Computer graphics often require drawing smooth curves with visually uniform spacing…
MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions
Problem
Computer graphics often require drawing smooth curves with visually uniform spacing between plotted points. Suppose a graphics programmer must render a plane curve given in parametric form by x(t) and y(t), where the parameter t represents time in an animation.
(a) Describe, in terms of an integral, how one can assign to each point on the curve a value that represents the distance traveled along the curve from the starting point.
(b) Explain why replacing the original parameter t with the distance-based parameter you described in part (a) results in a representation of the curve whose speed is constant.
(c) Discuss two practical advantages this re-parameterization offers when (i) placing equally spaced markers along the curve and (ii) controlling the motion of an object that moves along the curve in an animation.