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Viewing in 2D & 3D

2D Viewing

  1. world is infinite but screen is finite.
  2. 通过appropriate transformation可以得到通过screen看到的window

Window

  1. rectangular region, specified by a center (xCenter, yCenter) and size windowSize
  2. Screen referred to as Viewport is a discrete matrix of pixels specified by size screenSize (in pixels)

  3. Mapping the 2D world seen in the window onto the viewport is 2D viewing transformation, also called window-to-viewport transformation.

    • 注意对image，x轴正向朝右，y轴正向朝下，而2D world中y轴正向朝上，因此在缩放时要加一个负号
image-20241023082718628image-20241023082928371image-20241023082954229image-20241023083006043
  1. Aspect Ratio: 保高/保宽
image-20241023083410091

  1. OpenGL Commands

    1. gluOrtho2D( left, right, bottom, top ): Creates a matrix for projecting 2D coordinates onto the screen

    2. glViewport(x, y, width, height): Define a rectangle of pixels onto which the final image is mapped.

         - `(x, y)` specifies the lower-left corner of the viewport.

   - `(width, height)` specifies the size of the viewport rectangle.

  2. Modeling v.s. Viewing

    • Modeling变换改变了world中放置的物体，会改变场景
    • Viewing变换仅仅移动了相机，不改变场景

3D Viewing

To display a 3D world onto a 2D screen

  • Specification becomes complicated because there are many parameters to control

  • Additional task of reducing dimensions from 3D to 2D (projection)

  • 3D viewing is analogous(类似) to taking a picture with a camera

  1. PRP: Projection Reference Point = Eye Position
image-20241023085403993

  1. Persepective Projection: Foreshortening depends on distance from viewer 不能用来测量

  2. Engineering Viewing, Parallel Projection: DOP, Direction of Projection = Eye position at infinity

image-20241023090056922

  1. Oblique Projection(斜平行)

    • Projectors make an arbitrary angle with the projection plane
    • Angles in planes parallel to the projection plane are preserved
image-20241023090648765
  1. View Specification
image-20241023090904523
  • 通过一个绝对的upVector进行正交化得到一个和viewDirection垂直的向量
image-20241023091145228
  1. world-to-view transformation or view orientation matrix: 将世界坐标系变换到相机坐标系
image-20241023091852623
  1. Perspective Projection
    • prp: projection reference point or cop: center of projection
image-20241023092205744
  1. View Window
image-20241023092345942
  1. Perspective Viewing
image-20241023092408118
  1. Parallel Viewing
image-20241023092440059

  1. View Volume: 在此以外的内容看不到，不用考虑

    • Perspective projection

    image-20241023092539196

    • Parallel projection

    image-20241023092730773

Complete View Specification

  1. Specification in world coordinates

    • position of viewing (vrp), direction of viewing(-n),
    • up direction for viewing (upVector)

  2. Specification in view coordinates

    • view window : center (cx, cy), width and height,
    • prp : distance from the view plane,
    • front clipping plane : distance from view plane
    • back clipping plane : distance from view plane
image-20241023093047136

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