Distance from a Point to a Line

2 Key excerpts on "Distance from a Point to a Line"

• 

  eBook - ePub

  Mathematical Structures for Computer Graphics

  • Steven J. Janke(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  Projecting a vector (from the line to the plane) onto the normal vector again gives the distance between the line and the plane. Example 3.8 (Distance between Two Parallel Planes) We have two parallel planes with a common normal vector. One plane contains the point and the other contains the point. To find the distance between the planes, we calculate the coordinates of and then project it onto the normal vector. Notice that the order we subtract the points to get does not matter (Figure 3.8). Figure 3.8 Distance between parallel planes If we are thinking about computation efficiency, notice that, if we normalize the vector so that it has length, then no division is required to calculate the projection and hence the distance. Of course, it takes three divisions to normalize a vector, but if we will be using it many times, it might be worth it. Example 3.9 (Distance from a Line to a Plane) Instead of two planes, suppose we have one containing the point with normal. Consider now the line. The direction vector is perpendicular to the normal and hence parallel to the plane. By projecting a vector from to a point on the line, say, we can find the distance between the line and the plane. This time, normalize the vector first, giving In this case, picking any point on the line gives the same distance. If the line was not parallel to the plane, this would not be true. 3.2.4 Line to a Line In two dimensions, two lines either intersect or they are parallel. In three dimensions, it is also possible that they do not intersect and they are not parallel; these are skew lines. Suppose that for two nonintersecting lines we have found exactly where the lines come closest to each other. That is, we have a point on the first line and a point on the second line such that the distance between the two points is as close as possible for any points on the lines. Then the vector must be perpendicular to each line. If it is not, then suppose it is not perpendicular to the second line

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  eBook - ePub

  3D Math Primer for Graphics and Game Development

  • Fletcher Dunn, Ian Parberry(Authors)
  • 2011(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  The zero vector cannot be normalized. Mathematically, this is not allowed because it would result in division by zero. Geometrically, it makes sense because the zero vector does not define a direction—if we normalized the zero vector, in what direction should the resulting vector point?

  2.9.2   Geometric Interpretation

  In 2D, if we draw a unit vector with the tail at the origin, the head of the vector will touch a unit circle centered at the origin. (A unit circle has a radius of 1.) In 3D, unit vectors touch the surface of a unit sphere. Figure 2.16 shows several 2D vectors of arbitrary length in gray, beneath their normalized counterparts in black.

  Figure 2.16 Normalizing vectors in 2D

  Notice that normalizing a vector makes some vectors shorter (if their length was greater than 1) and some vectors longer (if their length was less than 1).

  2.10   The Distance Formula

  We are now prepared to derive one of the oldest and most fundamental formulas in computational geometry: the distance formula. This formula is used to compute the distance between two points.

  First, let’s define distance as the length of the line segment between the two points. Since a vector is a directed line segment, geometrically it makes sense that the distance between the two points would be equal to the length of a vector from one point to the other. Let’s derive the distance formula in 3D. First, we will compute the vector d from a to b . We learned how to do this in 2D in Section 2.7.3 . In 3D, we use

  d = b - a =

  [

  b x

  −

  a x

  b y

  −

  a y

  b y

  −

  a z

  ]

  .

  The distance between a and b is equal to the length of the vector d , which we computed in Section 2.8 :

  distance  ( a ,   b ) =

  ‖ d ‖

  =

  d x

  2

  +

  d y

  2

  +

  d z

  2

  .

  Substituting for d , we get

  distance  ( a ,   b ) =

  ‖ b - a ‖

  =

  (

  b x

  -

  a x

  ) 2

  + (

  b y

  -

  a y

  ) 2

  + (

  b z

  -

  a z

  ) 2

  .

  The 3D distance formula

  Thus, we have derived the distance formula in 3D. The 2D equation is even simpler:

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