What Is a Homogeneous Vector?

Basics

• Basic geometry involves points, lines and planes. A vector is a directed line; it moves in one direction. It resembles an arrow moving from one point to another point. For two vectors to be equal they must be the same size and go in the same direction. A vector that goes from one named point to another named point is fixed. It cannot be moved or rotated. Vectors that are fixed to one point can be rotated around that point. Understanding the basics of vectors is necessary to comprehending how they function in projective space.

Projective Space

• Projective space takes geometry into new territory beyond the concepts of Euclidean, or Cartesian, geometry. You are projecting what occurs when lines extend forever. You have an explanation for the idea that parallel lines meet at infinity. It is similar to the idea of two parallel railroad tracks meeting at the horizon. Homogeneous coordinates within projective space explain how homogeneous math, and thus homogeneous vectors, work.

Homogeneous Coordinates

• In Euclidean geometry, there are two coordinates, x and y. In homogeneous math there is a third coordinate, w. The homogeneous coordinates are expressed as x=x/w and y=y/w. Thus a point (1,2) in Euclidean or Cartesian becomes (1,2,1) in homogeneous. When the point approaches infinity in homogeneous, it becomes (1,2,0). The coordinates bear the name homogeneous because you can double or triple the values for the homogeneous coordinates and each point will represent the same point in Euclidean space. Changing the scale, or number, of an homogeneous coordinate does not change its location.

Equivalent Vectors

• Homogeneous vectors share the same characteristic as homogeneous coordinates. Two homogeneous vectors are equal is they are multiples of each other. This trait does not occur in a regular vector; double the size and the vectors are not equal. This trait results from the principles of homogeneous math in projective space.