几何

这节课

• Introduction to geometry
  • Examples of geometry
  • Various representations of geometry (几何的各种表示)

Many Ways to Represent Geometry

(隐式) Implicit Representations of Geometry

Based on classifying points (基于分类点)

• Points satisfy some specified relationship (点满足某种特定的关系)

E.g. sphere: all points in 3D, where $x^2+y^2+z^2=1$

More generally, $f(x,y,z)=0$

Implicit Surface – Sampling Can Be Hard (隐式表面采样可能很难)

$f(x,y,z)=(2-\sqrt{x^2+y^2}{)}^2+z^2-1$

Some tasks are hard with implicit representations (有些任务很难使用隐式表示)

Implicit Surface – Inside/Outside Tests Easy (隐式表面-内部/外部测试容易)

$f(x,y,z)=x^2+y^2+z^2-1$

Implicit representations make some tasks easy (隐式表示使一些任务变得容易)

More Implicit Representations in Computer Graphics

Algebraic Surfaces (Implicit) (代数曲面(隐式))

Surface is zero set of a polynomial in x, y, z (曲面是x, y, z的多项式的零集)

Constructive Solid Geometry (Implicit)

Combine implicit geometry via Boolean operations (复杂的几何体通过简单几何体进行集合运算（交并补）得到，该操作被称之为CSG)

Distance Functions (Implicit) (距离函数)

Instead of Booleans, gradually blend surfaces together using Distance functions: giving minimum distance (could be signed distance) from anywhere to object

An Example: Blending (linear interp.) a moving boundary (一个例子:混合(线性插值)一个移动的边界)

Blending Distance Functions (Implicit)

Can blend any two distance functions d1, d2:

Level Set Methods (Also implicit) (水平集)

Closed-form equations are hard to describe complex shapes Alternative: store a grid of values approximating function (封闭形式的方程很难描述复杂的形状 备选方案:存储一个网格值逼近函数) Surface is found where interpolated values equal zero Provides much more explicit control over shape (like a texture) (找到插值值为零的曲面 对形状(如纹理)提供更显式的控制)

Level Sets from Medical Data (CT, MRI, etc.)

Level sets encode, e.g., constant tissue density

Level Sets in Physical Simulation

Level set encodes distance to air-liquid boundary

Fractals (Implicit) (分型)

Exhibit self-similarity, detail at all scales

"Language" for describing natural phenomena

Hard to control shape! (展示自相似性，所有尺度的细节 描述自然现象的“语言” 难以控制形状!)

Implicit Representations - Pros & Cons (隐式表示-优点和缺点)

Pros:

• compact description (e.g., a function) (紧凑的描述(例如，一个函数))
• certain queries easy (inside object, distance to surface) (某些查询很容易(对象内部，到表面的距离))
• good for ray-to-surface intersection (more later) (适用于光线与表面的相交)
• for simple shapes, exact description / no sampling error (对于简单的形状，精确的描述/无采样误差)
• easy to handle changes in topology (e.g., fluid) (易于处理拓扑变化(例如，流体))

Cons:

• difficult to model complex shapes (难以模拟复杂的形状)

(显式) Explicit Representations in Computer Graphics

Many Explicit Representations in Graphics

Point Cloud (Explicit) (点云)

Easiest representation: list of points (x,y,z)

Easily represent any kind of geometry

Useful for LARGE datasets (>>1 point/pixel)

Often converted into polygon mesh

Difficult to draw in undersampled regions

Polygon Mesh (Explicit) (多边形网格)

Store vertices & polygons (often triangles or quads) (存储顶点和多边形(通常是三角形或四边形))

Easier to do processing / simulation, adaptive sampling (更容易做处理/模拟，自适应采样)

More complicated data structures (更复杂的数据结构)

Perhaps most common representation in graphics(也许在图形中应用最广泛)

The Wavefront Object File (.obj) Format (波前对象文件(.obj)格式)

Commonly used in Graphics research

Just a text file that specifies vertices, normals, texture coordinates and their connectivities (只是一个文本文件，指定顶点，法线，纹理坐标和它们的连接)

下图定义了一个立方体，有八个顶点（v）,六个面（vn）多个纹理坐标（vt）表示，然后使用f表示他们之间的关系(f(v/vt/vn))