Oval Calculator

Name: Oval - Geometry Calculator
Author: Jürgen Kummer

Calculations at an oval. An oval is mathematically elusive, the word means egg-shaped. Usually, an oval is seen as a round, convex shape without vertex and with one symmetry axis. This special oval is formed by two circles with radiuses R and r and a distance a of their centers. Around both circles, a third one is laid, which touches each of them in one point. The center of this is at the intersection of the two lines through touch points and centers of the other two circles. The area formed by this, a flat projection of a spherical triangle, is added to the two circles. This is repeated on the other side of both circles.

Formulas:

ρ = \frac{a^{2} + R^{2} - r^{2}}{2 \cdot (R - r)}

A = \frac{1}{2} \cdot {a (R - r) + π (r^{2} + R^{2}) - \frac{a^{3}}{(R - r)} + \frac{[a^{2} + {(R - r)}^{2}] \cdot (a^{2} - 3 r^{2} + 2 r R + R^{2})}{2 \cdot {(R - r)}^{2}} \cdot \arctan [\frac{2 a (R - r)}{a^{2} - {(R - r)}^{2}}]}

If a ≤ R - r: A = π * R²
(identical to a circle with radius R, there is no joining circle)

If R = r: ρ=∞, A = π * R² + 2 * a * R
(identical to a stadium with circle radius R and rectangle length a)

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

pi:
π = 3.141592653589793...

Radiuses and distance have a one-dimensional unit (e.g. meter), the area has this unit squared (e.g. square meter).

The oval above was discovered around the year 1500 by Albrecht Dürer, who wanted to construct an egg shape on paper using only a compass and ruler. A similar shape that cannot be constructed but is easier to calculate is the egg profile made of two semi-ellipses. An ellipse can also be considered an oval shape, which, unlike the other two, is the same in both lengthwise directions. The ellipse can therefore be considered a regular oval. Shapes that are pointed at one end but otherwise oval are the leaf of a rose and a half lemniscate of Bernoulli.
All of these oval shapes are axially symmetrical to the line of their largest diameter. Only the ellipse is symmetrical to another axis, which is perpendicular to the first one.

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