TY - GEN
T1 - Minkowski roots of complex sets
AU - Farouki, R. T.
AU - Gu, Weiqung
AU - Moon, Hwan Pyo
N1 - Publisher Copyright:
© 2000 IEEE.
PY - 2000
Y1 - 2000
N2 - An n-th Minkowski root ⊗1/n script A of a given complex set script A is defined by the property {z1z2⋯zn|zi∈⊗1/n script A}≡script A, i.e., the set of all products of n independently chosen values from ⊗1/nscript A is identical to script A. Hence, the n-th Minkowski power of ⊗1/nscript A yields the original set script A. Minkowski root extractions are fundamental operations in the Minkowski geometric algebra of complex sets: depending on the nature of script A, subtle issues concerning the existence, uniqueness and minimality or maximality of ⊗1/nscript A may arise. For a domain script A with a smooth boundary that is strictly logarithmically convex, we show that each connected component of the "ordinary" root script A1/n={z|zn∈script A} is a Minkowski n-th root. ⊗1/nscript A has a more intricate structure, however when ∂script A has logarithmic inflections. For example, if script A is a circular disk, ⊗1/nscript A is (a single loop of) the "n-th order" ovals of Cassini or lemniscate of Bernoulli when 0∉script A or 0∈∂script A, respectively. But when 0 is in the interior of script A, a composite curve (portions of the Cassini ovals and a higher-order curve) is required to describe ⊗1/nscript A.
AB - An n-th Minkowski root ⊗1/n script A of a given complex set script A is defined by the property {z1z2⋯zn|zi∈⊗1/n script A}≡script A, i.e., the set of all products of n independently chosen values from ⊗1/nscript A is identical to script A. Hence, the n-th Minkowski power of ⊗1/nscript A yields the original set script A. Minkowski root extractions are fundamental operations in the Minkowski geometric algebra of complex sets: depending on the nature of script A, subtle issues concerning the existence, uniqueness and minimality or maximality of ⊗1/nscript A may arise. For a domain script A with a smooth boundary that is strictly logarithmically convex, we show that each connected component of the "ordinary" root script A1/n={z|zn∈script A} is a Minkowski n-th root. ⊗1/nscript A has a more intricate structure, however when ∂script A has logarithmic inflections. For example, if script A is a circular disk, ⊗1/nscript A is (a single loop of) the "n-th order" ovals of Cassini or lemniscate of Bernoulli when 0∉script A or 0∈∂script A, respectively. But when 0 is in the interior of script A, a composite curve (portions of the Cassini ovals and a higher-order curve) is required to describe ⊗1/nscript A.
KW - lemniscate of Bernoulli
KW - logarithmic convexity
KW - minimal and maximal roots
KW - Minkowski geometric algebra
KW - Minkowski roots
KW - ordinary roots
KW - ovals of Cassini
UR - http://www.scopus.com/inward/record.url?scp=0005263798&partnerID=8YFLogxK
U2 - 10.1109/GMAP.2000.838260
DO - 10.1109/GMAP.2000.838260
M3 - Conference contribution
AN - SCOPUS:0005263798
T3 - Proceedings - Geometric Modeling and Processing 2000: Theory and Applications
SP - 287
EP - 300
BT - Proceedings - Geometric Modeling and Processing 2000
A2 - Martin, Ralph
PB - Institute of Electrical and Electronics Engineers Inc.
T2 - Geometric Modeling and Processing 2000, GMP 2000
Y2 - 11 April 2000 through 12 April 2000
ER -