Oval
An oval is a curve resembling a squashed circle but, unlike the ellipse, without a precise mathematical definition. The word oval derived from the Latin word "ovus" for egg. Unlike ellipses, ovals sometimes have only a single axis of reflection symmetry (instead of two).
The particular variant illustrated above can be constructed with a compass by joining together arcs of different radii such that the centers of the arcs lie on a line passing through the join point (Dixon 1991). Albrecht Dürer used this method to design a Roman letter font. If the separation between left and right caps is a and the radii are R and r, respectively, with r<R and R-r<a, then the center (0,y) and radius rho of the joining circle are
Call the three circles C_R, C_r, and C_rho. Let the upper point of intersection of C_r and C_rho be (x_0,y_0), let the angle between the vertical dashed line and the line through (x_0,y_0) be theta, and let the angle between the horizontal radius of C_r and dashed line through (x_0,y_0) be phi. Then
and half the area enclosed by the oval is the sum of the areas of leftmost quarter-circle, the sector of C_rho, and the sector of C_r minus the area of the triangular portion of the sector of C_rho lying below the x-axis, so
)/(2(R-r)^2)tan^(-1)((2a(R-r))/(a^2-(R-r)^2))].](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/Oval/Inline49.svg){kind=link}
As expected, this formula reduces to the area of a circle
| A=piR^2 |
(10)
|
for a->R-r, and to the area of a stadium
| A=pir^2+2ar |
(11)
|
for R->r.
See also
Cartesian Ovals, Cassini Ovals, Cundy and Rollett's Egg, Egg, Ellipse, Lemon Surface, Lens, Lune, Moss's Egg, Ovoid, Rounded Rectangle, Stadium, Superellipse, Tangent Circles, Thom's EggsExplore with Wolfram|Alpha
More things to try:
References
Critchlow, K. Time Stands Still. London, England: Gordon Fraser, 1979.Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.Dixon, R. "The Drawing Out of an Egg." New Sci., July 29, 1982.Dixon, R. Mathographics. New York: Dover, pp. 3-11, 1991.Pedoe, D. Geometry and the Liberal Arts. London, England: Peregrine, 1976.Referenced on Wolfram|Alpha
OvalCite this as:
Weisstein, Eric W. "Oval." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Oval.html