Education Technology

Bezier Curves (Advanced)

Activity Overview

It starts with an envelope, progressed to a curve and then a whole world of mathematics opens up. From the humble parabola, the curve is transformed and explored using algebra, parametric equations, rotations, calculus (including implicit differentiation); this activity has got the lot! Hours of enjoyment.


Students explore a simple Bezier curve in the form of a parabola, however the control points on the Bezier curve mean the parabola can be dilated, translated and rotated quickly and easily. Students soon discover a world of mathematics that can be used to help unwind the complexity of a parabola that has been rotated. The world of mathematics nicely links several areas of the Specialist Mathematics curriculum. 


  • Bezier Curve
  • Parametric Equations
  • Cartesian Equation
  • General Equation for a conic
  • Implicit differentiation
  • Chain rule
  • Axis of Symmetry
  • Rotation

About the Lesson

Students start with a simply geometric construction reminiscent of the string graphs, the result is a simple Bezier curve. From these humble beginnings students use a dilation to create a locus that can be transformed and flexed like all good Bezier curves. The curve can be modelled with parametric equations which students use to determine the cartesian representation. A combination of the chain rule and implicit differentiation help to find the gradient of the relations, but where is the turning point? So much mathematics in this beautiful activity.