planospiral

A planospiral is a type of spiral curve where the distance from the origin to any point on the curve increases linearly with the angle swept out from the origin. In polar coordinates, the equation of a planospiral can be expressed as r = a + bθ, where r is the radial distance, θ is the angle, and a and b are constants. The constant 'a' determines the starting radius of the spiral, and the constant 'b' controls how tightly the spiral winds. If 'a' is zero, the spiral begins at the origin. If 'b' is zero, the curve becomes a circle.

Planospirals are also known as Archimedean spirals, though this term is often used more broadly. The key characteristic of a planospiral is the uniform spacing between successive turns. As the angle θ increases, the radius r increases by a constant amount for every full rotation (2π radians). This constant increase in radius per unit angle is directly proportional to the constant 'b'.

These curves have various applications in science and engineering. For example, they can be used to model the path of a spacecraft as it spirals inward or outward, or to describe the shape of certain biological structures. In mathematics, the planospiral serves as a fundamental example of a simple, non-transcendental spiral. The rate at which the radius grows with respect to the angle is constant, differentiating it from other spiral types like the logarithmic spiral, where the radius grows exponentially with the angle.