# Die Geometrie der Beruehrungstransformationen by Lie Sophus, Scheffers G.

By Lie Sophus, Scheffers G.

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Plane Geometry and its Groups

San Francisco 1967 Holden-Day. eightvo. , 288pp. , index, hardcover. effective in VG DJ, a number of small closed tears.

Additional info for Die Geometrie der Beruehrungstransformationen

Example text

It is as if their descent from POLY cannot be suppressed! You should have noticed the same phenomenon in many of your own programs. 19 shows a POLY skeleton in dotted lines underlying the elaborate surface structures of NEWPOLY and INSP!. Can we understand this phenomenon? The key observation is this: Between successive vertices of the POLY skeleton, the program does the same thing. We might say that the program is just a decorated version of the underlying POLY. That the paths of NEWPOLY and INSPI consist of a collection of identical pieces is evident from the pictures they draw.

Direction. ) By assumption, the turtle returns to its initial heading after some number (say, n) repetitions of the POLY step. ) Draw a dotted line connecting the turtle's initial position Po to its position Pn after n repetitions. ISa). Now let the turtle continue for n more repetitions of the POLY step. 15b). Continuing with n more repetitions, and n more, and so on, we see that the turtle must run off infinitely far in the direction of the dotted line. Moreover, at no point can the turtle's path stray very far from the line, since the turtle must get back to it at the end of every n POLY steps.

To put Cartesian coordinates into our computer framework, imagine a "Cartesian turtle" whose moves are directed by a command called SEncy. SEncy takes two numbers as inputs. These numbers are interpreted as x and y coordinates, and the turtle moves to the corresponding point. RECTANGLE (WIDTH, HEIGHT) SETXY (WIDTH, 0) SETXY (WIDTH, HEIGHT) SETXY (0, HEIGHT) SETXY (0, 0) You are probably familiar with the uses of coordinates in geometry: studying geometric figures via equations, plotting graphs of numerical relationships, and so on.