2021/07/20 by Thierry Dana-Picard, Dana-Picard, Thierry, Zoltán Kovács +1
Agricultural and Biological Sciences · #53-08 #53A04 #Algebraic Geometry (math.AG) #FOS: Mathematics #History and Overview (math.HO) #Leaf Properties and Growth Measurement #Plant nutrient uptake and metabolism #Soybean genetics and cultivation
paper · pdf · doi:10.48550/arxiv.2107.14592
openalex publication_date 2021/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The non-uniqueness of a rational parametrization of a rational plane curve may influence the process of computing envelopes of 1-parameter families of plane curves. We study envelopes of family of circles centred on a regular trifolium and its offsets, paying attention to different parametrizations. We use implicitization both to show that two rational parametrizations of a curve are equivalent, and to determine an implicit equation for the envelope under study. The derivation of an implicit equation of an offset follows another path, leading to new developments of the package GeoGebra Discovery. As an immediate symbolic result, we obtain that in the general case the offset curve of a regular trifolium is an algebraic curve of degree 14. We illustrate this fact by providing a GeoGebra applet that computes such curves automatically and visualizes them in a web browser.