Logarithmic spiral
The nautilus shape is often associated with a spiral or even the golden ratio. In order to create our own, we will create a logarithmic spiral and use its properties to structure the nautilus shell.
Spiral definition
The logarithmic, or equiangular spiral is the locus of points corresponding to the locations over time of a point moving further away as it revolves around a fixed point. As such, in polar coordinates \((r, \theta)\) it can be described as follows.
– Logarithmic spiral equation
- \(r = a + e^{b * \theta}\)
In which \(a > 0\) and \(b \ne 0\) are real constants, \(e\) is the base of natural logarithms, \(r\) is the length of the radius from the centre of the spiral and \(\theta\) is the amount of rotation of the radius.
The spiral has the property that the angle \(\phi\) between a radius vector to a point on the curve and the tangent at this point is a constant defined by the following formula, this will help us shape the nautilus afterwards.
– Tangent oN a Logarithmic spiral
- \(tan( \phi) = 1 / b\)
↳ Spiral demo (three.js)
Spiral function
– References
- Tangent on a logarithmic spiral, Wolfram
- Is the nautilus shell spiral a golden ratio ?, Golden Number
Based upon equations by Johan Gielis and research of Paul Bourke, we will learn how to generate 2D supershapes. Then, we will design a nautilus shell by placing them along our logarithmic spiral.
2D supershape definition
The superformula is a generalization of both circle/ellipse and superellipse. In the Cartesian coordinate system, these shapes are described as the set of all points \((x, y)\) on the curve that satisfy the following equations.
– Circle/ellipse equation
- \((x / a)^2 + (y / b)^2 = 1\)
– Superellipse equation
- \(|x / a|^n + |y / b|^n = 1\)
Where \(a\), \(b\) and \(n\) are positive numbers excluding 0. The superformula can be used to describe many complex shapes and curves that are found in nature. In polar coordinates \((r, \phi)\), it can be described as follows.
– Supershape equation
- \(r = \left(\left|\frac{cos(m / 4 * \phi)}{a}\right|^{n_2} + \left|\frac{sin(m / 4 * \phi)}{b}\right|^{n_3}\right)^{-1 / n_1}\)
In which \(a\) and \(b\) (both excluding 0) are ratios from the centre of the supershape. While \(m\) adds rotational symmetry to the shape, curves are repeated in sections of the circle of angle \(2\pi / m\). Finally, choosing different values for \(n_1\), \(n_2\) and \(n_3\) generates different curves allowing us to create symmetric and asymmetric shapes.
Normal on a given point
Last but least, here's a quick tip to calculate normal at a given point, let's call it \(Q\), in the supershape. To do this, we add add the normalized vector from the previous to the current point on the curve, \(\vec{\scriptstyle{PQ}}\), to the one from the current to the next point on the curve, \(\vec{\scriptstyle{RQ}}\).
– Normal vector tip
- \(\hat{n} = |\vec{\scriptstyle{PQ}}| + |\vec{\scriptstyle{RQ}}|\)
↳ Supershape demo (three.js)
Supershape function
– References
- Supershapes (Superformula), Paul Bourke
- 2D supershapes, The Coding Train
- What's a vector ?, 3Blue1Brown
In the next chapter, we will design our nautilus in 3D by interpolating supershapes along our logarithmic spiral.
