Adv. Appl. Cliﬀord Algebras 27 (2017), 1977–1995
2016 Springer International Publishing
published online June 24, 2016
Applied Cliﬀord Algebras
A Hierarchical Representation and
Computation Scheme of Arbitrary-
dimensional Geometrical Primitives
Based on CGA
Wen Luo, Yong Hu, Zhaoyuan Yu
, Linwang Yuan and
Abstract. Resolving the conﬂicts between the high dimensionality of
geometry representation and the linear organization and storage of geo-
metrical objects in computer memories plays a key role in spatial data
structure constructions. In this paper, a new data structure MVTree is
developed based on geometric algebra to support the uniﬁed organiza-
tion and computation of geometrical primitives. The MVTree is a tree-
like data structure which has a dimensional hierarchical structure gener-
ated by outer product. Multidimensional geometrical primitives is repre-
sented as the combination of blades stored in the nodes of MVTrees.The
geometric computation between diﬀerent geometrical objects is operated
with GA operators with a judgement-based hierarchical computation.
Applications of the MVTree are demonstrated by a topological relation
computation and a Delaunay-TIN intersection. The results suggest that
the MVTree structure can support uniﬁed representation of arbitrary-
dimensional geometrical primitives, and can integrate data organization
and computation in a unitary structure as well. The application of the
new MVTree data structure can not only reduce the complexity of data
architectures but also inherit the power of geometric algebra computing
to improve the processing ability of computer graphic software.
Mathematics Subject Classiﬁcation. Primary 99Z99; Secondary 00A00.
Keywords. Data structure, Geometric algebra, Geometric computing,
Topological relations, Triangle intersection.