【Games101】Transformation

• Linear algebra related

1. Vector point multiplication → Obtain direction
2. Vector cross product → Get around & domestic and foreign
3. linear transformation ： A transformation that keeps the mesh parallel and equidistant

The properties of linear transformation ：

1. All the lines are still straight （ Including diagonals ）
2. The origin is still the origin , No movement

• How do we understand Transformation（ Transformation ）？

         ordinary , We can understand it as a series of steps , Through these steps , It can realize the transformation from three-dimensional space to two-dimensional image .

One 、 A two-dimensional / Three dimensional transformation  

• The zoom （Scale）

•   symmetry （Reflection）

• shear （Shear）

•   rotate （Rotation）

•   Affine transformation = linear transformation + Translation transformation

  Homogeneous coordinates ：

         because 2D The transform has no offset , So it's impossible to say 2D Pan operation , The main function of homogeneous coordinates is to introduce offset , Translate transform (Translate) Unified classification 2D Transformation , You don't want the translation transform to be different from the others 2D Transformation .

         At the same time, homogeneous coordinates can distinguish points from vectors , According to the translation invariance of the vector , Vectorial w Shaft for 0, Dot w Shaft for 1.

         With homogeneous coordinates , The translation transformation is expressed as follows ： 

•   Combination transformation

        （1） Properties based on rotation calculation （ Rotate around the origin ）, When multiple transformation matrices are calculated, they are usually rotated first and then translated .

         （2） For an image that is not at the origin , How to perform rotation transformation ？

         First, translate the image to the origin , The original position will be translated after the transformation operation .

• Summary of basic transformation matrix

  Two 、MVP Transformation

        MVP The transformation consists of three transformations , namely Model Transformation、View / Camera Transformation、Projection Transformation.

• Model transformation (placing objects)

         The objects in the world coordinate system , Using a change matrix, their vertex coordinates are transformed from local to local Local Coordinate system （ relative ） Switch to the world Global Coordinate system （ absolute ）, the DCC Model coordinates in are converted to model coordinates in the engine .

• View transformation (placing camera)

         Put the world coordinate system （ The center of the world is the origin ） To the camera coordinate system （ Camera position is the origin ）, Realize the transformation from the third perspective to the first perspective .

        （1） How do we define a camera ？

         We define the position of the camera as , Orientation is defined as , Up （ It's like an antenna ） Defined as .

        （2） How to achieve M→V Transformation ？

         Pan the camera to the origin ,、、 Rotate to 、、.

         among ,/ It's the camera up （ The antenna ）,/ Is the viewing direction of the camera , Cross multiply to get /.

         thus 、、 The model space coordinate system is converted to 、、 Observation space coordinate system .

       （3） How to calculate the target rotation matrix ？         

         Rotation matrix solution ： First find the rotation matrix from the origin to the camera , Then inverse the rotation matrix from the camera to the origin .

        

         notes ： The inverse of the rotation matrix = Transpose matrix of rotation matrix

• Projection transformation

         The space observed by the camera is usually a visual cone , The projection transformation transforms the viewing cone into a 1×1×1 Standard Cube , To facilitate subsequent operations .

         There are two forms of projection transformation , Perspective projection （Perspective Projection） And orthogonal projection （Orthographic Projection）.

        （1） Orthographic projection （Orthographic Projection）

         How to understand orthogonal projection ？

         We set the space of all objects as a cube , Translate and zoom the cube , It becomes a point centered on the origin 1×1×1 Standard Cube .

          The transformation matrix is as follows ：

        （2） Perspective projection （Perspective Projection）

         We think that perspective projection presents a phenomenon of near large and far small , That is, with most cameras 、 The imaging principle of the human eye is the same .

         How to realize perspective projection ？

1. First convert perspective projection to orthogonal projection , That is to realize the extrusion from the far plane to the near plane
2. Do another orthogonal projection

          How to realize extrusion from far plane to near plane ？

         First, get the screen aspect ratio , And then calculate through similar triangles .

          The resulting matrix is as follows ：

Reference resources ：

GAMES101- Introduction to modern computer graphics - Yan Lingqi _ Bili, Bili _bilibili

GAMES101: Introduction to modern computer graphics