# Plane Curve: Definition, Examples

A plane curve is a curve in a two-dimensional plane. In other words, the points of the curve are all on the same plane. In comparison, a space curve’s points do not necessarily all lie on a single plane [1].

## Plane Curve Equations

A plane curve can be defined with the parametric equation (which means that both x and y are defined as functions of a parameter, usually t) [2]
x = x(t), y = y(t),
Where coordinates (x, y) are expressed as functions of t on the closed interval t1 ≤ t ≤ t2. x(t) and y(t) are continuous functions, with a sufficient number of continuous derivatives; if there are r continuous derivatives, then the curve is class r. In vector notation, a parametric plane curve can be specified by a vector-valued function r = r(t).

The equation for a plane curve can also be expressed as rectangular coordinates f(x, y) = 0, or polar coordinates f(r, θ) = 0.

Plane curves can show a variety of interesting features, including:

• Asymptotes,
• Cusps (where a curve is tangent to itself),
• Acnodes (isolated points),
• Nodes (where the curve intersects itself).

## Types of Plane Curve

The simplest plane curves arise from algebraic equations (those involving addition, subtraction, multiplication, division, roots, and raising to powers). If the curve can be described by a polynomial equation, they are called algebraic curves.

• Simple plane curves are non intersecting. In other words, they do not cross their own paths. If a curve intersects itself, then it’s not simple.
• A closed plane curve has no endpoints; it completely encloses an area. For example, a circle or ellipse; the Lamé curve is closed when n in its Cartesian equation is a positive integer. If a curve has endpoints (like a parabola), then it is an open curve.

A smooth plane curve is given by a pair of parametric equations on the closed interval [a, b]; derivatives for x(t) and y(t) exists and are continuous on [a, b] and the first derivatives are not simultaneously zero on that interval [3].

## References

[1] Montoya, D. & Naves, D. On Plane and Space Curves. Retrieved January 15, 2022 from: https://web.ma.utexas.edu/users/drp/files/Spring2020Projects/DRP_Final_Project%20-%20Daniel%20Naves.pdf
[2] Patrikalakis, N. et al. (2009). 1.1.1 Plane Curves. Retrieved January 15, 2022 from: https://web.mit.edu/hyperbook/Patrikalakis-Maekawa-Cho/node5.html
[3] Length of Curve and Surface Area.

## Cassini Oval

A Cassini oval is a plane curve defined as the set of points in the plane with the products of distances to two fixed points (loci) F1 and F2 is constant [1]; as a formula, the distance is (F1, F2) = 2a [2].

Cassini ovals are generalizations of lemniscates. The ovals are similar to ellipses, but instead of adding distances to loci, you multiply them.

The curve is symmetric with respect to the x-axis, the y-axis, and the origin.

## Equations

The general equation is (x2 + y2 + a2)2 – 4a2x2 = b4.
The polar equation is r4 − 2a2r2cos 2θ + a4.

## Cassini Oval Shapes

The ovals can take on three different shapes, shown in the following image:

• When a = b, the oval resembles an infinity symbol ∞, which is a Leminiscate of Bernoulli.
• When a < b, the oval takes on the shape of an ellipse or peanut shell.
• If a > b, the oval splits into two ellipses shaped like eggs, with the narrow ends of the “eggs” facing each other. The shapes are mirror-images of each other.

The following image shows the various shapes the Cassini oval can take on, in one image:

## Area of Cassini Ovals

The area of a Cassini oval can be found in several ways, including numerical integration and elliptic integrals. What follows is a summary, you can find more detail on pages 235-236 of this PDF.

The oval is symmetric to both the x- and y-axes, so we can find the area of one quarter of an oval, and multiply by 4:

You can also calculate the area with the following elliptic integral, where E(x) is the complete elliptic integral of the second kind:

## History of the Cassini Oval

Cassini ovals were first studied by 15th Century Giovanni Cassini, who used the ovals to model the sun’s orbit around the Earth [3]. The orbits of the planets around the sun, orbits of satellites around a planet, and electron orbits in atoms, can all be modeled by a Cassini Oval. This may come as a surprise; a common misconception is that planetary orbits are highly elliptical [4].

Cassini Ovals have a wide variety of uses, including developing radar and sonar systems, modeling human blood cells, and fuel tank optimization.

## References

“Three different shapes” image created with Desmos.
[1] Mümtaz, K. A Multi Foci Closed Curve: Cassini Oval, Its Properties and Applications. Institutional Archive of the Naval Postgraduate School. 2013. PDF.
[2] Hellmers, J. et al. Simulation of light scattering by
biconcave Cassini ovals using the nullfield method with discrete sources. Journal of
Optics A: Pure and Applied Optics, Pure Appl. Opt. Vol 8. pp. 1-9. 2006.
[3] Gibson, K. The ovals of Cassini. Lecture Notes. 2007. Retrieved February 11, 2012 from: https://mse.redwoods.edu/darnold/math50c/CalcProj/sp07/ken/CalcPres.pdf
[4] Morgado, B. & Soares, V. Kepler’s ellipse, Cassini’s oval and the trajectory of planets. Eur. J. Phys. 35 (2014) 025009.