UC Davis Computer Graphics

• UC Davis Computer Graphics
  • Ref
  • Affine Transformation
  • Curves in plane
  • move objects in space
  • Rotations around an arbitrary axis
  • the camera transform
  • Shading and Texturing
  • Visible surface algorithm
  • Ray Tracing
  • Subdivision Surface
  • Shadows part
  • Camera motion
  • Branches of CG

Ref

• 视频 https://www.youtube.com/watch?v=0NbD-c0Ctdk&list=PL_w_qWAQZtAZhtzPI5pkAtcUVgmzdAP8g&index=2

Affine Transformation

vector space

Curves in plane

• subdivision curve, Chaikin's Algorithm
  • control point 1/4 3/4 again and again go througn mid-point
    • 反过来 pick some points tan line
  • 3d does not work
• Bezier Curve attract point
• B-spline curve continuous->derivative equal bezier vs spline 使用n = 3的贝塞尔曲线原因是要限定相邻的两个分段多项式，在交接点位置的一阶导数相等（斜率相等）和二阶导数相等（斜率的变化率相等）

move objects in space

• a point in space
• translation
• scaling center is not in origin, then scaling will cause translation
• rotate
• 3d rotate around z axis

Rotations around an arbitrary axis

• model-view-controller easy to debug
• Rotate around rotate 一个点 一个向量(arbitrary axis）rotate some angles 怎么旋转：把这个点平移到原点，向量旋转到和eg. z axis重合，rotate some angles，再反着做回来 translation 投影 rotate -theta, rotate fi reverse
• 4d projective point parabola 抛物线 project down as a circle rational splines 有理样条
• simulate a camera

the camera transform

viewing transform

• pinhole camera
  • camera point
  • direction vector / center of attention
  • up direction
  • angle of view
  • near and far distances 找到一种转换方式把3d的世界转换成视角的2d图像 pyramid -> cube(image space)-> square(screen space) (修正 image space 正负坐标反了) (修正+n， +f) （0, n*tan(a/2), -n）-> (1,1) image space
• real world

Clipping

• find the real interesting spaces - "in" ps. polygon convex, 连接任意两条边的线段都在区域内

  non-convex -> split into convex pieces
  

• glu lookat vs glu perspective
  • glulookat

    void gluLookAt (
    GLdouble eyex,    GLdouble eyey,     GLdouble eyez, 
    GLdouble centerx,     GLdouble centery,     GLdouble centerz, 
    GLdouble upx,     GLdouble upy,     GLdouble upz);
    

    定义了观察者的做坐标，视野中心点在世界坐标系的位置 & 相机的朝向
    • glu perspective

      void gluPerspective (
      GLdouble fovy, 
      GLdouble aspect, 
      GLdouble zNear, 
      GLdouble zFar);
      

      定义了相机的内在参数，可视角大小， 物体显示在canvas上的x和y方向上的比例，第三个参数zNear,定义距离相机（人眼）最近处物体截面相距的距离。这个值越大，表示观测点距离物体距离越远，看起来物体就比较小，反之则比较大。如果物体运动到距离观测点的距离小于了设定的zNear,则物体不会被绘制在画板上。 第四个参数zFar,定义可观测到的物体的最远处截面相距相机的距离。如果物体运动到距离观测点的距离大于了设定的zFar,则物体不会被绘制的画板上。
• do clipping in image space

Depth Buffers and Ray Tracing

？something is in front of another image space judge by z coordinates

• depth buffer 存储最近的物体的深度 如果object的深度大于z buffer的值，不绘制
• Ray Tracing
• painter's algorithm 画家算法首先将场景中的多边形根据深度进行排序，然后按照顺序进行描绘

Color and Shading

• color
• lights
  • background light (ambient light) is a constant
  • Lambert's Law
  • reflect base color n is glossness l input, v output h= l+v/2, h 和 n (法线) 夹角越小，反射的光越强

Quaternions

rotations structure data on disk? extension of complex 不满足交换律

• Unit quaternions rotate around vector v, rotate theta angle ps. 欧拉旋转: Gimbal Lock问题的核心还是在于我们采用了固定的旋转顺序 四元数: chrome-extension://cdonnmffkdaoajfknoeeecmchibpmkmg/assets/pdf/web/viewer.html?file=https%3A%2F%2Fkrasjet.github.io%2Fquaternion%2Fquaternion.pdf

Shading and Texturing

• shading 着色 b means base color -> texturing 10 hyper parameters direct light / spot light -> l change or not n bump mapping ploygons
  • flat shading (平面着色) one color
    • Gouraud Shading -> average color
  • phong shading -> average normal
• texturing 纹理 texture mapping -> interpolate

The hierarchical modeling system

how to model things model one thing -> transform to another polygon -> commands not only drawing things -> but also building things

Scan Conversion

• device space split polygon into trapezoids edge track endpoint other staff: light sources <- inverse of viewing space texture coordinates R/G/B inks
• Warnock's algorithm

Curves and surfaces

• Chaikin's Algorithm Chaikin's Curve -> control points 1/4 3/4 (fix ratio), initial point may not on the curve 3d surface -> two pairs of control points -> subdivision algorithms
• Bezier Patch
  • Bezier curve 1/2 initial point is on the curve
    • control points coplanar
    • convex hull property 凸包， curve is not getting away from u
    • variation diminishing 变差缩减性，不会来回摆动
    • slope at p0 is equal to p0p1, first derivative
    • how about second derivative
• NURBS non uniform rational Bspline
  • NUBS limit degree of curves
• Bezier surface four corners on the surface Bezier curves as boundary

Visible surface algorithm

• Painter's algorithm paint all from back to front <- sort polygons
  • Binary Tree Sort
    • BSP - trees moving camera position, just move subtree left to right
• Warnock's algorithm

Ray Tracing

• how it do eye position -> pixel trace to the nearest point
• first paper

  reflect  shadow refract
  

• forward & backward ray tracing backward ray tracing may miss things
• calculate the nearest point

Subdivision Surface

• Chaikin's Curve
• Chaikin's Surface/Doo-Sabin Surface 不同方向上做1/4 3/4（Chaikin's points），连接，划分成一个个小的mesh
• how to save such data ? -> split edge a pointer -> start point a pointer -> next edge a pointer -> neighbor subEdge
• Catmull-Clark Subdivision loop subdivision (triangle meshes) vertex point (half way half way)& edge point (mid point) Face point: F = average of all points defining the face Edge point: E = average of tow adjoining face points and the two vertices Vertex point: V = (average of all face points + average of all edge poins + constant origin vectors )/ 4
• how to model sphere -> triangle data structure: 3 vertexes + 3 pointers to neighbors

Shadows part

• ground plane (need a point and a normal) avoid sharp shadows -> texture with fuzzy edge polygons too complex -> general bounding box
• shadow volumes count numbers of into a shadow volumes (+1) and out of a shadow volumes (-1) sum = zero : no shadows ps. umbra penumbra
• depth buffer shadows
  • for each light source, create a "picture" (image space, z buffer)
  • for each eye point, create a "picture" (z buffer)
  • reconcile these pictures

Camera motion

• simulate camera motion path as smooth curve
  • Lagrange interpolation spline
  • Ferguson curve
  • Catmull 二次导等于间隔点连线 c1 连续 curve
  • ps

Branches of CG