# A treatise on the analytical geometry of the point, line, by Casey J.

By Casey J.

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25 The waves on the ocean can be approximated by a sinusoidal surface as: surface ¼ A (1 þ cos (2p)). If there is 3 ft between crests of the waves and the waves are 1 ft high in height, how much volume (ft3) of water is there in a single wave that is 10 feet long? 5 km sÀ1 at an altitude of 320 nautical miles. (a) How long does the satellite take to make one revolution around the Earth? (b) How long does it take the satellite to travel between LA and NY (3000 miles)? (c) How much energy is expended if this satellite (mass ¼ 1kg) crashes into a huge asteroid (mass asteroid ) mass satellite)?

It was also known by Ptolemy (1600 B C ), but was not then formally developed mathematically. Note that the reflection case (to be discussed later) illustrates a point about Fermat’s principle: the minimum time may actually be a local, rather than a global, minimum. The global minimum distance from A to B is still just a straight line between the two points! In fact, light starting from point A will reach point B by both routes – the direct route and the reflected route. We will discover that the reflected route forces the angle of incidence to equal the angle of reflection.

13. 13 Longitudinal shift of an on-axis point with a PPP. 12, is Dz0 ¼ t À t cos I : n cos I0 (2:29) This means that the image location is moved further (positive direction) down the z-axis with the introduction of a PPP of thickness t and index n. 14. The reduced thickness of a PPP is easily measured by focusing with a linear tracking microscope on the front surface and then measuring the distance at which the rear surface comes into focus by moving t/n. 15, while viewing the back surface, the apparent position you are looking at is really above the back surface by about 1/3 the thickness.