Special cases of this spiral include the golden spiral and the Fibonacci spiral, which approximates the golden spiral. It was first described by Descartes, but was studied in depth by Jacob Bernoulli who called it “the marvellous spiral”. (1) can be transformed into the following implicit cartesian equation: (3) arctan ( y x) x 2 + y 2 ( x 0). The logarithmic spiral is a self-similar spiral, which often appears in nature. Let us consider the simplest Archimedean spiral with polar equation: (1) r. The curve was named after the ancient Roman lituus (a curved augural staff or war-trumpet) by Roger Cotes in a collection of papers published in 1722. The curvature of Archimedes' spiral is kappa(theta)(2+theta2)/(a(1+theta2)(3/2)), (2) and the arc length is s(theta) 1/2a(thetasqrt(1+theta2)+sinh(-1)theta) (3) 1/2athetasqrt(1+theta2)+ln(theta+sqrt(1+theta2)). Hence, the angle is inversely proportional to the square of the radius. The lituus is a spiral with a polar equation: Note: You may want to keep page 2.2 always with the Archimedean spiral for illustration and. This curve was used by Archimedes (287BC. Compare the equations of the function and the rose curve. In the centre, it is at an infinite distance from the pole for θ starting at 0, r starts at infinity. Conon of Samos (about 280BC - about 220 BC) invented a spiral,the polar equation of which is written as r a q. It is the opposite of an Archimedean spiral and thus has the following polar equation: Hyperbolic Spiralįirst conceived by Pierre Varignon in 1704, the hyperbolic spiral is a transcendental curve, meaning that “ it is an analytic function that does not satisfy a polynomial equation“. These integrals, and hence the Euler spiral, can be used to describe the energy distribution of Fresnel’s diffraction at a single slit in wave theory. The parameter form consists of two equations with Fresnel’s intervals, which can only be solved approximately. Fermat’s Spiralįermat’s spiral is a parabolic spiral that obeys the following polar equation:Īlso known as a Cornu Spiral, it is a curve whose curvature grows as the distance from the origin increases “the curvature of a circular curve is equal to the reciprocal of the radius”. It is described by the following polar equation:īy changing parameter a the spiral will turn and parameter b controls the distance between successive turnings. As this passes through A(pi/4,pi/2), we have. This is a spiral named after the famous Greek mathematician Archimedes, who was the first to describe it in his book On Spirals. The polar equation of Archimedes' spiral is of the form. Other spirals falling into this group include the hyperbolic spiral (c 1), Fermat's spiral (c 2), and the lituus (c 2).
The normal Archimedean spiral occurs when c 1. In this blog post, I will discuss two-dimensional spirals (note that there also exist a variety of three dimensional spirals). Sometimes the term Archimedean spiral is used for the more general group of spirals. There are many different types of spirals depending on the formulae that create them. A spiral is a curve which “emanates from a point, moving farther away as it revolves around the point”. Here is the final version of the definition: ģd printing. The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance. Here is the definition is you want to play with it: Ī little trick is added here to improve visual output, 80’s style hyperbolic spiral: The same parameters, this time animating number, fixing a = 10)Ī hyperbolic spiral with a very small domain of angles = 3, just 6 points performing break-dance with constant increase on a (-10 to 10) and n = -1 This is a static sunflower (fermat’s) spiral (angle = 360, number = 100, a = 10, n = 2)Īn animated archimedean spiral (angle = 360, number = 100, a is -10 to 10, n = 1) According to Woldfram Mathworld ( here) constant n = -2 is named lituus, while n = -1 gives a hyperbolic spiral, n = 1 is a regular archimedes spiral and finally n = 2 will give a fermat’s spiral (described also here) These are totally boring curves when we see them static, however when Grasshopper animates the parameters, interesting natural movements appper: Constant a determines how fast the spiral will turn, whereas constant n is the 1/n power of the angle variable that gives unique names to the spirals. It is just a polar point construct, mapped onto a range of angles and number of points.
This is a small exercise of Grasshopper drawing various archimedean spirals.