Appendix A: Exercise
1. Find the equal-time curve for reflection and the one for refraction.
“To proceed then to these figures [Figure A-1 for reflection and Figure A-2 for refraction], let us suppose first that it is desired to find a surface CDE which shall reassemble at a point B rays coming from another point A; and that the summit of the surface shall be the given point D in the straight line AB. I say that, whether by reflection or by refraction, it is only necessary to make this surface such that the path of the light from the point A to all points of the curved line CDE, and from these to the point of concurrence (as here the path along the straight lines AC, CB, along AL, LB, and along AD, DB), shall be everywhere traversed in equal times, by which principle the finding of these curves becomes very easy.
“So far as relates to the reflecting surface, since the sum of the lines AC, CB ought to be equal to that of AD, DB, it appears that DCE ought to be an ellipse; and for refraction, the ratio of the velocities of waves of light in the media A and B being supposed to be known, for example that of 3 to 2 (which is the same, as we have shown, as the ratio of the sines in the refraction), it is only necessary to make DH equal to 3/2 of DB; and having after that described from the center A, some arc FC, cutting DB at F, then describe another from center B with its semi-diameter BX equal to 2/3 of FH; and the point of intersection of the two arcs will be one of the points required, through which the curve should pass. For this point, having been found in this fashion, it is easy forthwith to demonstrate that the time along AC, CB, will be equal to the time along AD, DB.
“For assuming that the line AD represents the time which the light takes to traverse this same distance AD in air, it is evident that DH, equal to 3/2 of DB, will represent the time of the light along DB in the medium, because it needs here more time in proportion as its speed is slower. Therefore the whole line AH will represent the time along AD, DB. Similarly the line AC or AF will represent the time along AC; and FH being by construction equal to 3/2 of CB, it will represent the time along CB in the medium; and in consequence the whole line AH will represent also the time along AC, CB. Hence it appears that the time along AC, CB, is equal to the time along AD, DB. And similarly it can be shown if L is another point in the curve CDE, that the times along AL, LB are always represented by the line AH, and therefore equal to the said time along AD, DB.”
- Huygens, C., 1690, Traité de la Lumière [Treatise on Light, in which are explained the causes of that which occurs in reflection and in refraction, and particularly in the strange refraction of Iceland Crystal]: The Hague. Republished by Macmillan and Company, London, 1912.
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Also in this chapter
- Wavefronts and raypaths
- d’Alembert’s solution
- One-dimensional waves
- Sinusoidal waves
- Phase velocity
- Wave pulses
- Geometric seismology
- The speed of light
- Huygens’ principle
- Reflection and refraction
- Ray theory
- Fermat’s principle
- Fermat’s principle and reflection and refraction