This report is meant to be a script of a tutorial on Clifford (or Geometric) algebra. It is there... more This report is meant to be a script of a tutorial on Clifford (or Geometric) algebra. It is therefore not complete in the description of the algebra and neither completely rigorous. The reader is also not likely to be able to perform arbitrary calculations with Clifford algebra after reading this script. The goal of this text is to give the reader a feeling for what Clifford algebra is about and how it may be used. It is attempted to convey the basic ideas behind the use of Clifford algebra in the description of geometry in Euclidean, projective and conformal space
In this paper, we explore the geometric objects of conformal geometric algebra based on their IPN... more In this paper, we explore the geometric objects of conformal geometric algebra based on their IPNS (inner product null space) representation in some detail. Spheres of dimension 1 , 2 and three are objects of conformal geometric algebra. Usually, points in conformal geometric algebra are represented as ordinary spheres with zero radius, but what about circles with zero radius? We expect many practical applications of these points with additional orientation information.
This work reviews some current engineering applications of geometric algebra and observes the pot... more This work reviews some current engineering applications of geometric algebra and observes the potential of this mathematical language to become a basis for a wide range of computational engineering applications. Geomet- ric algebra unifles many other mathematical concepts like quaternions and projective geometry and is able to easily deal with geometric objects, oper- ations and transformations. For computational engineering, not
ABSTRACT Early in the development of computer graphics it was realized that projective geometry w... more ABSTRACT Early in the development of computer graphics it was realized that projective geometry was well suited for the representation of transformations. Now, it seems that another change of paradigm is lying ahead of us based on geometric computing using conformal geometric algebra.Due to its geometric intuitiveness, elegance and simplicity, the underlying conformal geometric algebra appears to be a promising mathematical tool for computer graphics and animations.In this tutorial paper we introduce into the basics of the conformal geometric algebra and show its advantages based on two computer graphics applications.First, we will present an algorithm for the inverse kinematics of a robot that you are able to comprehend without prior knowledge of geometric algebra. We expect that here you will obtain the basic knowledge for developing your own algorithm afterwards.Second, we will show how easy it is in conformal geometric algebra, to fit the best suitable object in a set of points, whether it is a plane or a sphere.
What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the... more What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the geometric meaning of the algebraic expressions and operators. Geometric algebra is a mathematical framework to easily describe geometric concepts and oper- ations. It allows us to develop algorithms fast and in an intuitive way. It is based on the work of Hermann Grassmann [A2] and
What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the... more What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the geometric meaning of the algebraic expressions and operators. Geometric algebra is a mathematical framework to easily describe geometric concepts and oper- ations. It allows us to develop algorithms fast and in an intuitive way. It is based on the work of Hermann Grassmann [A2] and
This paper presents some basics for the analysis of point clouds using the geometrically intuitiv... more This paper presents some basics for the analysis of point clouds using the geometrically intuitive mathematical framework of conformal geometric algebra. In this framework it is easy to compute with osculating circles for the description of local curvature. Also methods for the fitting of spheres as well as bounding spheres are presented. In a nutshell, this paper provides a starting point for shape analysis based on this new, geometrically intuitive and promising technology.
Geometric Algebra (GA) is a mathematical framework that allows a compact and geometrically intuit... more Geometric Algebra (GA) is a mathematical framework that allows a compact and geometrically intuitive descrip-tion of geometric relationships and algorithms. In this paper a translation, rotation and scale invariant algorithm for registration of color images and other multichannel data is introduced. The use of Geometric Algebra allows to generalize the well known Fourier Transform which is widely used for the registration of scalar fields. In contrast to the original algorithm our algorithm allows to handle vector valued data in an appropriate way. As a proof of concept the registration results for artificial, as well as for real world data, are discussed.
... Each certain location has a minimal and a maxi-mal principal curvature with orthogonal princi... more ... Each certain location has a minimal and a maxi-mal principal curvature with orthogonal principal di-rections. ... data (Gross and Pfister, 2007), although it requires some effort to reconstruct the surface from these (noisy) point clouds (Hornung and Kobbelt, 2006; Mederos et al ...
We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computi... more We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computing based on Conformal Geometric Algebra (GA). The main goal of Gaalop is to realize implementations that are most likely faster than conventional solutions. We describe the concepts, the state-of-the-art as well as the future perspectives of Gaalop dealing with optimized software implementations, hardware implementations as well as mixed solutions.
This report is meant to be a script of a tutorial on Clifford (or Geometric) algebra. It is there... more This report is meant to be a script of a tutorial on Clifford (or Geometric) algebra. It is therefore not complete in the description of the algebra and neither completely rigorous. The reader is also not likely to be able to perform arbitrary calculations with Clifford algebra after reading this script. The goal of this text is to give the reader a feeling for what Clifford algebra is about and how it may be used. It is attempted to convey the basic ideas behind the use of Clifford algebra in the description of geometry in Euclidean, projective and conformal space
In this paper, we explore the geometric objects of conformal geometric algebra based on their IPN... more In this paper, we explore the geometric objects of conformal geometric algebra based on their IPNS (inner product null space) representation in some detail. Spheres of dimension 1 , 2 and three are objects of conformal geometric algebra. Usually, points in conformal geometric algebra are represented as ordinary spheres with zero radius, but what about circles with zero radius? We expect many practical applications of these points with additional orientation information.
This work reviews some current engineering applications of geometric algebra and observes the pot... more This work reviews some current engineering applications of geometric algebra and observes the potential of this mathematical language to become a basis for a wide range of computational engineering applications. Geomet- ric algebra unifles many other mathematical concepts like quaternions and projective geometry and is able to easily deal with geometric objects, oper- ations and transformations. For computational engineering, not
ABSTRACT Early in the development of computer graphics it was realized that projective geometry w... more ABSTRACT Early in the development of computer graphics it was realized that projective geometry was well suited for the representation of transformations. Now, it seems that another change of paradigm is lying ahead of us based on geometric computing using conformal geometric algebra.Due to its geometric intuitiveness, elegance and simplicity, the underlying conformal geometric algebra appears to be a promising mathematical tool for computer graphics and animations.In this tutorial paper we introduce into the basics of the conformal geometric algebra and show its advantages based on two computer graphics applications.First, we will present an algorithm for the inverse kinematics of a robot that you are able to comprehend without prior knowledge of geometric algebra. We expect that here you will obtain the basic knowledge for developing your own algorithm afterwards.Second, we will show how easy it is in conformal geometric algebra, to fit the best suitable object in a set of points, whether it is a plane or a sphere.
What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the... more What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the geometric meaning of the algebraic expressions and operators. Geometric algebra is a mathematical framework to easily describe geometric concepts and oper- ations. It allows us to develop algorithms fast and in an intuitive way. It is based on the work of Hermann Grassmann [A2] and
What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the... more What mathematicians often call Clifford algebra is calledgeometric algebra if the focus is on the geometric meaning of the algebraic expressions and operators. Geometric algebra is a mathematical framework to easily describe geometric concepts and oper- ations. It allows us to develop algorithms fast and in an intuitive way. It is based on the work of Hermann Grassmann [A2] and
This paper presents some basics for the analysis of point clouds using the geometrically intuitiv... more This paper presents some basics for the analysis of point clouds using the geometrically intuitive mathematical framework of conformal geometric algebra. In this framework it is easy to compute with osculating circles for the description of local curvature. Also methods for the fitting of spheres as well as bounding spheres are presented. In a nutshell, this paper provides a starting point for shape analysis based on this new, geometrically intuitive and promising technology.
Geometric Algebra (GA) is a mathematical framework that allows a compact and geometrically intuit... more Geometric Algebra (GA) is a mathematical framework that allows a compact and geometrically intuitive descrip-tion of geometric relationships and algorithms. In this paper a translation, rotation and scale invariant algorithm for registration of color images and other multichannel data is introduced. The use of Geometric Algebra allows to generalize the well known Fourier Transform which is widely used for the registration of scalar fields. In contrast to the original algorithm our algorithm allows to handle vector valued data in an appropriate way. As a proof of concept the registration results for artificial, as well as for real world data, are discussed.
... Each certain location has a minimal and a maxi-mal principal curvature with orthogonal princi... more ... Each certain location has a minimal and a maxi-mal principal curvature with orthogonal principal di-rections. ... data (Gross and Pfister, 2007), although it requires some effort to reconstruct the surface from these (noisy) point clouds (Hornung and Kobbelt, 2006; Mederos et al ...
We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computi... more We present Gaalop (Geometric algebra algorithms optimizer), our tool for high performance computing based on Conformal Geometric Algebra (GA). The main goal of Gaalop is to realize implementations that are most likely faster than conventional solutions. We describe the concepts, the state-of-the-art as well as the future perspectives of Gaalop dealing with optimized software implementations, hardware implementations as well as mixed solutions.
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