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

Use the cross product to determine whether the point is on the left or right side of the line

In geometry it is often necessary to determine whether a point is to the left or right of a line. There are several ways to solve this problem, one of the most straightforward is to utilize cross products.

Cross product formula

To calculate the cross product of two vectors, we use the following formula:

<code>叉积 = (x1 * y2) - (x2 * y1)</code>

Where (x1, y1) and (x2, y2) are the coordinates of the two points that define the vector.

Use cross product to determine left or right side

Consider a line defined by points a and b, and a point c whose position we want to determine relative to the line. We can do this by computing the cross product of vectors a-c and b-c:

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

Interpretation of results

• If the result is greater than 0, then point c is to the left of line ab.
• If the result is less than 0, then point c is to the right of line ab.
• If the result is equal to 0, then point c is collinear with ab (i.e. it lies on the line itself).

Horizontal line

When line ab is a horizontal line, if point c is above the line, the above formula returns True; if point c is below the line, it returns False.

The above is the detailed content of How Can We Determine if a Point Lies to the Left or Right of a Line Using Cross Products?. For more information, please follow other related articles on the PHP Chinese website!