Graphics Reference
In-Depth Information
11
Barycentric Coordinates
Cartesian coordinates are a fundamental concept in mathematics and are cen-
tral to computer graphics. Such rectangular coordinates are just offsets rela-
tive to some origin. Other coordinate systems also exist such as polar, spherical
and cylindrical coordinates, and they, too, require an origin. Barycentric co-
ordinates, on the other hand, locate points relative to existing points, rather
than to an origin and are known as local coordinates . The German mathe-
matician August Mobius (1790-1868) is credited with their discovery.
' barus ' is the Greek entomological root for ' heavy ', and barycentric coor-
dinates were originally used for identifying the centre of mass of shapes and
objects. It is interesting to note that the prefixes ' bari ', ' bary 'and' baro 'have
also influenced other words such as baritone, baryon (heavy atomic particle)
and barometer.
Although barycentric coordinates are used in geometry, computer graphics,
relativity and global time systems, they do not appear to be a major topic in
a typical math syllabus. Nevertheless, they are important and I would like to
describe what they are and how they can be used in computer graphics.
The idea behind barycentric coordinates can be approached from different
directions, and I have chosen mass points and linear interpolation. But before
we begin this analysis, it will be useful to investigate a rather elegant theorem
known as Ceva's Theorem, which we will invoke later in this chapter.
11.1 Ceva's Theorem
Giovanni Ceva (1647-1734) is credited with a theorem associated with the
concurrency of lines in a triangle. It states that: In triangle ∆ ABC , the lines
AA ,BB and CC ,where A ,B and C are points on the opposite sides facing
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