CUARTO AÑO
4/ PRINCIPALES LÍNEAS PARA EL DESARROLLO CURRICULAR
Figure 7. Bifurcations of the family of wave fronts near an A4 singularity
4. Some examples of caustics and wavefronts in the transversal setting
In this section we include some concrete exmaples of the Minkowski set and the wave fronts for some pairs of curves and surfaces. In particular we deomonstrate how the singularities of the wavefronts sweep out the caustic.
Example 3.34 of anA2 singularity. If we have a surfaceM in the adapted coordinate system
given by z(x, y) = 1 2x 2+1 2y 2 −1
and a curve N where α(t) = t2 and β(t) =t2 +t3 then there is an A
2 point on the base chord
atλ= 2 corresponding to a cuspidal edge on the wavefront, see figure 8, which sweeps out the smooth surface of the caustic, see figure 9.
Figure 8. Example 3.34 of the wave front at an A2 singularity
Figure 9. Example 3.34 of the caustic at anA2 singularity
Example 3.35 of an A3 singularity in the non-flattening case.
If we have a surfaceM in the adapted coordinates given by z(x, y) = 1 2x 2+1 2y 2 −1
and a curveN where α(t) =t2 and β(t) =−1 2t
2 then there is an A
3 point on the base chord at
λ= 12, see figure 10, and the caustic is a cuspidal edge, see figure 11.
Figure 10. Wave front at an A3 singularity (example 3.35).
Figure 11. Caustic at an A3 singularity (example 3.35)
Example 3.36 of an A2 singularity in the flattening case.
If we have a surfaceM in the adapted coordinate system given by z(x, y) = 1
3x
3+xy
−1
and a curveN where α(t) = t2 andβ(t) = t3 then every point on the base chord belongs to the
caustic, see figure 12. At most points on the chord the wave fronts have cuspidal edges and the caustic is smooth. At the values λ=−32 ±12√21 the momentary wave fronts are diffeomorphic to a swallowtail and the caustic is diffeomorphic to a cuspidal edge, see figure 13.
The surface M is divided into two regions: points that share a parallel tangent plane with points on the curve and those that do not. The points that share a parallel tangent plane have a double cover and each corresponds to two tangent planes on either side oft= 0 on the curve.
4. SOME EXAMPLES OF CAUSTICS AND WAVEFRONTS IN THE TRANSVERSAL SETTING 61
The two regions are divided by a curve on the surface called the parallel tangents boundary curve (PTBC) (for similar objects see [31]).
Each point on the curve N corresponds to a curve Ct on the surface M containing points with parallel tangent planes. The envelope of these curves gives the PTBC. In this example the PTBC is given by the equationy=−4
3x
2 see figure 14. Asλ moves away from zero, the front,
which starts as double cover of the surface, splits into two sheets separated by a cuspidal edge along the PTBC.
Figure 12. Wave fronts along a flattening chord
Figure 13. Caustic Σ in the flattening case. The blue dot shows the A3 point on the flat-
tening chord l(a, b)
Example 3.37 of a bifurcation of an A∗
3 singularity of beaks type.
If we have a surfaceM in the adapted coordinate system given by z(x, y) =−1
2x
2
−12y2−x2y+y3−1 and a curve N where α(t) = t2 and β(t) = 1
2t
2 then we get an A∗
3 front bifurcation of beaks
Figure 14. The surface M with the PTBC and the curve Cti labelled
Example 3.38 of a bifurcation of an A∗
3 singularity of lips type.
If we have a surfaceM in the adapted coordinate system given by z(x, y) = 1 2x 2+1 2y 2+x2y+y3 −1
and a curveN where α(t) = t2 and β(t) = −1 2t
2 then we get an A∗
3 bifurcation of lips type at
λ= 12 see figure 16.
Figure 15. Example 3.37 of the bifurcation of the family of wave fronts near an A∗
3 beaks singularity
Remark. The singularities of types A∗
3 occur when the wave front at λ0 is tangent to the
4. SOME EXAMPLES OF CAUSTICS AND WAVEFRONTS IN THE TRANSVERSAL SETTING 63
Figure 16. Example 3.37 of the bifurcations of wave fronts of A∗
3 lips. Also
shown is its interaction with the caustic
swallowtail beaks or swallowtail lips see figure 17. Although the two caustics in this figure are diffeomorphic to each other, the bifurcations of the momentary wave fronts for the two cases are not diffeomorphic. This is an example of how studying spatio-temporal events with both space-time and time-space equivalences can yield more information than just one equivalence on its own (see [35]).
Figure 17. The caustics which correspond to lips (left) and beaks (right) at an A∗
Example 3.39 of an A4 singularity.
If we have a surfaceM in the adapted coordinates given by z(x, y) = 1 2x 2+1 2y 2 −1
and a curve N where α(t) = t2 and β(t) = −1 2t2 −
1 2t4 +
1
7t5 then there is 3-dimensional
generalised swallowtail point on theλ = 12 front, see figure 18, and the caustic is diffeomorphic to a swallowtail, see figure 19.
Figure 18. Bifurcations of wave fronts of an A4 singularity (example 3.39).