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Method for Generating High Quality of Normal Vectors from Polygonal Data

IP.com Disclosure Number: IPCOM000118070D
Original Publication Date: 1996-Sep-01
Included in the Prior Art Database: 2005-Mar-31
Document File: 4 page(s) / 93K

IBM

Koide, A: AUTHOR

Abstract

Generating normal vectors from polyhedral data now becomes more and more important in 3D graphic applications. This is because recent advances in 3D digitizers and 3D imaging devices yields huge volumes of polyhedral data without normal vectors while the visual quality of them in shading display is strongly dependent on the quality of their estimated normal vectors.

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Method for Generating High Quality of Normal Vectors from Polygonal
Data

Generating normal vectors from polyhedral data now becomes more
and more important in 3D graphic applications.  This is because
recent advances in 3D digitizers and 3D imaging devices yields huge
volumes of polyhedral data without normal vectors while the visual
quality of them in shading display is strongly dependent on the
quality of their estimated normal vectors.

This invention describes two rapid and high-quality methods for
generating normal vectors on vertices from polyhedral data.  The
proposed methods give exact normal vectors in case their vertices
form spherical surfaces.
1.  Notations and Base Equation
Let d_j be displacement vectors from a vertex x, whose normal
vector is to be calculated, to adjacent vertices xj (see Eq.
(1)).  Suppose that the vertices x and x_j lie on a spherical
surface with a radius R.  Then the displacement vectors
should satisfy (see Eq. (2)).  Here n is the true normal
vector of the vertex x.  The expansion of the quadratic
power of (2) leads to a useful equation (Eq.  (3)).
2.  A Sum Method with a Special Weight Form
a.  Claim
This invention claims as a method for generating normal
vectors that we compute approximate normal vector n by a
sum of external products (see Eq. (4)) with a special
form of weights (see Eq. (5)).  Our special form of
weights (5) gives exact normal vector for spherical
surfaces and gives higher quality of approximate vectors
for general surfaces than the conventional forms of
weights (See Eq. (6) or (7)).
b.  Proof
This proves that this special form of weights (5)
gives exact normal vector for each vertex of a spherical
surface.  Any external product d_j*d_(j+1) can be
expanded to Eq. (8) by using a unit vector e.  Here oej
is given by Eq. (9).  The identity (8) is easily proved
by its inner product with the linear independent vectors
e, d_j, and d_(j+1).  Let us replace...