Conic Sections
Tilt a torch or lamp against a flat surface and the edge of the light forms a curve — a circle, ellipse, parabola or hyperbola, depending on the angle. These conic sections aren't named explicitly in the IB AI/AA syllabus, but fitting a curve to measured data connects directly to quadratic modelling and regression.
Introduction
Shine a torch at a wall at different angles and the boundary of the light traces one of four curves: a circle, an ellipse, a parabola, or a hyperbola, depending on the angle between the beam and the surface. These are conic sections, cross-sections of a cone, but they are not covered by name in the AI or AA syllabus. This exploration instead treats the problem as a curve-fitting exercise: photograph or measure the light boundary at a fixed angle, then find and justify the best-fitting quadratic model for the data. As an extension, arches and cooling towers in architecture use these same curve families for structural reasons that are worth investigating separately from the curve-fitting itself.
Guiding Questions
- Shine a light at a fixed angle onto a flat surface and photograph the boundary of the lit region. What type of curve does it look like?
- Measure coordinates along the boundary and fit a quadratic model to the data. How good is the fit, and what does that tell you?
- Find one example of this curve family in architecture (an arch, a dish, a cooling tower) and fit the same kind of model to a photograph of it.
- Compare the light-boundary curve at two different angles. How does changing the angle change the parameters of your fitted model?