知识点 - 计算几何基础
讲义
点
我们把点 (mathbf r) 看成从 (mathbf 0) 到 (mathbf r)的向量 (vec{mathbf r})
#define ftype long double
struct point2d {
ftype x, y;
point2d() {}
point2d(ftype x, ftype 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*=(ftype t) {
x *= t;
y *= t;
return *this;
}
point2d& operator/=(ftype 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*(ftype t) const {
return point2d(*this) *= t;
}
point2d operator/(ftype t) const {
return point2d(*this) /= t;
}
};
point2d operator*(ftype a, point2d b) {
return b * a;
}
点乘
- (mathbf a cdot mathbf b = mathbf b cdot mathbf a)
- ((alpha cdot mathbf a)cdot mathbf b = alpha cdot (mathbf a cdot mathbf b))
- ((mathbf a + mathbf b)cdot mathbf c = mathbf a cdot mathbf c + mathbf b cdot mathbf c)
若有单位向量:
我们定义 (mathbf r = (x;y;z)) 表示 (r = x cdot mathbf e_x + y cdot mathbf e_y + z cdot mathbf e_z).
因为
所以对 (mathbf a = (x_1;y_1;z_1)) 和 (mathbf b = (x_2;y_2;z_2)) 有
ftype dot(point2d a, point2d b) {
return a.x * b.x + a.y * b.y;
}
ftype dot(point3d a, point3d b) {
return a.x * b.x + a.y * b.y + a.z * b.z;
}
一些定义:
- Norm of (mathbf a) (长度的平方): (|mathbf a|^2 = mathbf acdot mathbf a)
- Length of (mathbf a): (|mathbf a| = sqrt{mathbf acdot mathbf a})
- Projection of (mathbf a) onto (mathbf b)(投影): (dfrac{mathbf acdotmathbf b}{|mathbf b|})
- Angle between vectors(夹角): (arccos left(dfrac{mathbf acdot mathbf b}{|mathbf a| cdot |mathbf b|} ight))
- 从上一点说明点乘的正负可以用来判断锐角(acute)钝角(obtuse)直角(orthogonal).
ftype norm(point2d a) {
return dot(a, a);
}
double abs(point2d a) {
return sqrt(norm(a));
}
double proj(point2d a, point2d b) {
return dot(a, b) / abs(b);
}
double angle(point2d a, point2d b) {
return acos(dot(a, b) / abs(a) / abs(b));
}
叉乘
定义:
先定义三重积triple product (mathbf acdot(mathbf b imes mathbf c)) 为上面这个平行六面体的体积,于是我们可以得到(mathbf b imes mathbf c)的大小和方向。
性质:
- (mathbf a imes mathbf b = -mathbf b imes mathbf a)
- ((alpha cdot mathbf a) imes mathbf b = alpha cdot (mathbf a imes mathbf b))
- (mathbf acdot (mathbf b imes mathbf c) = mathbf bcdot (mathbf c imes mathbf a) = -mathbf acdot( mathbf c imes mathbf b))
- ((mathbf a + mathbf b) imes mathbf c = mathbf a imes mathbf c + mathbf b imes mathbf c).
对任意的 (mathbf r) 有:[mathbf rcdot( (mathbf a + mathbf b) imes mathbf c) = (mathbf a + mathbf b) cdot (mathbf c imes mathbf r) = mathbf a cdot(mathbf c imes mathbf r) + mathbf bcdot(mathbf c imes mathbf r) = mathbf rcdot (mathbf a imes mathbf c) + mathbf rcdot(mathbf b imes mathbf c) = mathbf rcdot(mathbf a imes mathbf c + mathbf b imes mathbf c) ]这证明了第三点 ((mathbf a + mathbf b) imes mathbf c = mathbf a imes mathbf c + mathbf b imes mathbf c) - (|mathbf a imes mathbf b|=|mathbf a| cdot |mathbf b| sin heta)
因为
于是我们可以算出 (mathbf a = (x_1;y_1;z_1)) 和 (mathbf b = (x_2;y_2;z_2)) 的叉乘结果:
用行列式表达的话:
二维的叉乘 (namely the pseudo-scalar product)可以被定义为
一个直观理解方式是为了计算(|mathbf a| cdot |mathbf b| sin heta) 将 (mathbf a)转 (90^circ)得到(widehat{mathbf a}=(-y_1;x_1)).于是(|widehat{mathbf a}cdotmathbf b|=|x_1y_2 - y_1 x_2|).
point3d cross(point3d a, 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);
}
ftype triple(point3d a, point3d b, point3d c) {
return dot(a, cross(b, c));
}
ftype cross(point2d a, point2d b) {
return a.x * b.y - a.y * b.x;
}
直线与平面
一个直线可以被表示为一个起始点(mathbf r_0) 和一个方向向量(mathbf d) ,或者两个点(mathbf a) , (mathbf b).对应的方程为
一个平面可以被三个点确定: (mathbf a), (mathbf b) , (mathbf c)。或者一个初始点(mathbf r_0)和一组在这个平面里的向量(mathbf d_1) , (mathbf d_2)确定:
直线交点
(l_1:mathbf r = mathbf a_1 + t cdot mathbf d_1) 带入 (l_2:(mathbf r - mathbf a_2) imes mathbf d_2=0)
point2d intersect(point2d a1, point2d d1, point2d a2, point2d d2) {
return a1 + cross(a2 - a1, d2) / cross(d1, d2) * d1;
}
三个平面交点
给你三个平面的初始点 (mathbf a_i) 和法向量 (mathbf n_i) 于是得到方程:
用克拉默法则解:
point3d intersect(point3d a1, point3d n1, point3d a2, point3d n2, point3d a3, 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);
}
模板
两个流派,一个是向量表示直线即两个点(mathbf a) , (mathbf b),另一个是直线方程即(a_1 x + b_1 y + c_1 = 0)