By J. Dennis Lawrence
Suitable for college kids and researchers in geometry and desktop technological know-how, the textual content starts via introducing common houses of curves and kinds of derived curves. next chapters practice those homes to conics and polynomials, cubic and quartic curves, algebraic curves of excessive measure, and transcendental curves. a complete of greater than 60 distinct curves are featured, every one illustrated with a number of CalComp plots containing curves in as much as 8 assorted variations. Indexes offer tables of derived curves, curve names, and a 95-item advisor to additional reading.
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Extra resources for A Catalog of Special Plane Curves
4. Assume the division ring k is ordered. (Whenever this assumption is made, readers who are not familiar with ordered division rings or ordered fields should use the field of real numbers or the field of rational numbers for k. A reference for ordered division rings is E. Artin, Geometric Algebra, pp. , q) be parallel line segments of X and put r = (p, q)l(x, y). If r > 0, (x, y) and (p, q) are said to have the same orientation] if r < 0, they have opposite orientation. Assume (x, y) and (p, q) have the same orientation.
9. PARALLELISM 29 Hence we assume that U Φ W and prove that S n S' φ 0. Since U and W are distinct (n — l)-dimensional linear subspaces of F, U + W = V. Con sequently, the vector A = x^y can be written as A = B + C where B e U and CeW. Then ^ = ΛΛ; = ( 5 + C)x^>(-C)y = Bx and this point clearly belongs to both S and S'. ") Done. The following proposition states that Euclid's famous fifth parallel axiom holds for arbitrary n (= dimension of X) and all affine subspaces of X, no matter what their dimension may be.
4. ) 5. Find a set of parametric equations for the line through the ponts x and y in each of the following cases. a. x = (2, - l , 7 ) a n d j = (6, 4, - 3 ) (n = 3). b. x = (8, 6, - 1 ) and j = ( - 1 , 0 , 4) (n = 3). c. x = (6, - 1 , 0 , 4 , - 2 ) a n d j = (2, - 1 , - 1 , 3 , 0 ) (n = 5). 6. Parametric equations for affine subspaces S(x, U) of dimension d > 1 can be given just as easily as for lines. , Bd be a coordinate system for U where the coordinates of Bt and (bu, . . , bni) for / = 1, .
A Catalog of Special Plane Curves by J. Dennis Lawrence