一:点乘介绍
1. 向量点乘:
2. 向量点乘的性质:
3. 向量点乘公式:
4. 向量的点乘的属性:
(1):向量与自身做点乘,会得到向量长度的平方:
(2):向量长度,为向量与自身点乘后再求平方根:
(3):向量投影,将a向量投影到向量b上:
(4):向量夹角:
二:叉乘介绍:
1. 向量叉乘:
2. 向量叉乘公式:
3. 向量叉乘的属性:
判断三个向量是否共面:
三:应用1 - 求两直线的交点:
(1)2D直线方程:
(2)将直线1带入直线2中: ,叉乘等于0,意味着两向量共线。
(3)求交点:
四:应用2 - 求三个平面的交点:
(1):三个平面方程:
(2):三个平面方程,三个未知数,利用克拉默法则求解即可。
三:实现
cpp
#ifndef _POINT_H_
#define _POINT_H_
#include <iostream>
#include <cmath>
class Point2D
{
public:
float x, y;
Point2D() {}
Point2D(float x, float y) : x(x), y(y) {}
Point2D &operator+=(const Point2D &t)
{
x += t.x;
y += t.y;
return *this;
}
Point2D &operator-=(const Point2D &t)
{
x -= t.x;
y -= t.y;
return *this;
}
Point2D &operator*=(float t)
{
x *= t;
y *= t;
return *this;
}
Point2D &operator/=(float t)
{
x /= t;
y /= t;
return *this;
}
Point2D operator+(const Point2D &t) const
{
return Point2D(*this) += t;
}
Point2D operator-(const Point2D &t) const
{
return Point2D(*this) -= t;
}
Point2D operator*(float t) const
{
return Point2D(*this) *= t;
}
Point2D operator/(float t) const
{
return Point2D(*this) /= t;
}
float dot(const Point2D& b) const
{
return x * b.x + y * b.y;
}
friend std::ostream &operator<<(std::ostream &out, const Point2D &t)
{
out << '(' << t.x << ',' << t.y << ')';
return out;
}
};
Point2D operator*(float a, const Point2D &b)
{
return b * a;
}
float dot(const Point2D& a, const Point2D& b)
{
return a.dot(b);
}
float norm(const Point2D& a)
{
return dot(a, a);
}
double abs(const Point2D& a) {
return sqrt(norm(a));
}
double proj(const Point2D& a, const Point2D& b)
{
return dot(a, b) / abs(b);
}
double angle(const Point2D& a, const Point2D& b)
{
return acos(dot(a, b) / abs(a) / abs(b));
}
float cross(const Point2D& a, const Point2D& b)
{
return a.x * b.y - a.y * b.x;
}
Point2D intersect(const Point2D& a1, const Point2D& d1, const Point2D& a2, const Point2D& d2)
{
return a1 + cross(a2 - a1, d2) / cross(d1, d2) * d1;
}
class Point3D: public Point2D
{
public:
float z;
Point3D() {}
Point3D(float x, float y, float z) : Point2D(x, y), z(z) {}
Point3D &operator+=(const Point3D &t)
{
x += t.x;
y += t.y;
z += t.z;
return *this;
}
Point3D &operator-=(const Point3D &t)
{
x -= t.x;
y -= t.y;
z -= t.z;
return *this;
}
Point3D &operator*=(float t)
{
x *= t;
y *= t;
z *= t;
return *this;
}
Point3D &operator/=(float t)
{
x /= t;
y /= t;
z /= t;
return *this;
}
Point3D operator+(const Point3D &t) const
{
return Point3D(*this) += t;
}
Point3D operator-(const Point3D &t) const
{
return Point3D(*this) -= t;
}
Point3D operator*(float t) const
{
return Point3D(*this) *= t;
}
Point3D operator/(float t) const
{
return Point3D(*this) /= t;
}
float dot(const Point3D &t) const
{
return x * t.x + y * t.y + z * t.z;
}
friend std::ostream &operator<<(std::ostream &out, const Point3D &t)
{
out << '(' << t.x << ',' << t.y << ',' << t.z << ')';
return out;
}
};
Point3D operator*(float a, Point3D b)
{
return b * a;
}
float dot(const Point3D& a, const Point3D& b)
{
return a.dot(b);
}
float norm(const Point3D& a)
{
return dot(a, a);
}
double abs(const Point3D& a) {
return sqrt(norm(a));
}
double proj(const Point3D& a, const Point3D& b)
{
return dot(a, b) / abs(b);
}
double angle(const Point3D& a, const Point3D& b)
{
return acos(dot(a, b) / abs(a) / abs(b));
}
Point3D cross(const Point3D& a, const Point3D& b)
{
return Point3D(a.y * b.z - a.z * b.y,
a.z * b.x - a.x * b.z,
a.x * b.y - a.y * b.x);
}
float triple(const Point3D& a, const Point3D& b, const Point3D& c)
{
return dot(a, cross(b, c));
}
Point3D intersect(const Point3D& a1, const Point3D& n1, const Point3D& a2, const Point3D& n2, const Point3D& a3, const Point3D& n3)
{
Point3D x(n1.x, n2.x, n3.x);
Point3D y(n1.y, n2.y, n3.y);
Point3D z(n1.z, n2.z, n3.z);
Point3D d(dot(a1, n1), dot(a2, n2), dot(a3, n3));
return Point3D(triple(d, y, z),
triple(x, d, z),
triple(x, y, d)) / triple(n1, n2, n3);
}
#endif