Normal Vectors

1
Normal Vectors

• For smooth, flat surfaces, the vector normal to
  the surface exists at every point
• Plane ax by cz d 0
• Also, n ? (p - p0) 0where p0 is a point on
  the plane, p is any point (x,y,z) on the plane
  and n is the normal to the plane

2
Normal Vectors

• Given three noncollinear points, p0, p1, p2
• and vectors p2 - p0 and p1 - p0 are parallel to
  the plane
• Normal n (p2 - p0 ) ?(p1 - p0 )
• Order of the vectors in the cross product is
  important, since it determines pointing in or out
• OpenGL does not determine the normal automatically

3
Normal Vectors

• For curved surfaces, there are different methods
  for representation (bezier, b-spline, NURBS)
• Unit Sphere
• Also, f(p) p ? p 1 0

4
NV Implicit Surfaces

• Normal of a surface defined implicitly

Normal vector
Gradient
5
NVParameterized Surfaces

• Normal vectors to a parameterized surface

n
n
6
NVParameterized Surfaces 2
Compute
7
NV OpenGL

• We usually have to work with a collection of
  vertices and approximate a normal vector
• OpenGL takes the do-it-yourself approach to
  normals
• The user program must determine the normal
• We associate a normal with a vertex through the
  function glNormal3f(nx, ny, nz)
• All subsequent glVertex calls use that normal