A bow **curve **is a quartic (4th degree) curve defined by the Cartesian equation [1]:

x^{4} + y^{3} – x^{2}y = 0.

It is also defined by the parametric equation

x = t – t

^{3}, y = t

^{2}– t

^{4}.

and the polar equation

## Other Bow-Shaped Curves

The curve defined above is sometimes called the*quartic bow curve*, presumably to distinguish it from other, similar curves shaped like bows.

Some authors use the name “bow curve” without clarifying which type of bow-shape they are referring to, which can get confusing. For example, Guo [2] mentions a shape constructed with a “bow curve” on page 174. However, the image the paragraph refers to (figure 3.3. on page 175) is clearly a *longbow curve*, parameterized by the equations [3]

x(θ) = a (sin^{2}θ + 1) cost(θ); y(θ) = a (sin^{2}θ + 1).

Make sure you know which type of “bow” curve you’re being asked to analyze or draw.

## Tangent Lines of the Bow Curve

The bow curve has three tangent lines (i.e., a triple point) at the origin. A triple point is a point on a curve where three branches of the curve intersect; in other words, it’s a point traced three times when the curve is traversed.

## The Spectahedron

The curve can fit inside a third order spectrahedron—a shape that can be represented as a linear matrix inequality. In two dimensions, the spectrahedron looks like a trapezoid:

## Uses

This curve has limited, if an, practical uses. Its main use is purely academic, appearing in some texts as either to study interesting curve behavior or triple points.

## References

Graphs created with Desmos.

[1] Ingalls, C. Bow Curve. Retrieved January 1, 2022 from: https://people.math.carleton.ca/~cingalls/studentProjects/Katie’s%20Site/html/Bow%20Curve.html

[2] Guo, F. et al. Semidefinite Representation of Non-Compact Convex Sets.

[3] M243: Calculus II (Sp 2019).