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Affine Spaces

An affine space $ \bf {A}$ , has two types of entities, i.e., points and vectors. All the vectors form a vector space, denoted as $ \bf {V_A}$ , on its own. $ \bf {A}$ has the following properties.

  1. There is a unique vector related to a pair of points, defining two operations,

    1. Add a vector to a point to get another point.

      $\displaystyle p + \vec{v} = q.$ (1)

    2. Subtracting two points to get a vector,

      $\displaystyle q-p = \vec{v}.$ (2)

  2. Affine combination of points into another point.

    $\displaystyle \sum_{i=0}^n{\lambda_i p_i} = q,$ (3)

    when

    $\displaystyle \sum_{i=0}^n{\lambda_i} = 1. $

    In NURBS community, we actually almost always put an extra restriction on equation (3). That is, all the coefficients $ \lambda_i$ are required to be non-negative, and such an affine combination is called convex hull combination, or more commonly interpolation (contrast to extrapolation<).

  3. Combination of points into a vector.

    $\displaystyle \sum_{i=0}^n{\lambda_i p_i} = \vec{v},$ (4)

    when

    $\displaystyle \sum_{i=0}^n{\lambda_i} = 0. $

Based on these properties, it can be proved that an affine space has (addition and subtraction) algebra for its points and vectors in the common sense.

The usual Euclidean spaces are affine spaces with the affine operations defined component-wise. We use $ {\bf A}^r,$ to represent an affine space of Euclidean $ r$ -space. The related vector space, $ \bf V_A$ is just $ {\bf R}^r$ . Also in this tutorial, $ \bf A$ and $ \bf A_i$ is used to represent a general affine space of any dimension; similarly $ \bf V$ and $ \bf V_i$ for a general vector space.

Example 1   In Euclidean 1-space, the same notation $ 1$ , could be a point, a vector, or just a scalar(i.e. an element of the field on which the vector space is defined). The ambiguity can usually be resolved based on the context in most cases. When this is impossible, vector $ 1$ is explicitly written as $ \vec{1}$ .


next up previous
Next: Linear and Affine Mappings Up: Affine Spaces and Affine Previous: Affine Spaces and Affine
Xianming Chen 2006-10-02