Differential Geometry Introduction

in Study on Computer Vision, Differential Geometry

What is Geometry like Fundamentally

  1. The study of shapes, sizes, patterns, and positions.
  2. The study of spaces where soem quantity (lengths, angles, etc.) can be measured.

Many Ways to digitally encode geometry

  • Explicit : Immediately where this shape or point is, and how they connected
    • point cloud
    • polygon mesh
    • subdivision, Nurbs
  • Implicit : am i in the shape or not
    • Level set
    • algebraic surface
    • L-system
    • constructive solid geometry
    • blobby surface
    • fractals

Implicit Representations of Geometry

  • Points aren’t known directly, but satisfy some relationship
  • unit sphere is all points such that x^2 + y^2 + z^2 = 1
  • More generally f(x,y,z) = 0

Explicit Representations of Geometry

  • All points are given directly
  • points on sphere are (cos(u)sin(v), sin(u)sin(v), cos(v)), for 0 <= u <= 2pi and 0 <= v <=pi.

Differential Geometry Introduction

Roughly Speaking, classical differential geometry is the study of local properties of curves and surface. By local properties we mean those properties which depend only on the behavior of the curve or surface in the neighborhood of a point. (Differential Calculus), so the curves and the surfaces considered in dg will be defined by functions which can be diffeerentiated a certain number of times. The other aspect is the so-called global differential geometry. All the influence impacted by the local properties of curve and surface is so called global differential geometry.

What is Curve

What is Surface

Surface & Curve

What is Normal

What is Curvature

What is Freeform Surface Modeling

What is PCA(Principal Component Analysis)

Pagination

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