Understanding of homogeneous coordinates

 

When Euclidean geometry studies space , One part is the description of parallel lines ： Two parallel lines on the same plane do not intersect , The following is an aerial view .

 

 

It is equivalent to drawing two parallel lines on a piece of paper , They must be infinite and disjoint . But if we take this piece of paper , When we look at it from the side, we will find , The two lines are not parallel （ In fact, the essence is parallel ）

 

This visual effect change caused by the viewpoint change , It is related to the structure of our eyes , There is a term in geometry called ： Projection space （ Everything we see in the world is projected on our retina ）.

So there is this problem ： How to solve the parallel line of Euclidean geometry , This problem does not hold in projective geometry , So we have the concept of homogeneous coordinates . Homogeneous coordinates are hard to say , It is equivalent to adding a variable to the coordinate system of Euclidean geometry , Use this variable w Add the original coordinates , It is used to solve the problem that two parallel lines in the projection space become non parallel. Suppose that we put the x The coordinates are all variable w , Every point is displaced , Let's pick up the paper and look at it from the side , Suppose it happens to be this w The law of the value of can make the two lines projected onto our retina become parallel ：

  Then we return to the aerial view , These two blue lines are like this ：

 

The points of these two lines can be used coordinate x, y To represent the , We can describe it as (x , y ) => (x/w, y/w), (x/w, y/w) Homogeneous coordinates .

The geometric meaning of homogeneous coordinates is , It represents only one direction , There is no beginning or end . So in Euclidean geometry, parallel lines are parallel lines that have no intersections , In projection geometry, two parallel lines have intersections , Where this intersection occurs , Using variables w To help describe is (x, y, w).