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Cross product

lastmod 2024-03-07created 2016-08-2912 edits

WARNING

This page is about the ‘cross product’, where multiplying vectors gives a vector. For the ‘outer product’, where multiplying vectors gives a matrix, see that page.

Definition of the cross product Right-hand rule

The vector product, or cross product, is a type of binary operation between vectors in 3-dimensional space in mathematics. Unlike the scalar product, whose result is a scalar, its result is a vector. It appears in physics formulas such as angular momentum and the Lorentz force.

Uses

  • Used to find a vector perpendicular to both vectors.
  • Using the property above, you can find the relative position of a point to a line (left side, right side).

Relative position of a point to a line

Using the right-hand rule of the cross product (Cross-Product), this uses the property that the result becomes negative or positive depending on the positions of Vector(p1 - p0) and Vector(p2 - p0).

\(\begin{align} & \vec{v_1} = p1 - p0 = [p1_x - p0_x, p1_y - p0_y, 0] \\ & \vec{v_2} = p2 - p0 = [p2_x - p0_x, p2_y - p0_y, 0] \\ & \vec{v_1} \times \vec{v_2} = [0, 0, ((p1_x - p0_x) * (p2_y - p0_y) - (p1_y - p0_y) * (p2_x - p0_x))] \\ \end{align}\)

C++ code sample: 

/**
 * Obtain the relative position of point to line.
 *
 * @translate{ko, 라인을 기준으로 점의 상대적 위치를 확인한다.}
 *
 * @param p0    [in] Point 0 of line.
 * @param p1    [in] Point 1 of line.
 * @param check [in] Check point.
 *
 * @return If a positive number, the right side of the vector. @n
 *         if a negative number, the left side of the vector. @n
 *         if 0 is on the line.
 *
 * @remarks
 *  Example:
 *  @code
 *    |
 *    |         * p2
 *    |
 *    |  p0 *------>* p1
 *    |
 *   -+------------------
 *    |
 *  @endcode
 *  \f[
 *   \vec{v_1} = p1 - p0 = [p1_x - p0_x, p1_y - p0_y, 0]
 *   \vec{v_2} = p2 - p0 = [p2_x - p0_x, p2_y - p0_y, 0]
 *   \vec{v_1} \times \vec{v_2} = [0, 0, ((p1_x - p0_x) * (p2_y - p0_y) - (p1_y - p0_y) * (p2_x - p0_x))]
 *  \f]
 *  외적(Cross-Product)의 오른손 법칙을 사용하여, Vector(p1 - p0)과 Vector(p2 - p0)의 위치에 따라 음수 또는 양수로 변하는 성질을 이용한다.
 */
template <typename T, typename Point = libtbag::geometry::BasePoint2<T> >
inline T getRelativePositionOfPointToLine(Point const & p0, Point const & p1, Point const & check)
{
    return ((p1.x - p0.x) * (check.y - p0.y)) - ((p1.y - p0.y) * (check.x - p0.x));
}

enum RelativePosition
{
    RELATIVE_POSITION_RIGHT,
    RELATIVE_POSITION_LEFT,
    RELATIVE_POSITION_ON
};

template <typename T, typename Point = libtbag::geometry::BasePoint2<T> >
inline RelativePosition getRelativePositionOfPointToLine2(Point const & p0, Point const & p1, Point const & check)
{
    T const RESULT = getRelativePositionOfPointToLine<T, Point>(p0, p1, check);
    if (RESULT > 0) {
        return RELATIVE_POSITION_RIGHT;
    } else if (RESULT < 0) {
        return RELATIVE_POSITION_LEFT;
    } else {
        return RELATIVE_POSITION_ON;
    }
}

See also

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References

  1. 2DGeometry_-line_point_position-_cross_product.pdf ↩

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