Parameter Formulas and Equations for Simple Ovals and Applications
Many tools for drawing a polycentric oval subject to geometrical or aesthetical constraints have been presented in Chap. 3. In this chapter we derive formulas which on the other hand allow to calculate all important parameters when the value of three independent ones is known, as well as the limitations the given parameters are subject to. These formulas allow for a deeper insight in the properties of any oval, as will be shown on the chosen forms of Chap. 6. Cases included are numbered consistently with the construction numbers of Chap. 3. Also included is the proof of the formulas yielding the solution to the frame problem presented in Sect. 3.3 as well as formulas for the length of an oval and for the area surrounded by it. Section 4.4 is devoted to concentric ovals and their properties. Finally in Sect. 4.5 we tackle the problem of finding a closed form for the equation of a simple oval, and suggest a way of representing a generic one by means of Geogebra.