1. ALGEBRAIC CURVES AND CONTROL THEORY by Bill Wolovich 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. Brown University Providence, RI

2. 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:

3. Defines a Quartic (4th Degree) Algebraic Curve Defines a General Algebraic or Implicit Polynomial (IP) Curve of degree n. The curve is monic if

4. Some Examples of Quartic Curves

5. Theorem 1: Any closed, bounded algebraic curve of (even) degree n can be uniquely represented as: where is a polynomial of degree n-2, and each conic factor Theorem 2: Any closed, bounded quartic curve can be uniquely represented as the product of two ellipses and a circle; i.e.

6. Anrepresentation 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.

7. A Euclidean (Rotation and Translation) Transformation

8. If the centers of the 2 ellipses and the circle are defined by, the (unique) Euclidean transformation matrix E which maps is given by:

9. 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:

10. A complete set of Euclidean invariants for arepresentation. The ratios and the distances are useful invariants for object recognition, as we now show.

11. Red Quartic IP Fits to Blue (a) Airplane, (b) Butterfly, (c)Guitar, (d) Tree, (e) Mig 29, and (f) Hiking Boot

12. Object recognition based on the elliptical ratio invariants

13. Discrimination between the boot and the tree using

14. CURRENT WORK Theorem 3: Any non-degenerate algebraic curve can be uniquely expressed as a sum of line products. For Quartic Curves: Motion of Planar Algebraic Curves Using

15. We next homogenize the lines in the decomposition via:

16. Now suppose the curve undergoes an unknown rigid motion defined by: withskew-symmetric; i.e.

17. The motion of the homogenized line coefficients is defined by which implies that the line parameters of the original curve satisfy the coupled Riccati equations: