Activity Overview
Students start by considering the motion of a point on the wheel of a bicycle. With the aid of interactive diagrams in the TI-Nspire document, students go on to explore the case when the wheel rotating inside another resulting in a hypocycloid. Students generate the parametric form of the equation, use compound and double angle formulas to generate the equivalent Cartesian equations and use calculus to analyse features of the curves, including gradient and arc length.
Objectives
- Parametric form and graphs of a cycloids and epicycles
- Converting parametric equations to Cartesian
- Application of chain rule to related rates of change and implicit differentiation
- Derivatives of explicit and implicit functions
- Application of integration to arc lengths of curves
- Use of trigonometric identities including compound and double angle formulas
Vocabulary
- Parameter
- Cartesian
- Implicit
- Cycloid
- Hypocycloid
- Locus
About the Lesson
The equation for a point on a circle rotating inside another can produce a hypocycloid. The equation is easy to generate in parametric form. Once students have generated the equation, they are provided with a set of scaffolded questions to determine the equivalent Cartesian equation thereby demonstrating the power of parametric equations. The curve is then studied using calculus with some delightfully elegant results in parametric form which would not be seen if studying the curve only in Cartesian form. The use of differential and integral calculus for both Parametric and Cartesian forms once again highlights the power of parametric equations.
The extension activity looks at a geometric construction of a ladder sliding down a wall and the resulting equation, the same hypocycloid!