Basic geometric objects

Affine space consists of points and vectors existing independently of any specific reference system. There are some operations relating points and vectors:
  1. Subtracting one point from another gives a unique vector:

    This also defines point-vector addition: given a point Q and a vector v there is a unique point P such that

  2. The usual head-to-tail rule for vector addition applies:

    Summing a point and a vector times a scalar defines a line in affine space:

Affine combinations

The affine combination of two points

is defined to be the point

. The point Q divides the segment connecting the two original points in a ratio proportional to the two coefficients.

As long as the coefficients still sum to 1, this can be generalized to an arbitrary number of points:

Euclidean Spaces

A Euclidean space is an affice space with an inner product defined. An inner product is a function from pairs of vectors to the reals:

that satisfies three properties.

Symmetry:
Bi-linearity:
Positive definiteness:

Euclidean spaces have some other useful concepts:

length of a vector
distance between two points
angle between two vectors
perpendicular vectors
parallel vectors

Frames for Affine Spaces

If O is any point in space, and v_i is a basis for the vectors in the space, then

is called a frame for the space. The frame is called Cartesian if the basis vectors are orthonormal (of unit length and mutually pairwise perpendicular). Given a frame, any point P can be written uniquely with respect to that frame as

where the p_i are real coefficients. Similarly, any vector u can be written uniquely as

The last two equations can be rewritten in matrix form to make the similarities clearer:

This leads to the concept of barycentric coordinates.


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