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April 10, 2011

Teaching geometric optics of mirrors and lenses

Geometric optics

GeoGebra is very useful for illustrating the principles of geometric optics. The GeoGebra construction sequence of the ray diagram differ very little from textbook description. The only difference is that the GeoGebra page is interactive and works for other scenarios. The GeoGebra page allows the investigator to explore the effect on the image when the position of the object or focal length is varied. There is no need to reconstruction another ray diagram for every variation. One construction covers all scenarios! This is powerful demonstration of the generality of construction principle.

The constructions are fairly straightforward. The pages below include a construction step toolbar. Press the play button to playback the construction sequence. There are three different ways to use the GeoGebra pages for teaching. (1) The pages can be used as presentations in the lesson. (2) Students can use the pages as simulations of optics experiments. (3) The pages are benchmarks for what could be emulated by the GeoGebra. Students could construct similar pages as projects.

Concave mirror

Convex mirror

mirror_concave.png convex mirror

Converging lens

Diverging lens

converging lens diverging lens

The pages (click to open a separate page for the actual GeoGebra pages) were created for demonstration in a physics lesson and for students to investigate the properties of the mirrors and lenses. In addition to the mirrors and lenses, the following pages of the parabola and prism were created to illustrate the design principles of the optical elements. The mirrors are based on the mathematics of parabola. The lenses are based on the physics of the refraction described by Snell's law.


This dynamic geometry page shows that a parabola is a locus of a point P which is equidistant to the focus F and a point G on the directrix d. PG is parallel to the x-axis. The lengths of PF and PG are equal, therefore the FGP is an isosceles triangle. b, the perpendicular bisector FG is the tangent to the locus. This properties is important in Physics. A ray incident in parallel to the x-axis of the parabolic mirror is reflected to to the focus F. The reverse is also true because a ray is reversible.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

In this construction, F' is a reflection of F about y-axis. Therefore, the parabola will always passes through the origin. Move F on the x-axis will change the curvature of the parabola.


This simulates the refraction of light in a glass prism using Snell's law. The refractive index is adjustable and glass prism can be transformed into a trapezoidal or triangular prism. The page is capable of producing two totally internal reflections inside the prism. The page aims to illustrate the concept that the cross-section of a converging or diverging lens resembles a glass prism. Focusing the light at the focus is the result of the application of Snell's law.

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

It was a "Tour de force" of GeoGebra skill to produce the page. There are 105 construction steps. There is a maximum of two refraction and reflection points on each side of the trapezium. Thus, the physics simulated by the page is correct up to a limit. The page should not be used by students unsupervised and teachers should explore many different scenarios to prepare for any surprise. Snell's law is simulated using a string of commands "Rotate[R_1, If[abs(n sin(Angle[R', R, R_2])) ≤ 1, -asin(n sin(Angle[R', R, R_2])), Angle[R', R, R_2] - π], R]" for the refraction of light at the point R. R_1 and R_2 are points at the end of the interfacial normal at R. Each refraction/reflection point can accept an incident ray from another reflection points.

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