几何
表示形式
- 隐式表示
- 形式: $f(x,y,z) = 0$
- based on classifying points
- points satisfy some specified relationship
- pros
- compact description (e.g., a function)
- certain queries easy (inside object, distance to surface)
- good for ray-to-surface intersection (光线求交,more later)
- for simple shapes, exact description / no sampling error
- easy to handle changes in topology (e.g., fluid)
- cons
- Sampling can be hard
- difficult to model complex shapes
- many implicit representations in graphics:
- Algebraic surface
- 最直接的，用数学公式
- 不直观
- Constructive solid geomentry(CSG)
- 基本形状的布尔操作组合成复杂形状
- Combine implicit geometry via Boolean operations
- Distance Functions(距离场)
- giving minimum distance(could be signed distance) from anywhere to object.
- level set methods(s水平集)
- store a grid of values approximating function
- 例子：CT扫描,等高线
- fractals
- 自相似
- 递归
- Algebraic surface
- 显式表示
- 形式:$f:R^{2} \rightarrow R^{3}; (u,v) \rightarrow (x,y,z)$
- All points are given directly or via parameter mapping
- Pros
- Sampling is Easy
- cons
- Inside/outside Test Hard
- many explicit Representations in Graphics
- point cloud
- 一堆点
- 可以表示任何几何
- Useful for LARGE datasets (>>1 point/pixel)
- Often converted into polygon mesh
- Difficult to draw in undersampled regions(在采样不足的区域很难绘制)
- Polygon mesh
- store vertices & polygons(often triangles or quads)
- easier to do processing /simulation,adaptive sampling
- more complicated data structures
- perhaps most common representation in graphics
- obj文件
- subdivision surfaces
- NURBS
- Bezier surfaces
- point cloud
曲线 Curve
Bézier Curves 贝塞尔曲线
一条由四个点（其实是任意≥3个点）定义的曲线：
- p0和p3定义起点和终点
- p1和p2定义起点与终点的切线方向（与p0和p3一起）
Evaluating Bézier Curves (de Casteljau Algorithm)
例子：(quadratic Bezier 二次贝塞尔曲线)
计算方法：
一些性质：
- 仿射变换前后统一
- 凸包性质：形成的曲线一定在控制点形成的凸包内
piecewise Bezier Curves(分段贝塞尔曲线)
高阶bezier曲线不容易控制
instead,chain many low-order bezier curve
- Piecewise cubic bezier the most common technique
连续性
- $C^{0}$连续:连接处值相等
- $C^{1}$连续:连接处一阶可导
- …
- $G^{0}$连续:等于$C^{0}$
- $G^{1}$连续:
样条
- Spline (样条)：a continuous curve constructed so as to pass through a given set of points and have a certain number of continuous derivatives. （a curve under control）
样条曲线：一种连续曲线，构造成通过一组给定的点并具有一定数量的连续导数(控制下的曲线）
B-splines
- bezier曲线缺点
- 一旦确定特征多边形，就确定了曲线的阶次
- Bezier曲线拼接复杂（需要满足几何连续性，参数连续性等）
- Bezier曲线不能作局部修改（只能整体修改）
- basis splines 基函数样条
- 满足局部性
- 可能是图形学里面最复杂的一部分
- 是贝塞尔曲线的超集
- Further：B样条、NURBS（非均匀有理B样条）https://www.bilibili.com/video/av66548502
曲面
- 两个不同时间t（u,v）
- 4x4个点，四条4个控制点的贝塞尔曲线，取同一时间（比如说u）获得四个控制点，取时间v，即获得最后的曲面上的点
Mesh Subdivision
- Increase resolution
- 细分的应用场景：Displacement mapping 位移贴图 需要模型足够细致，于是需要细分（最好是动态细分）
Loop Subdivision
Loop是发明者名字，跟循环没关系
- 局限
需要三角形Mesh
步骤：
create more triangles (vertices)
Split each triangle into four
tune their positions （形状需要有改变）
Assign new vertex positions according to weights
New / old vertices updated differently 新老点分别改变
Catmull-Clark Subdivision(General Mesh)
适用任何Mesh
方法
- 顶点调整
Mesh Simplification
- Decrease resolution
- Try to preserve shape/appearance
- Goal: reduce number of mesh elements while maintaining the overall shape
Edge collapse:边坍缩
- 哪些边合并？如何合并？
- Quadric Error Metrics（⼆次误差度量）放在二次误差之和最小的地方
- Quadric Error: new vertex should minimize its sum of square distance(L2 distance) to previously related triangle planes.
- Quadric Error Metrics（⼆次误差度量）放在二次误差之和最小的地方
Simplification via Quadric Error
- Garland & Heckbert 1997.
- iteratively collapse edge with smallest score
- 有问题，一条边的操作会影响其它边，需要更新
- 数据结构：优先队列 or 堆
- 贪心算法，非全局最优
- 可以有的放矢
Mesh Regularizatiom
- （不会出现特别奇怪的三角形）
- Modify sample distribution to improve quality