In these examples learners are presented with multiple representations of the mathematical topic, such as orthotomic and the caustic curves.

The locus of the point B' is called the orthotomic curve and it is linked to the concept of a caustic curve [link to a video clip].

For a B on [C.sub.1] (light source at a point on [C.sub.1]), the reflection of B on a curve [C.sub.1] with respect to a moving point P on [C.sub.2], which we call B', the locus of the point B' is called the orthotomic curve.

The orthotomic curve of [C.sub.2] relative to B is shown in Figure 4(b) which can be experimented with by using a dynamic geometry software such as Geometry Expression ([26]) and verified by using a CAS such as Maple (See [27]).

Picking another light source C on [C.sub.1] we obtain another orthotomic curve of [C.sub.2] relative to C (shown in black or darker curve in Figure 4(c)).

The sharp corner (cusp) of the black orthotomic occurs at the inflection point of [C.sub.2].

Next, we find the reflection of a point on curve [C.sub.1] with respect to a moving point P on [C.sub.2]: We find the locus of the point P, which links to the concept of orthotomic and caustic curves.

It turns out the locus is the orthotomic curve of [C.sub.2] relative to P and the evolute of the orthotomic curve is called the caustic.

2 Generalizations in 2-D, orthotomic and caustic curves

An orthotomic curve is the set of reflections of a given point O with respect to all the tangents of a given curve not passing through O.

Then the orthotomic curve of [x(s), y(s)] relative P has a cusp at s = [s.sub.0], if and only if [x(s), y(s)] has an infection point at s = [s.sub.0].

(1) For a fixed point C on C1 (light source at a point on C1), the orthotomic curve of [C.sub.2] relative to C is shown in black, which can be experimented by using [Geometry Expression] (See [11) and verified by using [Maple] (See [12]).