However, note that the length equation is transcendental, and the inverse task (finding unknown dimensions while the length is among the known dimensions) requires numerical methods. These are all formulas that we need to find out unknown dimensions by known dimensions. Here is how number of turns n is related to angles:Īnd here is how diameters are related to angles (this follows directly from the spiral polar equation) Usually, it has a sleeve, hence the inner diameter and initial angle. If a spiral starts from zero angle (from the center), the formula is simplified:īut in real life, of course, a roll of material does not start from the center. The type of spirals is governed by the equation rr(s,), e.g., an Archimedean spiral is expressed as rs+a1, where a1>0. To find out the length, we need to integrate from the initial angle to the final angle. Now we have the dependence of the length dl on the angle dφ. Using the polar equation of a spiral, we can replace ρ with kφ, and dρ with kdφ Hence:Īn infinitesimal spiral segment dh can be replaced with an infinitesimal segment of a circle with radius ρ hence its length is ρdφ. Please be careful with unit control when you enter the known dimensions! 20 meters are not the same as 20 millimeters.Īn infinitesimal spiral segment dl can be thought of as hypotenuse of the dl, dρ, and dh triangle. Theory and formulas, as usual, can be found below the calculator. You can also solve an inverse problem (when you know the roll length) - calculate thickness and number of turnings using roll length and both diameters. For example, you can calculate roll length from inner and outer diameters and roll thickness or number of turnings. You can easily find out some of these objects' dimensions, like diameters and thickness, or a number of turnings, and, using the calculator below, calculate the missing ones. We can see spirals in everyday life in any objects that are in the rolled form: rolls of paper, tapes, films, and so on. These dimensions are related (see formulas below the calculator), and you can calculate any two if you know the other three. An example of this curve is the threads of a bolt.We have five spiral dimensions: outer diameter - D, inner diameter - d, thickness, separation distance or distance between arms - t, spiral length - L, number of turnings - n. Its points are such that it makes a constant angle with the cross sections of the cylinder. However, unlike the two dimensional plane curves of a spiral, a helix is a three dimensional space curve which lies on the surface of a cylinder. A helix is like a spiral in that it is a curve made by rotating around a point at an ever-increasing distance. Another spiral, the hyperbolic spiral, conforms to the equation r = a/ θ.Īnother type of curve similar to a spiral is a helix. This spiral discovered by Bonaventura Cavalieri (1598-1647) creates a curve commonly known as a parabola. The shell of a Nautilus and the seed patterns of sunflower seeds are both in the shape of a logarithmic spiral.Ī parabolic spiral can be represented by the mathematical equation r 2 = a 2 θ. Examples of the logarithmic spiral are found throughout nature.
Also, these spirals are different from a circle in that the length of the radii increases, while in a circle, the length of the radius is constant. However, unlike a circle, the angle at which its points cross its radii is not a right angle. These spirals are similar to a circle because they cross their radii at a constant angle. A logarithmic spiral is defined by the equation r = e a θ, where e is the natural logarithmic constant, r and θ represent the polar coordinates, and a is the length of the changing radius.
Another mathematician, Jakob Bernoulli (1654-1705), who made important contributions to the subject of probability, is also credited with describing significant aspects of this spiral. The logarithmic, or equiangular spiral was first suggested by Rene Descartes (1596-1650) in 1638.
The spiral of Archimedes conforms to the equation r = a θ, where r and θ represent the polar coordinates of the point plotted as the length of the radius a, uniformly changes. The simplest of all spirals was discovered by the ancient Greek mathematician Archimedes of Syracuse (287-212 B. The Archimedean spiral is the locus of points corresponding to the locations over time of a point moving away from a fixed point with a constant speed along a line which rotates with constant angular velocity.The famous Archimedean spiral can be expressed as a simple polar equation.
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