Lemniscate

A lemniscate is a fascinating type of curve that revolves around two distinct points known as foci. One of the most intriguing properties of a lemniscate is that for any point on the curve, the product of the distances to its foci remains constant. This unique characteristic makes lemniscates particularly useful in various modeling applications, including the creation of SDF-like (Signed Distance Function) entities.

Lemniscate of Bernoulli

The lemniscate of Bernoulli is a classic example of this type of curve. As illustrated in Figure 10.28, the product of the distances from any point on the curve to the two foci (F1 and F2) is constant.

Figure 10.28 Lemniscate of Bernoulli (Kmhkmh, CC BY 4.0, via Wikimedia Commons). The product of distances from F1 to P and from F2 to P is constant.

Lemniscate as an SDF-like Function

The lemniscate can be adapted into an SDF-like function, as shown in Figure 10.29. In this context, the foci are denoted as f1 and f2, and a numeric constant c is used. Although this SDF-like function does not represent the true distance to the curve, it behaves similarly to a smooth sign function, being negative inside the curve and positive outside. This property allows it to be used effectively for modeling purposes.

Figure 10.29 A lemniscate of Bernoulli as an SDF-like function

Applications in Design

Lemniscates can also be employed in design, such as in the creation of shapes like a Play button. As demonstrated in Figure 10.30, a Play button can be described using a multifocal lemniscate, which requires only six points and a constant. This is significantly more efficient than using a cubic Bézier spline, which would require at least three Bézier curves and twelve control points.

Figure 10.30 A Play button made of a multifocal lemniscate