Painting metaphor

   A qualified painter can present the scene he sees on the canvas with paint , And for unqualified painters , Give him a transparent canvas , He can copy the scenery through the canvas （ Similar to reddening ）. The latter has more general enlightenment significance for graphics .

Enlightenment of transparent canvas

   In the example of painting with transparent canvas , The plane on which the canvas is located is called Visual plane , The canvas itself is called View plane window , The view plane window can be divided into countless grid cells （ Scene pixels ）, The painter's eyes are called Projection center （COP）. There are three main components of painting metaphor that need to be further studied ： scene 、 View 、 Rendering process .

scene ： A scene is a collection of objects , Each object has its geometric and material properties . The coordinate system that describes the whole scene is called the world coordinate system （WC）

View ： View is the angle of viewing the scene , Its nature is similar to that of a camera

Rendering ： The rendering process is equivalent to the painter observing the scene through the canvas , Finally determine the scene color and fill

Other concepts

The vistas :

Anamorphosis ： Because the sampling of the image is discontinuous , Discretization produces what is called “ Heteromorphism ”. First, the boundary is not smooth due to the low sampling frequency in the scene , Second, it is caused by using a single color to represent multiple colors .

Projection 、 light 、 Brightness equation

Projection ： Implementation of camera model

Full camera description

In order to completely describe the camera model , The following concepts need to be redefined ：
View reference point VRP： World coordinate system （WC） A point in , Defines the origin of the new observation coordinate system .
The normal direction of the apparent plane VPN：WC A vector in , Its direction specifies the positive of the observation coordinate system z Axis direction .（ It can be regarded as the camera facing ）
View up vector VUV： by WC Vector in , Redefined the new coordinate system y Affirmative direction .
Distance from the apparent plane ： from VRP Start along VPN Distance to the viewing plane
Projection type ： The main projection types are parallel projection and perspective projection
Front clipping plane and rear clipping plane ： Some planes parallel to the viewing plane , Placed in and VPR Where the distance is a certain value .

Projection

The process of projection is the process of using some method to represent three-dimensional objects on a two-dimensional plane . There are two basic ways of projection , Parallel projection and perspective projection , Then there are many subtypes .

Parallel projection ： The projected rays are straight lines parallel to each other , The projection of the object is not affected by the change of distance , This projection method is called the parallel projection method . Parallel projection will give people “ Not true ” Feelings , Because it is inconsistent with the results of our daily observation , But in the field of engineering, parallel projection plays an important role , Because the parallel projection keeps the real side of the object from some angle .

Perspective projection ： The figure is obtained by projecting an object onto a single projection plane from a projection center , The effect it brings is closer to the natural visual effect .

Normative framework ： Normative framework is a basic arrangement of projection , It is used to capture the basic elements of perspective projection and parallel projection . For perspective projection , The normative framework refers to COP stay （0,0,-1） It's about , Visual plane and UV plane , And the view plane window is UV In the plane UV Upper range （-1,1） Square window . For parallel projection , The standard frame refers to the projection direction （0,0,-1）, The viewing plane window is the same as the perspective projection .

   Converting to a canonical framework requires a linear transformation of the original space , Using the knowledge of linear algebra mentioned in the introduction , It can complete the transformation from the scene to the standard perspective framework .

Stereoscopic view ： Stereoscopic views form stereoscopic images by constructing left and right eye images respectively 3D Images . There are two ways to produce 3D Image method . One is the idea of three-dimensional overlap , Regard both eyes as having the same view plane and view plane method , Only the projection center is different , There is a common overlap area on the view plane . This method may bring distortion and visual discomfort . Another method is to regard the two eyes as independent view reference points , FA Xiang is also different , This method will generate a non flat virtual image plane , It may also cause discomfort .

Clipping polygon algorithm

   When we finish the projection of the figure to the viewing plane , We need to further judge which parts of the polygon on the view plane can be used by us " notice " Of , That is, you need to find the part of the polygon that falls into the view plane window , This involves the clipping of polygons .

Sutherland-hodhman Algorithm

A two-dimensional
   Initialize an empty vertex sequence $P$ , Traverse all edges of the polygon in order , For each edge , Perform the following operations according to different conditions of the edge
（1） Line segment $\overrightarrow{p_0{p}_1}$ Entered the clipping area ： set up $p$ Is the intersection of line segment and boundary , take $p$ and $p_1$ In turn $P$ Tail of .
（2） Line segment $\overrightarrow{p_0{p}_1}$ Left the crop area ： set up $p$ Is the intersection of line segment and boundary , take $p$ Send in $P$ Tail of .
（3） Line segment $\overrightarrow{p_0{p}_1}$ Inside the crop region ： skip
（4） Line segment $\overrightarrow{p_0{p}_1}$ Outside the crop region ： take $p_1$ Send in $P$ Tail of .

The three dimensional
   It is easy to generalize the algorithm from 2D to 3D , The idea is to cut the polygon separately for each plane , Then the resulting polygon （ If it exists ） Pass to the next boundary .

Weiler-Atherton Algorithm

  Sutherland-hodhman In some cases, the algorithm may generate degenerate polygons （ As shown in the figure below, using this algorithm will generate a vertex sequence set i-b-l-4-5-k-b-j-9-0 Examples come from textbooks ） While using Weiler-Atherton The algorithm can avoid outputting degenerate polygons .

Algorithm flow
① Number the intersection of polygon and clipping window , Traverse the points on the boundary of polygon and clipping window in order , Draw the sequence diagram according to the number .（ among i , l, k, j For the public point ）
② Determine that the edge of the polygon to which the common point belongs is “ Leave ” The window is still “ Get into ” window , In the above example i, k by “ Leave the vertex ”, l, j by “ Go to the top ” . Vertex properties are related to the direction of boundary traversal , Traverse direction changes , The properties of common points will be reversed .（ But with ① Synchronization complete ）

③ Define sequence set $P$, Repeat the following steps ： From the public point “ Go to the top ” set out , On the left side of the figure （ Along the polygon boundary ） Forward , encounter “ Leave the vertex ” until , Add the passed vertices $P$ . After traversing all common vertices, the loop ends .

Local illumination and ray tracing

   When building scenarios , In order to determine the color of the object surface , We need to consider the following types of light ： The object itself emits light 、 The mirror reflects light 、 Diffuse light and ambient light . The color of the object surface can be obtained by combining the effects produced by these four .

   In this chapter , We treat objects as material entities that reflect light , In order to depict the color of objects , We focus on the surface of the object . In computer graphics, we consider an extreme case ： There are only two types of reflections on the surface of the object ： Completely specular and Full diffuse reflection .

Lambert's law
   Lambert's law is needed to calculate diffuse reflection . The law states ： For diffuse reflectors , The intensity of reflected light in any direction at a point on the surface is proportional to the cosine of the incident angle ： $L\propto \cos \theta$.

Ray tracing

forward Forward propagation algorithm

   From the perspective of computer graphics , We replace our eyes with a plane of pixels . Consider the light emitted directly or by reflection on all objects , If a certain beam of light can be seen by us , It means that the photon will hit a pixel on the view window , And increase the brightness of this point to a value greater than zero . Traverse all objects , Until all pixels are adjusted , Creation can generate images in a certain view .

backward Backward propagation algorithm

   Forward propagation algorithm is unrealistic to simulate the interaction between photon and object in computer , Because in reality, the reflected light we see is only a small part of all the light , This means that a large number of photons must be projected to build the view we need . Besides , We cannot guarantee that the surface of an object is completely covered by photons , To solve this problem , Backward propagation algorithm can be used .
   The backward propagation algorithm only considers which light will enter the eyes and be seen by us , This can be achieved by shooting a beam of light backward from the eye position towards the scene , According to the light the object receives in this case , Restore true colors .