VIC: Astroid

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!