【图形学】投影和消隐简介

投影

正交投影

对于物体上任意一点的三维坐标P(x,y,z),投影后的三维坐标为 $P^\prime(x^\prime,y^\prime,z^\prime)$ ,那么正交投影的方程为 $\begin{cases} x^\prime=x\\y^\prime=y\\z^\prime=0 \end{cases}$

斜投影

在这里插入图片描述

如图所示，空间中一点 $P_1(x,y,z)$ 在xOy面上的斜投影坐标 $P_2(x^\prime,y^\prime,0)$ 正交投影点 $P_3(x,y,0)$ ,那么有 $\begin{cases} x^\prime=x-L\cos\beta=x-z\cot\alpha\cos\beta\\ y^\prime=y-L\sin\beta=y-z\cot\alpha\sin\beta \end{cases}$

透视投影

透视投影有三个坐标系，世界坐标系，观察坐标系，屏幕坐标系
在这里插入图片描述

在这里插入图片描述

观察坐标系到投影坐标系的转换可以用相似求出
在这里插入图片描述

$\begin{cases} x^\prime=n\cdot \frac{x}{z}\\ y^\prime=n\cdot \frac{y}{z} \end{cases}$

建立投影类

class CProjection
{
public:CProjection();~CProjection();void SetViewPoint(double R);CPoint3 GetViewPoint();CPoint2 ObliqueProjection(CPoint3 WorldPoint);//正交投影CPoint2 OrthogonalProjection(CPoint3 WorldPoint);//斜二测投影CPoint2 PerspectiveProjection(CPoint3 WorldPoint);//透视投影
private:CPoint3 m_viewPoint;double R, d;
};CProjection::CProjection()
{R = 1200; d = 800;m_viewPoint.m_x = 0;m_viewPoint.m_y = 0;m_viewPoint.m_z = R;
}CProjection::~CProjection()
{
}void CProjection::SetViewPoint(double R)
{this->R = R;
}CPoint3 CProjection::GetViewPoint()
{return m_viewPoint;
}CPoint2 CProjection::ObliqueProjection(CPoint3 WorldPoint)//斜二测投影
{CPoint2 ScreenPoint;ScreenPoint.m_x = WorldPoint.m_x - 0.3536 * WorldPoint.m_z;ScreenPoint.m_y = WorldPoint.m_y - 0.3536 * WorldPoint.m_z;return ScreenPoint;
}CPoint2 CProjection::OrthogonalProjection(CPoint3 WorldPoint)//正交投影
{CPoint2 ScreenPoint;ScreenPoint.m_x = WorldPoint.m_x ;ScreenPoint.m_y = WorldPoint.m_y ;return ScreenPoint;
}CPoint2 CProjection::PerspectiveProjection(CPoint3 WorldPoint)
{CPoint2 ScreenPoint;CPoint3 ViewPoint;ViewPoint.m_x = WorldPoint.m_x;ViewPoint.m_y = WorldPoint.m_y;ViewPoint.m_z = m_viewPoint.m_z - WorldPoint.m_z;ScreenPoint.m_x = d * ViewPoint.m_x / ViewPoint.m_z;ScreenPoint.m_y = d * ViewPoint.m_y / ViewPoint.m_z;return ScreenPoint;
}

在这里插入图片描述

消隐，背面剔除算法

给定视点位置或视线方向后，确定场景中哪些物体表面是可见的、哪些物体表面是不可见的，即是消隐。
背面剔除算法主要是针对凸多面体，其表面要么可见，要么不可见。算法要给出测试每个表面是否可见的表达式。
在这里插入图片描述
如图所示，根据表面的外法向量 $\vec N$ 和式向量 $\vec V$ 的夹角 $\theta$ 来进行可见性判定。
$\vec N=\vec {P_2P_4} \times \vec {P_3P_4}$
给定视点坐标 $O(x_v,y_v,z_v)$ 后，视向量表示为 $\vec V=(x_v-x_4,y_v-y_4,z_v-z_4)$
将法向量 $\vec N$ 归一化为单位向量 $\vec n$ ,将视向量归一化为单位向量 $\vec v$ ,则有
$\vec n \cdot \vec v=\cos\theta$
可见性的判定如下：

当 $\cos\theta>0$ 时，表面可见，绘制多边形的边界线
当 $\cos\theta=0$ 时，表面多边形退化为一条直线
当 $\cos\theta<0$ 时，表面多边形不可见
为了实现背面剔除算法，这里设计一个三维向量类CVector3

这里给的例子是一个立方体，那如果是一个贝塞尔曲面拟合的物体(比如球)，那么用递归曲面片的方式绘制曲面，对于每一个细分曲面片又可以看成一个平面四边形，求出法向量即可。
在这里插入图片描述

三维向量类CVector3

class CVector3
{
public:CVector3();virtual ~CVector3();CVector3(double x, double y, double z);//绝对向量CVector3(const CPoint3& p);CVector3(const CPoint3& p0, const CPoint3& p1);//相对向量double Magnitude();//计算向量的模CVector3 Normalize();//归一化向量friend CVector3 operator + (const CVector3& v0, const CVector3& v1);//运算符重载friend CVector3 operator - (const CVector3& v0, const CVector3& v1);friend CVector3 operator * (const CVector3& v, double t);friend CVector3 operator * (double t, const CVector3& v);friend CVector3 operator / (const CVector3& v, double t);friend double DotProduct(const CVector3& v0, const CVector3& v1);//计算向量的点积friend CVector3 CrossProduct(const CVector3& v0, const CVector3& v1);//计算向量的叉积
private:double m_x, m_y, m_z;
};
CVector3::CVector3(void)
{m_x = 0.0, m_y = 0.0, m_z = 1.0;//指向z轴正向
}CVector3::~CVector3(void)
{
}CVector3::CVector3(double x, double y, double z)//绝对向量
{m_x = x;m_y = y;m_z = z;
}
CVector3::CVector3(const CPoint3& p)
{m_x = p.m_x;m_y = p.m_y;m_z = p.m_z;
}
CVector3::CVector3(const CPoint3& p0, const CPoint3& p1)//相对向量
{m_x = p1.m_x - p0.m_x;m_y = p1.m_y - p0.m_y;m_z = p1.m_z - p0.m_z;
}
double CVector3::Magnitude(void)//向量的模
{return sqrt(m_x * m_x + m_y * m_y + m_z * m_z);
}
CVector3 CVector3::Normalize(void)//归一化为单位向量
{CVector3 vector;double magnitude = sqrt(m_x * m_x + m_y * m_y + m_z * m_z);if (fabs(magnitude) < 1e-6)magnitude = 1.0;vector.m_x = m_x / magnitude;vector.m_y = m_y / magnitude;vector.m_z = m_z / magnitude;return vector;
}
void CVector3::IntoOut()
{if (m_x * m_y < 0&&m_x*m_z>0) {m_x = -m_x, m_z = -m_z;}else if (m_x * m_z < 0 && m_x * m_y>0) {m_x = -m_x, m_y = -m_y;}else if (m_y * m_z < 0 && m_y * m_z>0) {m_y = -m_y, m_z = -m_z;}
}
CVector3 operator + (const CVector3& v0, const CVector3& v1)//向量的和
{CVector3 vector;vector.m_x = v0.m_x + v1.m_x;vector.m_y = v0.m_y + v1.m_y;vector.m_z = v0.m_z + v1.m_z;return vector;
}CVector3 operator - (const CVector3& v0, const CVector3& v1)//向量的差
{CVector3 vector;vector.m_x = v0.m_x - v1.m_x;vector.m_y = v0.m_y - v1.m_y;vector.m_z = v0.m_z - v1.m_z;return vector;
}CVector3 operator * (const CVector3& v, double t)//向量与常量的积
{CVector3 vector;vector.m_x = v.m_x * t;vector.m_y = v.m_y * t;vector.m_z = v.m_z * t;return vector;
}CVector3 operator * (double t, const CVector3& v)//常量与向量的积
{CVector3 vector;vector.m_x = v.m_x * t;vector.m_y = v.m_y * t;vector.m_z = v.m_z * t;return vector;
}CVector3 operator / (const CVector3& v, double scalar)//向量数除
{if (fabs(scalar) < 1e-6)scalar = 1.0;CVector3 vector;vector.m_x = v.m_x / scalar;vector.m_y = v.m_y / scalar;vector.m_z = v.m_z / scalar;return vector;
}double DotProduct(const CVector3& v0, const CVector3& v1)//向量的点积
{return(v0.m_x * v1.m_x + v0.m_y * v1.m_y + v0.m_z * v1.m_z);
}CVector3 CrossProduct(const CVector3& v0, const CVector3& v1)//向量的叉积
{CVector3 vector;vector.m_x = v0.m_y * v1.m_z - v0.m_z * v1.m_y;vector.m_y = v0.m_z * v1.m_x - v0.m_x * v1.m_z;vector.m_z = v0.m_x * v1.m_y - v0.m_y * v1.m_x;return vector;
}