# You Gotta Know These Geometric Curves

- The circle is a curve consisting of the points that are an equal distance (called the radius) from a particular point (called the center). The irrational constant pi can be defined as the ratio of the circle’s circumference (the circle-specific term for perimeter) to its diameter (equal to twice the radius); the circle’s area is pi times the radius squared. A sector is the wedge-like shape bounded by two radii and an arc of the circumference; a chord is a straight line segment joining two points on the circumference. Cyclic shapes are shapes for which it is possible to draw a circle that goes through all the shape’s vertices; all triangles and all rectangles (including squares) are cyclic, but many quadrilaterals are not. The Cartesian equation (
*x*−*h*)^{2}+ (*y*−*k*)^{2}=*r*^{2}gives a circle of radius*r*centered at (*h*,*k*). - The circle is the simplest of the conic sections, which are the class of curves formed by the intersection of a plane with a double cone (two cones with the same axis, joined at the apex). Another definition of conic sections is that, for any point
*P*on the curve, the distance from*P*to a point called the focus is equal to a constant multiple of the distance from*P*to a line called the directrix; the constant multiple is the eccentricity, a dimensionless number that describes how “squashed” the curve is or how far away it is from being a circle. For example, for a circle, the focus is the center, the directrix is a “line at infinity,” and the eccentricity is therefore zero. The next three entries are the other three conic sections, other than degenerate conic sections (a single point and various configurations of lines, are considered degenerate conics, meaning they can arise from mathematical definitions of conic sections but are not meaningful curves). - Ellipses are conic sections that resemble ovals. Their eccentricity is less than 1, and some definitions treat a circle as a special case of the ellipse. On an ellipse, the
*sum*of the distances to two foci is constant. Their major axis and minor axis are the line segments within the ellipse that lie on their axes of symmetry, with the former being longer than the latter. In Cartesian coordinates, an ellipse has an equation of the form (*x*−*h*)^{2}+ (*y*−*k*)^{2}= 1, where (*h*,*k*) is the center and*a*and*b*are the semimajor and semiminor axes (half the length of the major and minor axes, in either order depending on which is greater). The orbits of planets trace out ellipses, per Kepler’s first law. - Parabolas are conic sections that are open, meaning the curve does not “come back to meet itself.” They all have the same eccentricity, 1, meaning all parabolas are similar to one another and, for any point on the parabola, the distance to the directrix equals the distance to the single focus. The standard equation for an upward-opening parabola is
*x*^{2}= 4*a**y*, with the focus being at (0,*a*). In Cartesian coordinates, the graph of any quadratic single-variable polynomial is a parabola. In a constant gravitational field, the path of a projectile is an arc of a parabola. - Hyperbolas are conic sections with eccentricity greater than 1. On a hyperbola, the
*difference*of the distances to two foci is constant. Hyperbolas have two branches, each of which tends towards a pair of asymptotes. In Cartesian coordinates, an ellipse has an equation of the form (*x*−*h*)^{2}− (*y*−*k*)^{2}= 1, where (*h*,*k*) is the center. - The cycloid is the curve traced out by a point on the circumference of a circle as it rolls once along a straight line without slipping. Portions of a cycloid provide the solution to the brachistochrone problem, which is to identify the curve between two points so that a particle sliding frictionlessly down the curve will slide as fast as possible. The cycloid also provides the solution for the tautochrone problem, which seeks a curve for which the time taken for a particle to descend the curve does not depend on the initial position.
- The catenary is the shape of a uniformly heavy chain (
*catena*in Latin) suspended from two points at the same height in a constant gravitational field. The equations (in Cartesian coordinates) for catenaries use the hyperbolic cosine function. All catenaries are similar to each other. Though catenaries look similar to parabolas,*y*scales exponentially with*x*rather than quadratically as for the parabola. - The Witch of Agnesi is a bell-shaped curve given by the graph of
*y*= 1 / (1 +*x*^{2}). When rescaled, it becomes the probability density function for the Cauchy or Lorentzian distribution, which is well-defined but nevertheless has no well-defined mean. It was popularized by the mathematician Maria Agnesi, the first woman to receive a position as a math professor (though she apparently did not take it up). Its unusual name comes from a translator misreading its Italian name “versiera” as “avversiera,” meaning “witch.” - There are several types of spirals, all of which wind outwards from a central area and cross axes infinitely many times. One of the most familiar types of spirals is the Archimedean spiral, characterized in polar coordinates by its radius being a linear function of the polar angle (
*r*=*k**θ*). Other types include the logarithmic spiral, in which the radius is proportional to the exponential of a multiple of the polar angle (*r*=*a**e*^{k θ}), exemplified by the shell of the nautilus. - The cardioid, whose name references its heart-shaped appearance, can be defined similarly to the cycloid (see above) except that the circle now rolls around another identical circle. The cardioid has a cusp where the point on the former circle meets the latter circle. A cardioid generated from circles of radius
*a*, and with its cusp at*θ*= 0, is given by the polar equation*r*= 2*a*(1 − cos*θ*). - The lima¸on, whose name comes from a French term for “snail,” was described by Étienne Pascal, the father of Blaise Pascal. It is defined similarly to the cardioid, except that the point can now lie anywhere within the outer circle. Depending on their parameters, lima¸ons can have a cardioid-like “dimple” or an inner loop formed by the curve intersecting itself.
- Lemniscates are a class of curves with a figure-8 shape. The infinity symbol is a lemniscate. There are several types of lemniscates, including the lemniscate of Booth, which is a special case of the hippopede, and the lemniscate of Bernoulli, along which the
*product*of a point’s distances to two foci is constant. Its name comes from the Greek for “ribbon.”

This article was contributed by NAQT editor Joseph Krol.