三维投影

三维投影是将三维空间中的点映射到二维平面上的方法。由于目前绝大多数图形数据的显示方式仍是二维的，因此三维投影的应用相当广泛，尤其是在计算机图形学，工程学和en:drafting中。

正交投影

正交投影是一系列用于显示三维物体的轮廓、细节或精确测量结果的变换方法。通常又被称作plan、截面图、鸟瞰图或立面图。

当视平面的法向（即摄像机的朝向）平行于笛卡尔坐标系三根坐标轴中的一根，数学变换定义如下： 若使用一个平行于y轴（侧视图）的正交投影将三维点$a_{x}$, $a_{y}$, $a_{z}$投影到二维平面上得到二维点$b_{x}$, $b_{y}$，可以使用如下公式

$b_{x}=s_{x}a_{x}+c_{x}$

$b_{y}=s_{z}a_{z}+c_{z}$

其中向量s是一个任意的缩放因子，而c是一个任意的偏移量。这些常量可自由选自，通常用于将视口调整到一个合适的位置。该投影变换同样可以使用矩阵表示（为清晰起见引入临时向量d）

${\begin{bmatrix}{d_{x}}\\{d_{y}}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&0&1\\\end{bmatrix}}{\begin{bmatrix}{a_{x}}\\{a_{y}}\\{a_{z}}\\\end{bmatrix}}$

${\begin{bmatrix}{b_{x}}\\{b_{y}}\\\end{bmatrix}}={\begin{bmatrix}{s_{x}}&0\\0&{s_{z}}\\\end{bmatrix}}{\begin{bmatrix}{d_{x}}\\{d_{y}}\\\end{bmatrix}}+{\begin{bmatrix}{c_{x}}\\{c_{z}}\\\end{bmatrix}}.$

虽然正交投影产生的图像在一定程度上反映了物体的三维特性，但此类投影图像和实际观测到的并不相同。特别是对于相同长度的平行线段，无论离虚拟观察者（摄像机）远近与否，它们都会在正交投影中显示为相同长度。这会导致较近的线段看起来被缩短了。

透视投影

See also 变换矩阵

已隱藏部分未翻譯内容，歡迎參與翻譯。

The perspective projection requires greater definition. A conceptual aid to understanding the mechanics of this projection involves treating the 2D projection as being viewed through a camera viewfinder. The camera's position, orientation, and field of view control the behavior of the projection transformation. The following variables are defined to describe this transformation:

• $\mathbf {a} _{x,y,z}$ - the point in 3D space that is to be projected.
• $\mathbf {c} _{x,y,z}$ - the location of the camera.
• $\mathbf {\theta } _{x,y,z}$ - The rotation of the camera. When $\mathbf {c} _{x,y,z}$=<0,0,0>, and $\mathbf {\theta } _{x,y,z}$=<0,0,0>, the 3D vector <1,2,0> is projected to the 2D vector <1,2>.
• $\mathbf {e} _{x,y,z}$ - the viewer's position relative to the display surface.

Which results in:

• $\mathbf {b} _{x,y}$ - the 2D projection of $\mathbf {a}$.

First, we define a point $\mathbf {d} _{x,y,z}$ as a translation of point $\mathbf {a}$ into a coordinate system defined by $\mathbf {c}$. This is achieved by subtracting $\mathbf {c}$ from $\mathbf {a}$ and then applying a vector rotation matrix using $-\mathbf {\theta }$ to the result. This transformation is often called a camera transform (note that these calculations assume a left-handed system of axes):

${\begin{bmatrix}\mathbf {d} _{x}\\\mathbf {d} _{y}\\\mathbf {d} _{z}\\\end{bmatrix}}={\begin{bmatrix}1&0&0\\0&{\cos -\mathbf {\theta } _{x}}&{\sin -\mathbf {\theta } _{x}}\\0&{-\sin -\mathbf {\theta } _{x}}&{\cos -\mathbf {\theta } _{x}}\\\end{bmatrix}}{\begin{bmatrix}{\cos -\mathbf {\theta } _{y}}&0&{-\sin -\mathbf {\theta } _{y}}\\0&1&0\\{\sin -\mathbf {\theta } _{y}}&0&{\cos -\mathbf {\theta } _{y}}\\\end{bmatrix}}{\begin{bmatrix}{\cos -\mathbf {\theta } _{z}}&{\sin -\mathbf {\theta } _{z}}&0\\{-\sin -\mathbf {\theta } _{z}}&{\cos -\mathbf {\theta } _{z}}&0\\0&0&1\\\end{bmatrix}}\left({{\begin{bmatrix}\mathbf {a} _{x}\\\mathbf {a} _{y}\\\mathbf {a} _{z}\\\end{bmatrix}}-{\begin{bmatrix}\mathbf {c} _{x}\\\mathbf {c} _{y}\\\mathbf {c} _{z}\\\end{bmatrix}}}\right)$

Or, for those less comfortable with matrix multiplication. Signs of angles are inconsistent with matrix form:

${\begin{array}{lcl}d_{x}&=&\cos \theta _{y}\cdot (\sin \theta _{z}\cdot (a_{y}-c_{y})+\cos \theta _{z}\cdot (a_{x}-c_{x}))-\sin \theta _{y}\cdot (a_{z}-c_{z})\\d_{y}&=&\sin \theta _{x}\cdot (\cos \theta _{y}\cdot (a_{z}-c_{z})+\sin \theta _{y}\cdot (\sin \theta _{z}\cdot (a_{y}-c_{y})+\cos \theta _{z}\cdot (a_{x}-c_{x})))+\cos \theta _{x}\cdot (\cos \theta _{z}\cdot (a_{y}-c_{y})-\sin \theta _{z}\cdot (a_{x}-c_{x}))\\d_{z}&=&\cos \theta _{x}\cdot (\cos \theta _{y}\cdot (a_{z}-c_{z})+\sin \theta _{y}\cdot (\sin \theta _{z}\cdot (a_{y}-c_{y})+\cos \theta _{z}\cdot (a_{x}-c_{x})))-\sin \theta _{x}\cdot (\cos \theta _{z}\cdot (a_{y}-c_{y})-\sin \theta _{z}\cdot (a_{x}-c_{x}))\\\end{array}}$

This transformed point can then be projected onto the 2D plane using the formula (here, x/y is used as the projection plane, literature also may use x/z):

${\begin{array}{lcl}\mathbf {b} _{x}&=&(\mathbf {d} _{x}-\mathbf {e} _{x})(\mathbf {e} _{z}/\mathbf {d} _{z})\\\mathbf {b} _{y}&=&(\mathbf {d} _{y}-\mathbf {e} _{y})(\mathbf {e} _{z}/\mathbf {d} _{z})\\\end{array}}$

Or, in matrix form using homogeneous coordinates:

${\begin{bmatrix}\mathbf {f} _{x}\\\mathbf {f} _{y}\\\mathbf {f} _{z}\\\mathbf {f} _{w}\\\end{bmatrix}}={\begin{bmatrix}1&0&0&-\mathbf {e} _{x}\\0&1&0&-\mathbf {e} _{y}\\0&0&1&0\\0&0&1/\mathbf {e} _{z}&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {d} _{x}\\\mathbf {d} _{y}\\\mathbf {d} _{z}\\1\\\end{bmatrix}}$

and

${\begin{array}{lcl}\mathbf {b} _{x}&=&\mathbf {f} _{x}/\mathbf {f} _{w}\\\mathbf {b} _{y}&=&\mathbf {f} _{y}/\mathbf {f} _{w}\\\end{array}}$

The distance of the viewer from the display surface, $\mathbf {e} _{z}$, directly relates to the field of view, where $\alpha =2\cdot \tan ^{-1}(1/\mathbf {e} _{z})$ is the viewed angle. (Note: This assumes that you map the points (-1,-1) and (1,1) to the corners of your viewing surface)

Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.

图示

已隱藏部分未翻譯内容，歡迎參與翻譯。

File:Perspective Transform Diagram.png

To determine which screen x coordinate corresponds to a point at Ax,Az multiply the point coordinates by:

${\text{screen x coordinate}}(Bx)={\text{model x coordinate}}(Ax)\times {\frac {{\text{distance from eye to screen}}(Bz)}{{\text{distance from eye to point}}(Az)}}$

the same works for the screen y coordinate:

${\text{screen y coordinate}}(By)={\text{model y coordinate}}(Ay)\times {\frac {{\text{distance from eye to screen}}(Bz)}{{\text{distance from eye to point}}(Az)}}$

(where Ax and Ay are coordinates occupied by the object before the perspective transform)

参看

• 计算机图形学
• 三维计算机图形
• 显示卡
• T&L
• 材质贴图
• 透视
• en:Homography
• 齐次坐标

深入阅读

• Kenneth C. Finney. 3D Game Programming All in One. Thomson Course. 2004: 93. ISBN 159200136X.