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The term oval when used to describe curves in geometry is not well-defined, except in the context of projective geometry. Many distinct curves are commonly called ovals or are said to have an "oval shape". Generally, to be called an oval, a plane curve should resemble the outline of an egg or an ellipse. In particular, these are common traits ...
Additionally, if a spherical wavefront is refracted through a spherical lens, or reflected from a concave spherical surface, the refracted or reflected wavefront takes on the shape of a Cartesian oval. The caustic formed by spherical aberration in this case may therefore be described as the evolute of a Cartesian oval. [5]
A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two fixed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle.
A two-dimensional orthographic projection at the left with a three-dimensional one at the right depicting a capsule. A capsule (from Latin capsula, "small box or chest"), or stadium of revolution, is a basic three-dimensional geometric shape consisting of a cylinder with hemispherical ends. [1]
A Cartesian oval is the set of points for each of which the weighted sum of the distances to two given foci is constant. If the weights are equal, the special case of an ellipse results. A Cassini oval is the set of points for each of which the product of the distances to two given foci is constant.
That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape. Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points.
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In geometry, a Cassini oval is a quartic plane curve defined as the locus of points in the plane such that the product of the distances to two fixed points is constant. This may be contrasted with an ellipse , for which the sum of the distances is constant, rather than the product.