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Cams and Followers
Published in Eric Constans, Karl B. Dyer, Introduction to Mechanism Design, 2018
The path followed by the center of the roller is known as the pitch curve, and is found by offsetting the cam profile by the radius of the follower, as shown in Figure 10.68. The term parallel curve has a strict mathematical definition: it is the locus of points that results from moving a perpendicular distance ρ from the original curve. The simplest parallel curve is a parallel line, which is found by offsetting the original line by a constant distance. A parallel curve to a circle is another circle, and a parallel curve to a square is another square.
Parallel curves
Published in International Journal of Mathematical Education in Science and Technology, 2022
Richard Dexter Sauerheber, Tony Stewart
The derivation of the actual function that is a parallel curve of x2 is also beyond the scope of this work. However, curve fitting was performed on points calculated for the parallel curve of distance of separation of 1 along the function y = x2. Points at position (a, c) for the constructed parallel curve were computed knowing that corresponding points (x, y) on the original curve connect the line segment that is perpendicular to both curves at that position. Thus, (x−a)2 + (y−c)2 = 1 and y = −(1/m)x + c + a/m must both be satisfied. Here, the separation distance remains 1 throughout the domain, while at the same time the slope of the parallel constructed curve at the point of intersection of the line segment joining the perpendicular curves is the same as the slope of x2 at the corresponding position, given by m = y’. Here line segments connecting any two corresponding points on the parallel curves would have a slope given by −1/m where m varies along the parabola as y’ = 2x. Earlier studies indicated that if a given curve is a polynomial function, then its parallel curves are usually not polynomials (Marsh, 2005, Analytic Properties of Plane Offset Curves, 1990). The points were however attempted to be curve-fitted with quadratic regression for instructional interest. A reasonably parallel curve to x2 with a 1 unit separation distance was generated (Figure 9) that differs from the translational shifted curve shown in Figure 2.