[Computer Animation] Point and Vector (Affine Geometry)

Point and Vector (Affine Geometry)

Animation

Function of time

 

Movement

 

1D Space

Uniform Speed	Non-Uniform Speed

 

2D or 3D Space

• Cartesian Coordinate System(데카르트 좌표계) 를 사용하여 표현한다.

• 함수가 만들어지는 절차

Sampling Global Points
In a New Coordinates System
Sampling Difference Vectors
In a New Coordinates System
Function

 

Points and Vectors

Points and Vectors

• A Point is a position specified with coordinate values.
• A Vector is specified as the difference between two points.

 

• If an Origin is specified, then a point can be represented by a vector from the origin.
• A Point is not a vector in Coordinate-Free concepts.

 

Vector and Affine Spaces

• Vector Space(벡터 공간)
  • Includes vectors and related operations.
  • No points
• Affine Space(아핀 공간)
  • Superset of vector space.
  • Includes vectors, points and related operations.

 

Vector Spaces

• A Vector Space consists of 
  • Set of Vectors, together with
  • Two Operations
    • Addition of vectors(벡터의 합)
    • Multiplication of vectors by Scalar Numbers. (스칼라 수에 의한 벡터의 곱)
• A Linear Combination(선형 결합) of vectors is also a vector.

 

Affine Spaces

• A Affine Space consists of
  • Set of Points, An Associated Vector Space, and
  • Two more operations
    • The difference between two points
    • The addition of a vector to a point
• They are Coordinate-Free Geometric Operations.

 

Coordinate-Free Operations

Example of Coordinate-Dependence

• 두 위치의 합(Sum)은 얼마일까?

 

If you assume coordinates, ...

• 합은 $(x_{1} + x_{2}, y_{1} + y_{2})$ 가 된다.
  • 정답일까?
  • 기하학적으로 의미가 있을까?

 

• Vector Sum
  • $(x_{1}, y_{q})$ and  $(x_{2}, y_{2})$ are considered as vectors from the origin to $p$ and $q$, respectively.

 

If you select a different origin, ...

• If you choose a different coordinate frame, you will get a different result.

 

Coordinate-Free Geometric Operations

• Addition
• Subtraction
• Scalar Multiplication
• Linear Combination
• Affine Combination

 

Addition

• Vector + Vector = Vector
• Point + Vector = Point

 

Subtraction

• Vector - Vector = Vector
• Point - Point = Vector
• Point - Vector = Point

 

Scalar Multiplication

• Scalar × Vector = Vector
  • 1 × Point = Point
  • 0 × Point = Vector
  • c × Point = (undefined)   if (c ≠ 0, 1)

 

Linear Combination of Points

 

Examples

• (p + q) / 2 : Midpoint of line pq
• (p + q) / 3 : Can you find a geometric meaning ?
  • 기하학적 의미가 없다.
• (p + q + r) / 3 : Center of gravity of Δpqr
• (p/2 + q/2 - r) : A Vector from r to the midpoint of q and p

 

Summary