二维交叉积定义 [英] 2d cross product definition

问题描述

在确定线段是否在多边形内中，我注意到接受的答案有一个不寻常的二维交叉定义:

In determine if line segment is inside polygon I noticed the accepted answer has an unusual 2d cross roduct definition of:

(u1, u2) x (v1, v2) := (u1 - v2)*(u2 - v1)

我从来没有遇到过这样的二维叉积的定义.任何人都可以启发我了解这种定义的起源吗?

I have never encountered a definition of the 2d cross product such as this one. Can anyone enlighten me as to where such definition originates from?

推荐答案

我建议你看看 外代数.它概括了叉积和行列式的概念.描述平面区域的动机示例"部分完全回答了您的问题.

I suggest you to take a look at Exterior Algebra. It generalizes the notion of cross product and determinant. The "Motivation examples" section describing areas in the plane answers exactly your question.

它适用于任何维度.3D 是一种特殊情况，其中两个向量的叉积结果也有 3 个分量.但是，在 2D 中，只有一个结果分量，而在 4D 中，则有 6 个.在 4D 中，您可以应用一种使用 3 个向量的叉积，这也为您提供 4 个分量.

It works in any dimension. 3D is a specific case where the result of the cross product of two vectors also has 3 components. However, in 2D, there is only one resulting component, and in 4D, there is 6. In 4D, you can apply a kind of cross product using 3 vectors, which give you also 4 components.

需要注意的是，虽然 3D 中叉积的结果有 3 个分量，但单位和含义是不同的.例如，x 分量具有面积单位，并表示 YZ 平面中的面积，这与标准"向量相反，其中 x 分量具有长度单位并且是坐标差.使用外部代数，这些差异变得更加清晰，因为符号也不同(dx vs dy^dz).

It is important to note that while the result of a cross product in 3D has 3 components, the units and the meaning are different. For example, the x-component has units of area, and represents the area in the YZ plane, as opposed to a "standard" vector where the x-component has unit of length and is a difference is coordinates. Using exterior algebra, these differences become clearer since the notation is also different (dx vs dy^dz).

注意:您引用的答案有误.而不是 (u1, u2) x (v1, v2) := (u1 - v2)*(u2 - v1)，它应该是 (u1, u2) x (v1, v2) := u1*v2 - u2*v1

Note: The answer that you referenced has a mistake. Instead of (u1, u2) x (v1, v2) := (u1 - v2)*(u2 - v1), it should be (u1, u2) x (v1, v2) := u1*v2 - u2*v1

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