Cross Product

Method: point.cross(that)

Returns: number

Description

In 2D, the cross product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\) is a scalar (the \(z\)-component of the 3D cross product with the vectors embedded in the \(xy\)-plane):

\[\mathbf{a} \times \mathbf{b} = a_x b_y - a_y b_x\]

This is equivalent to:

\[\mathbf{a} \times \mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\sin\theta\]

where \(\theta\) is the signed angle measured counter-clockwise from \(\mathbf{a}\) to \(\mathbf{b}\).

The magnitude equals the area of the parallelogram formed by the two vectors, and equivalently twice the area of the triangle they enclose.

Interpreting the sign

Turn direction test

Given three points \(A\), \(B\), \(C\), the cross product of \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) tells you the winding order:

\[\overrightarrow{AB} \times \overrightarrow{AC} = (B_x - A_x)(C_y - A_y) - (B_y - A_y)(C_x - A_x)\]

Positive — \(A \to B \to C\) turns left (counter-clockwise)
Zero — the three points are collinear
Negative — \(A \to B \to C\) turns right (clockwise)

Implementation

cross(that) {
  return this.x * that.y - this.y * that.x;
}

Usage in Design

The cross product is used to determine arc sweep direction (clockwise vs counter-clockwise) and to test the winding order of geometry — for example, when deciding which side of a line a point lies on, or when resolving arc direction during polyline arc-segment creation.