TOPICS
Search

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).

Oval

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

rho=(a^2+R^2-r^2)/(2(R-r))
(1)
y=((R-r)^2-a^2)/(2(R-r)).
(2)

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

x_0=(a(a^2-r^2+R^2))/(a^2+(R-r)^2)
(3)
y_0=(2a^2r)/(a^2+(R-r)^2)-r
(4)
theta=tan^(-1)(a/y)
(5)
phi=tan^(-1)((y_0)/(x_0-a)),
(6)

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

A=2(1/4piR^2+1/2rho^2theta+1/2r^2phi-1/2ay)
(7)
=1/2piR^2+rho^2theta+r^2phi-ay
(8)
=1/2[a(R-r)+pi(r^2+R^2)-(a^3)/(R-r)+([a^2+(R-r)^2](a^2-3r^2+2rR+R^2))/(2(R-r)^2)tan^(-1)((2a(R-r))/(a^2-(R-r)^2))].
(9)

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 Eggs

Explore with Wolfram|Alpha

References

Critchlow, K. Time Stands Still. London: 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: Peregrine, 1976.

Referenced on Wolfram|Alpha

Oval

Cite this as:

Weisstein, Eric W. "Oval." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Oval.html

Subject classifications