ALGEBRAIC CURVES AND CONTROL THEORY

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ALGEBRAIC CURVES AND CONTROL THEORY
by
Bill Wolovich
Brown University Providence, RI
Based on Chapter 3 of the book INVARIANTS FOR
PATTERN RECOGNITION AND CLASSIFICATION, World
Scientific Publishing, 2000, titled A New
Representation for Quartic Curves and Complete
Sets of Geometric Invariants by M. Unel and W.
A. Wolovich.
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The unit circle curve can be defined either
explicitly by the parametic equations x(t)
sin t and y(t) cos t, or implicitly by the
polynomial, or algebraic equation
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Some Examples of Quartic Curves
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An
representation of a quartic curve
The centers of the ellipses and the circle are
useful related points that map to one another
under Euclidean and affine transformations.
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A Euclidean (Rotation and Translation)
Transformation
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The centers of the ellipses and the circle also
can be used to define a canonical transformation
which maps a quartic curve to a canonical
(quartic) curve, namely
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A complete set of Euclidean invariants for a
representation.
The ratios and the distances
are useful invariants for object recognition, as
we now show.
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Red Quartic IP Fits to Blue (a) Airplane, (b)
Butterfly, (c)Guitar, (d) Tree, (e) Mig 29, and
(f) Hiking Boot
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Object recognition based on the elliptical ratio
invariants
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Discrimination between the boot and the tree
using
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CURRENT WORK
Motion of Planar Algebraic Curves Using
Theorem 3 Any non-degenerate algebraic curve
can be uniquely expressed as a sum of line
products.
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Now suppose the curve undergoes an unknown rigid
motion defined by
with
skew-symmetric i.e.
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