An algorithm that translates intrinsic equations of curves into intrinsic procedures

Abstract

The paper describes an algorithm which translates each intrinsic equation of a curve into an intrinsic turtle procedure to draw the same curve. This algorithm inputs an intrinsic equation (or function) of a plane curve of the type ρ = ρ (φ) (where rho is the curvature-radius and φ is the tangent-angle). It ‘outputs’ a new procedure whose ‘action’ part is a simple step of the form (in Logo or in C+ +, etc.): right 1 forward < ρ >. This algorithm can be represented by a general procedure Intrinsic_Graph that produces a graphical representation of the ‘input’ equation curve. Thus, we can use the intrinsic equation: ρ = sin (n φ) of any cycloidal curve (or any other curve) to draw it. The intrinsic algorithm demonstrated links classical mathematical topics with turtle geometry. It uses the turtle's fundamental property that it always looks in the direction of the tangent to the curve. Thus, it ‘directs’ the turtle to proceed in a ‘natural way’ along the curve and at the same time to draw it.