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

Efficiently Determining Whether a Point Lies Left or Right of a Line

Many applications require determining a point's position relative to a line. This article presents an efficient method for identifying whether a point falls to the left or right of a given line.

This task is readily accomplished using the cross product. Consider a line segment defined by points a and b, and a point c. The following formula determines the point's side:

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

A positive result indicates point c lies to the left of the line segment a–b. A negative result signifies it's to the right.

For a horizontal line, a positive result means c is above the line.

It's crucial to remember line orientation. If the line is defined from b to a, the formula becomes:

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

This approach provides a simple and effective way to categorize points based on their location relative to a line, streamlining data management and analysis.

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