Oval

The term Oval (Latin ovum = egg ) denotes a flat rounded convex figure, in the broadest sense similar to the profile of a bird's egg. Comprising circles and ellipses as special cases wherein any oval has to have, in contrast to this no axis of symmetry.

The use of the term is not always consistent, sometimes it is also used purely descriptive. In calculus, it can, however, formally define by means of plane curves, in this context, one also talks of Eikurven or Eilinien.

A three-dimensional rounded convex body ( generally a closed convex subset of ) is called ovoid. In this sense, an oval with its interior is then a two-dimensional ovoid.

In the synthetic geometry of the terms and Oval Ovoid be defined as a term for certain quadratic quantities in arbitrary projective geometries. An oval or ovoid, as will be discussed in this article is, in the projective completion of the real plane and the real space is always an oval or ovoid in terms of the synthetic definition. See Quadratic lot # ovoid.

Formal Definition and properties

The round shape of an oval obtained by asking for a closed curve smoothness and convexity. This leads to the following definition:

However, this definition does not capture all geometric figures that are sometimes referred to as ovals. To meet, for example, ovals, which are composed of different arcs, this definition does not because its second derivative does not exist on the entire curve. If you want to cover this kind of cases, so you have to compromise on the smoothness of the curve make ( or instead of). Occasionally, therefore only the convexity is required. However, this has the disadvantage that the definition then includes figures which are normally barely perceives as " egg-shaped ", such as convex polygons.

An oval in the sense of the above definition has the following properties:

• An oval is a Jordan curve, ie, it has no loops or loops.
• The oriented curvature of an oval has no sign change, that is, depending on the sense of the flow oriented curvature for each point of the oval is either negative or positive. This clearly means that it has no twists or dents. It can only run in a pure left turn or right turn.
• The inside of an oval is a convex set, and the oval forms its boundary.
• For ovals of the four vertex theorem holds, that is, the curvature of an oval has at least four extreme points.
• If a point of the oval a tangent, so is the whole oval on one side of the tangent.
• If one demands in addition that the curvature of the oval disappears on any section, that is, the curvature increases at most isolated points in the value zero, then there exists at each point of the oval, the above tangent. More generally, then for an arbitrary line that she has with the oval either no point ( Passante ), exactly one point ( tangent ) or exactly two points ( secant ) together.

Examples and constructions

Ovals can be constructed using completely different methods. A number of construction techniques can be obtained from the various Konstruktionensverfahren of ellipses, which are each slightly modified at an appropriate location. You can create an ellipse by cutting a plane with a cone ( see conics ). If one now uses instead of the cone certain other rotating body, such as a rotated hyperbola is obtained by ellipses various ovals. Another possibility is in its parametric form, or in their algebraic equation to replace the constant parameters A and B ( lengths of the semi-axes ) of an ellipse through features.

Can be defined as an ellipse, the amount of the points P for which the sum of the distances from the two foci F1 and F1 is constant. Substituting now this sum of the distances by a weighted sum, the set of points forms an oval, which only has a symmetry axis, lie on the one pointed and one blunt end. Such oval is designated as a Cartesian oval.

The constructing method of de La Hire generates an ellipse using two concentric circles. You now shifts the center of the outer circle a bit, but it keeps otherwise the remaining steps of the design process in, then you get a ( new ) oval. This has an axis of symmetry, when moving the center of the outer circle along the elliptical axes. If you move outside the center of the axles, the result is an oval without symmetry axes.

Construction with non-concentric circles

Ellipse:

Ellipse:

The solution set of an equation with two unknowns, or certain subsets of it can often be regarded as curves in the plane. With a suitable equation thereby obtained an oval. When such a dissolution profile is not oval, but having a convex loop, it can be produced by adding a correction term from the loop an oval.

Lamésches Oval:

Lemniscate:

Cassini oval:

Szegö curve:

Cartesian sheet:

Ovals can also be composed of circular arcs and straight line segments. However, these ovals have less smoothness as required in the above definition, as they are only in and out of. They are therefore still smooth in the sense of a continuous derivative, but have no more continuous curvature. The curvature is instead constantly to the sections and has at the interfaces of the circular arcs or line segments a discontinuity.