【数学】向量点乘、叉乘，应用及代码实现（C++）-3 代码实现

第一个版本代码，不用额外的库，手搓一些Utility函数，透彻了解原理：

#include<iostream>
#include<cmath>
using namespace std;

struct Point
{
    double x, y, z;
    // Overloading the multiplication operator
    Point operator*(double k) const 
    {
        return {k*x, k*y, k*z};
    }
    Point operator+(Point A) const
    {
        return {A.x + x, A.y + y, A.z + z};
    }
    bool operator==(Point A) const
    {
        if (A.x == x and A.y == y and A.z == z)
        {
            return true;
        }
        else
        {
            return false;
        }
    }
};

double dotProduct(Point A, Point B)
{
    return A.x * B.x + A.y * B.y + A.z * B.z;
}

Point crossProduct(Point A, Point B)
{
    return {A.y * B.z - A.z * B.y,
            A.x * B.z - A.z * B.x,
            A.x * B.y - A.y * B.x};
}

float calcNorm(Point A)
{
    return sqrt(A.x * A.x + A.y * A.y + A.z * A.z);
}

Point calcProjection(Point A, Point B, Point P)
{
    Point AB = {B.x - A.x, B.y - A.y, B.z - A.z};
    Point AP = {P.x - A.x, P.y - A.y, P.z - A.z};
    double dot_product = dotProduct(AB, AP);
    double k = dot_product / dotProduct(AB, AB);
    Point C = A + (AB * k);
    return C;
}

bool verifyProjection(Point A, Point B, Point P, Point C)
{
    Point AC = {C.x - A.x, C.y - A.y, C.z - A.z};
    Point AB = {B.x - A.x, B.y - A.y, B.z - A.z};
    Point PC = {C.x - P.x, C.y - P.y, C.z - P.z};
    double dot_product = dotProduct(PC, AB);
    Point cross_product = crossProduct(AC, AB);
    Point zero_vec = {0, 0, 0};
    if (dot_product == 0 and cross_product == zero_vec)
    {
        return true;
    }
    else
    {
        return false;
    }
}

float calcDistance(Point A, Point B, Point P)
{
    Point AB = {B.x - A.x, B.y - A.y, B.z - A.z};
    Point AP = {P.x - A.x, P.y - A.y, P.z - A.z};
    Point cross_product = crossProduct(AB, AP);
    float area_parallelogram = calcNorm(cross_product);
    return (area_parallelogram / calcNorm(AB));
}

int main()
{
    // Line segment AB
    Point A = {0, 0, 0};
    Point B = {4, 0, 0};
    // Point P
    Point P = {5, 8, 0};
    // Project P to AB and get point C
    Point C = calcProjection(A, B, P);
    cout << "Projection Point C: (" << C.x << ", " << C.y << ", " << C.z << ")" << endl;
    if (verifyProjection(A, B, P, C))
    {
        cout << "Correct!" << endl;
    }
    else
    {
        cout << "Incorrect." << endl;
    }
    cout << "Distance from P to AB is: " << calcDistance(A, B, P) << endl;
    return 0;
}

另外一个版本，通过使用Eigen库来避免自己写Utility函数，行数大大减少（君子生非异也，善假于物也。）：

#include <iostream>
#include <Eigen/Dense>
using namespace std;
using namespace Eigen;

Vector3d calcProjection(Vector3d A, Vector3d B, Vector3d P)
{
    Vector3d AB = B - A;
    Vector3d AP = P - A;
    float AC_norm = AB.dot(AP) / AB.norm();
    Vector3d C = A + AC_norm / AB.norm() * AB;
    return C;
}

bool verifyProjection(Vector3d A, Vector3d B, Vector3d P, Vector3d C)
{
    Vector3d AB = B - A;
    Vector3d PC = P - C;
    Vector3d AC = C - A;
    Vector3d zero_vec = {0, 0, 0};
    Vector3d cross_product = AB.cross(AC);
    float dot_product = PC.dot(AB);
    if (dot_product == 0 and cross_product == zero_vec)
    {
        return true;
    }
    else
    {
        return false;
    }
}

float calcDistance(const Vector3d A, const Vector3d B, const Vector3d P)
{
    Vector3d AB = B - A;
    Vector3d AP = P - A;
    Vector3d cross_product = AB.cross(AP);
    float area_parallelogram = cross_product.norm();
    return area_parallelogram / AB.norm();
}

int main()
{
    Eigen::Vector3d A = {0, 0, 0};
    Eigen::Vector3d B = {2, 0, 0};
    Eigen::Vector3d P = {1, 3, 0};

    Vector3d C = calcProjection(A, B, P);
    cout << "Projection point is: " << C.x() << ", " << C.y() << ", " << C.z() << endl;
    if (verifyProjection(A, B, P, C))
    {
        cout << "Verification passes!" << endl;
    }
    else
    {
        cout << "Verification failed." << endl;
    }
    cout << "Distance from P to AB is: " << calcDistance(A, B, P) << endl;
}

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