value calculated by this formula is greater than 0; if the point is on the left side of the straight line, the value is less than 0; if the points are collinear (that is, point C is on the straight line A--B) , the value is 0.

For a horizontal line, you can modify the formula to determine whether point C is above or below the line:

<code class="language-javascript">isAbove(a, b, c) {
  return (c.y - a.y) > 0;
}</code>

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