# Computations with Algebraic Curves

@inproceedings{Abhyankar1988ComputationsWA, title={Computations with Algebraic Curves}, author={Shreeram S. Abhyankar and Chandrajit L. Bajaj}, booktitle={ISSAC}, year={1988} }

We present a variety of computational techniques dealing with algebraic curves both in the plane and in space. Our main results are polynomial time algorithms (1) to compute the genus of plane algebraic curves, (2) to compute the rational parametric equations for implicitly defined rational plane algebraic curves of arbitrary degree, (3) to compute birational mappings between points on irreducible space curves and points on projected plane curves and thereby to compute the genus and rational…

## 39 Citations

### Square-root Parametrization of Plane Curves

- Mathematics
- 1994

By calculating the genus of an irreducible algebraic plane curve of degree n in terms of its singularities, we see that, counted properly, the curve can have at most \(\frac{{(n - 1)(n - 2)}} {2}\)…

### Geometric computations with algebraic varieties of bounded degree

- Mathematics, Computer ScienceSCG '90
- 1990

The main results are algorithms to compute the degree of an algebraic variety, and compute the rational parametric equations (a rational map from points on a hyperplane) for implicitly defined algebraic varieties of degrees two and three.

### Piecewise Rational Approximations of Real Algebraic Curves

- Mathematics
- 1997

We use a combination of both algebraic and numerical techniques to construct a C 1 -continuous, piecewise (m, n) rational ǫ-approximation of a real algebraic plane curve of degree d. At singular…

### Implicit Representation of Rational Parametric Surfaces

- Mathematics, Computer ScienceJ. Symb. Comput.
- 1992

### Piecewise Rational Approximations of Real

- Mathematics
- 1992

AbstractWe use a combination of both algebraic and numerical techniques to construct a C 1_ continuous, piecewise (m, n) rational t:-approximationof a real algebraic plane curve ofdegree d. At…

### NURBS Approximations of Real Algebraic Curves

- Mathematics
- 2007

We use a combination of both algebraic and numerical techniques to construct a C 1-continuous, piecewise (m; n) rational B-spline-approximation of a real algebraic plane curve of degree d. At…

### Efficient and Reliable Computation with Algebraic Numbers for Geometric Algorithms

- Computer Science, Mathematics
- 1998

The overall approach combines techniques from symbolic computation based on exact arithmetic with oating point arithmetic for reliable computations with algebraic numbers and is about one order of magnitude faster as compared to earlier implementations that produce reliable results.

### Numeric-symbolic algorithms for evaluating one-dimensional algebraic sets

- Computer Science, MathematicsISSAC '95
- 1995

This work presents algorithms based on a combination of numeric and symbolic techniques for evaluating one-dimensional algebraic sets in a subset of the real domain and presents techniques to handle singularities for well-conditioned inputs.

### ON THE THEORY OF RESOLVENTS AND ITS APPLICATIONS

- Mathematics, Computer Science
- 1999

A generalized algorithm to construct the generalized resolvents based on Wu-Rits's zero decomposition algorithm and a complete method of finding parametricequations for algebraic curves are presented.

### An exact and efficient approach for computing a cell in an arrangement of quadrics

- MathematicsComput. Geom.
- 2004

## References

SHOWING 1-10 OF 71 REFERENCES

### Automatic parameterization of rational curves and surfaces III: Algebraic plane curves

- MathematicsComput. Aided Geom. Des.
- 1988

### The computerization of algebraic geometry

- Mathematics, Computer ScienceEUROSAM
- 1979

The proper mathematical foundations for the treatment of algebraic functions are presented, and the formalism leads directly to the requirement for algorithms to find the genus of an algebraic curve, and to discover what function is associated with a given divisor.

### Precise Implementation of Cad Primitives using Rational Parameterizations of Standard Surfaces

- Mathematics
- 1984

We discuss the problem of computing the intersection curve of two algebraic surfaces, each of which possesses rational parameterization. The special case where the two surfaces are quadric is…

### Algebraic curves

- Mathematics
- 1950

This introduction to algebraic geometry examines how the more recent abstract concepts relate to traditional analytical and geometrical problems. The presentation is kept as elementary as A linear…

### Automatic parametrization of rational curves and surfaces II: cubics and cubicoids

- Mathematics, Computer Science
- 1987

### Implicit representation of parametric curves and surfaces

- Mathematics, Computer ScienceComput. Vis. Graph. Image Process.
- 1984

### On the worst-case arithmetic complexity of approximating zeros of polynomials

- Mathematics, Computer ScienceJ. Complex.
- 1987

### Generalized Polynomial Remainder Sequences

- Mathematics, Computer Science
- 1983

Habicht’s subresultant theorem allows new and simple proofs of many results and algorithms found in different ways in Computer algebra.