Circle and sphere blending with conformal geometric algebra
C. Doran. (2003)cite arxiv:cs/0310017Comment: 20 pages, 13 figures.
Abstract
Blending schemes based on circles provide smooth `fair' interpolations
between series of points. Here we demonstrate a simple, robust set of
algorithms for performing circle blends for a range of cases. An arbitrary
level of G-continuity can be achieved by simple alterations to the underlying
parameterisation. Our method exploits the computational framework provided by
conformal geometric algebra. This employs a five-dimensional representation of
points in space, in contrast to the four-dimensional representation typically
used in projective geometry. The advantage of the conformal scheme is that
straight lines and circles are treated in a single, unified framework. As a
further illustration of the power of the conformal framework, the basic idea is
extended to the case of sphere blending to interpolate over a surface.
Description
[cs/0310017] Circle and sphere blending with conformal geometric algebra
%0 Generic
%1 doran2003circle
%A Doran, Chris
%D 2003
%K geometricalgebra
%T Circle and sphere blending with conformal geometric algebra
%U http://arxiv.org/abs/cs/0310017
%X Blending schemes based on circles provide smooth `fair' interpolations
between series of points. Here we demonstrate a simple, robust set of
algorithms for performing circle blends for a range of cases. An arbitrary
level of G-continuity can be achieved by simple alterations to the underlying
parameterisation. Our method exploits the computational framework provided by
conformal geometric algebra. This employs a five-dimensional representation of
points in space, in contrast to the four-dimensional representation typically
used in projective geometry. The advantage of the conformal scheme is that
straight lines and circles are treated in a single, unified framework. As a
further illustration of the power of the conformal framework, the basic idea is
extended to the case of sphere blending to interpolate over a surface.
@misc{doran2003circle,
abstract = {Blending schemes based on circles provide smooth `fair' interpolations
between series of points. Here we demonstrate a simple, robust set of
algorithms for performing circle blends for a range of cases. An arbitrary
level of G-continuity can be achieved by simple alterations to the underlying
parameterisation. Our method exploits the computational framework provided by
conformal geometric algebra. This employs a five-dimensional representation of
points in space, in contrast to the four-dimensional representation typically
used in projective geometry. The advantage of the conformal scheme is that
straight lines and circles are treated in a single, unified framework. As a
further illustration of the power of the conformal framework, the basic idea is
extended to the case of sphere blending to interpolate over a surface.},
added-at = {2015-06-24T23:05:45.000+0200},
author = {Doran, Chris},
biburl = {https://www.bibsonomy.org/bibtex/2ced794d3267d397f2a87fd20a5e954ae/wolftype},
description = {[cs/0310017] Circle and sphere blending with conformal geometric algebra},
interhash = {d0a974649032d69a5891093e41773482},
intrahash = {ced794d3267d397f2a87fd20a5e954ae},
keywords = {geometricalgebra},
note = {cite arxiv:cs/0310017Comment: 20 pages, 13 figures},
timestamp = {2015-06-24T23:05:45.000+0200},
title = {Circle and sphere blending with conformal geometric algebra},
url = {http://arxiv.org/abs/cs/0310017},
year = 2003
}