Question

Consider the two curves represented by the equations that are not superellipses. State the type of curve produced each time.

1. |x/2|^2+|y/2|^2=1

2. |x/3|^2 + |y/2|^2 = 1

Answer 1

Number 1: In mathematics, algebraic curves are the simplest objects of Euclidean geometry that can not be defined by linear properties. Specifically, in Euclidean geometry, a plane algebraic curve is the set of the points of the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

For example, the unit circle is an algebraic curve, being the set of zeros of the polynomial x2 + y2 − 1

Various technical considerations have led to consider that the complex zeros of a polynomial belong to the curve. Also, the notion of algebraic curve has been generalized to allow the coefficients of the defining polynomial and the coordinates of the points of the curve to belong to any field, leading to the following definition.

In algebraic geometry, a plane affine algebraic curve defined over a field k is the set of points of K2 whose coordinates are zeros of some bivariate polynomial with coefficients in k, where K is some algebraically closed extension of k. The points of the curve with coordinates in k are the k-points of the curve and, all together, are the k part of the curve.

For example, (2,\sqrt{-3}) is a point of the curve defined by x2 + y2 − 1 =0 and the usual unit circle is the real part of this curve. The term "unit circle" may refer to all the complex points as well to only the real points, the exact meaning being usually clear form the context. The equation x2 + y2 + 1 = 0 defines an algebraic curve, whose real part is empty.

More generally, one may consider algebraic curves that are not contained in the plane, but in a space of higher dimension. A curve that is not contained in some plane is called a skew curve. The simplest example of a skew algebraic curve is the twisted cubic. One may also consider algebraic curves contained in the projective space and even algebraic curves that are defined independently to any embedding in an affine or projective space. This leads to the most general definition of an algebraic curve:

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one.
Number 2: In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context. There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space (usually a Euclidean space) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold. This article deals primarily with the first concept.

The canonical example of extrinsic curvature is that of a circle, which everywhere has curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature of a smooth curve is defined as the curvature of its osculating circle at each point.

More commonly this is a scalar quantity, but one may also define a curvature vector that takes into account the direction of the bend as well as its sharpness. The curvature of more complex objects (such as surfaces or even curved n-dimensional spaces) is described by more complex objects from linear algebra, such as the general Riemann curvature tensor.

The remainder of this article discusses, from a mathematical perspective, some geometric examples of curvature: the curvature of a curve embedded in a plane and the curvature of a surface in Euclidean space. See the links below for further reading.